This specification defines a core subset of Mathematical Markup Language, or MathML, that is suitable for browser implementation. MathML is a markup language for describing mathematical notation and capturing both its structure and content. The goal of MathML is to enable mathematics to be served, received, and processed on the World Wide Web, just as HTML has enabled this functionality for text.
The [[MATHML3]] specification has several shortcomings that make it hard to implement consistently across web rendering engines or to extend with user-defined constructions, e.g.:
This MathML Core specification intends to address these issues by being as accurate as possible on the visual rendering of mathematical formulas using additional rules from the TeXBook’s Appendix G [[?TEXBOOK]] and from the Open Font Format [[OPEN-FONT-FORMAT]], [[OPEN-TYPE-MATH-ILLUMINATED]]. It also relies on modern browser implementations and web technologies [[HTML]] [[SVG]] [[CSS2]] [[DOM]], clarifying interactions with them when needed or introducing new low-level primitives to improve the web platform layering.
Parts of MathML3 that do not fit well in this framework or are less fundamental have been omitted. Instead, they are described in a separate and larger [[MATHML4]] specification. The details of which math feature will be included in future versions of MathML Core or implemented as polyfills is still open. This question and other potential improvements are tracked on GitHub.
By increasing the level of implementation details, focusing on a workable subset, following a browser-driven design and relying on automated web platform tests, this specification is expected to greatly improve MathML interoperability. Moreover, effort on MathML layering will enable users to implement the rest of the MathML 4 specification, or more generally to extend MathML Core, using modern web technologies such as shadow trees, custom elements or APIs from [[HOUDINI]].
The term MathML element refers to any element in the MathML namespace. The MathML elements defined in this specification are called the MathML Core elements and are listed below. Any MathML element that is not listed below is called an Unknown MathML element.
The grouping elements are [^maction^], [^math^], [^merror^], [^mphantom^], [^mprescripts^], [^mrow^], [^mstyle^], [^semantics^] and unknown MathML elements.
The scripted elements are [^mmultiscripts^], [^mover^], [^msub^], [^msubsup^], [^msup^], [^munder^] and [^munderover^].
The radical elements are [^mroot^] and [^msqrt^].
The attributes defined in this specification have no namespace and are called MathML attributes:
maction
attributesmo
attributesmpadded
attributesmspace
attributesmunderover
attributesmtd
attributes<math>
ElementMathML specifies a single top-level or root
math element, which encapsulates each
instance of MathML markup within a document. All other MathML content
must be contained in a <math>
element.
The <math>
element accepts the attributes described
in as well as the
following attributes:
The
display
attribute, if present,
must be an
ASCII case-insensitive
match
to block
or inline
.
The user agent stylesheet
described in
contains rules for this attribute that affect the
default values for the [^math/display^]
(block math
or inline math
)
and math-style
(normal
or compact
) properties.
If the display
attribute is absent or has an invalid value, the User Agent
stylesheet treats it the same as inline
.
This specification does not define any observable behavior that is specific to the alttext attribute.
If the <math>
element does not have its computed
[^math/display^] property equal to
block math
or inline math
then it is laid out according to the CSS specification where
the corresponding value is described.
Otherwise the layout algorithm of the
[^mrow^] element is used to produce a
math content box. That math content box is used as the content for the layout of
the element, as described by CSS for display: block
(if the computed value is block math
) or
display: inline
(if the computed value is inline math
).
Additionally, if the computed
[^math/display^] property is equal to
block math
then that math content box is rendered
horizontally centered within the content box.
$$...$$
and inline mode $...$
correspond to
display="block"
and display="inline"
respectively.
In the following example, a [^math^] formula is rendered in display mode on a new line and taking full width, with the math content centered within the container:
As a comparison, the same formula would look as follows in inline mode. The formula is embedded in the paragraph of text without forced line breaking. The baselines specified by the layout algorithm of the [^mrow^] are used for vertical alignment. Note that the middle of sum and equal symbols or fractions are all aligned, but not with the alphabetical baseline of the surrounding text.
Because good mathematical rendering requires use of mathematical
fonts, the
user agent stylesheet
should set the
font-family
to the
math
value on the <math>
element instead of inheriting
it. Additionally, several CSS properties that can be set on
a parent container such as
font-style
, font-weight
,
direction
or text-indent
etc
are not expected to apply to the math formula and so the
user agent stylesheet
has rules to reset them by default.
In addition to CSS data types, some MathML attributes rely on the following MathML-specific types:
true
or
false
.
The following attributes are common to and may be specified on all MathML elements:
autofocus
class
data-*
dir
displaystyle
id
mathbackground
mathcolor
mathsize
nonce
scriptlevel
style
tabindex
on*
event handler attributes
The
id,
class,
style,
data-*
,
autofocus and
nonce and
tabindex
attributes have the same syntax and semantics as defined for
[^global/id^],
[^global/class^],
[^html-global/style^],
data-*,
[^htmlsvg-global/autofocus^],
[^htmlsvg-global/nonce^] and
[^htmlsvg-global/tabindex^]
attributes on HTML elements.
The
dir
attribute, if present,
must be an
ASCII case-insensitive match
to ltr
or rtl
.
In that case, the user agent is expected to treat the attribute as a
presentational hint setting the element's
direction
property to the corresponding value.
More precisely, an
ASCII case-insensitive match
to rtl
is mapped to rtl
while
an ASCII case-insensitive match to ltr
is mapped to ltr
.
rtl
in Arabic speaking world.
However, languages written from right to left often embed math
written from left to right and so the
user agent stylesheet resets
the
direction
property accordingly on the [^math^]
elements.
In the following example, the dir attribute is used to render "𞸎 plus 𞸑 raised to the power of (٢ over, 𞸟 plus ١)" from right-to-left.
All MathML elements support event handler content attributes, as described in event handler content attributes in HTML.
All event handler content attributes noted by HTML as being supported by all HTMLElements are supported by all MathML elements as well, as defined in the MathMLElement IDL.
The
mathcolor
and
mathbackground
attributes, if present, must
have a value that is a
<color>.
In that case, the user agent is expected to treat these attributes as a
presentational hint setting the element's
color and
background-color
properties to the corresponding values.
The mathcolor
attribute describes the foreground fill
color of MathML text, bars etc
while the mathbackground
attribute describes the background color of an element.
The
mathsize
attribute, if present, must
have a value that is a valid <length-percentage>.
In that case, the user agent is expected to treat the attribute as a
presentational hint setting the element's
font-size
property to the corresponding value.
The mathsize
property indicates the desired height
of glyphs in math formulas but also scales other parts (spacing, shifts,
line thickness of bars etc) accordingly.
displaystyle
and scriptlevel
attributes
The
displaystyle
attribute, if present, must have a value that is a boolean.
In that case, the user agent is expected to treat the attribute as a
presentational hint setting the element's
math-style
property to the corresponding value.
More precisely, an
ASCII case-insensitive match
to true
is mapped to normal
while
an ASCII case-insensitive match to false
is mapped to compact
.
This attribute indicates whether formulas should try to minimize
the logical height (value is false
) or not
(value is true
) e.g. by changing the size of content or
the layout of scripts.
The
scriptlevel
attribute, if present, must have value
+<U>
, -<U>
or <U>
where <U>
is an
unsigned-integer.
In that case
the user agent is expected to treat the scriptlevel
attribute as a
presentational hint setting the element's
math-depth
property to the corresponding value.
More precisely,
+<U>
, -<U>
and
<U>
are respectively mapped to
add(<U>)
add(<-U>)
and <U>
.
displaystyle
and scriptlevel
values
are automatically adjusted within MathML elements.
To fully implement these attributes, additional CSS properties must be
specified in the user agent stylesheet
as described in .
In particular, for all MathML elements a default
font-size: math
is specified to ensure that
scriptlevel
changes are taken into account.
In this example, an [^munder^]
element is used to attach a
script "A" to a base "∑". By default, the summation
symbol is rendered with the font-size inherited from its
parent and the A as a scaled down subscript.
If displaystyle is true, the summation symbol is drawn
bigger and the "A" becomes an underscript.
If scriptlevel is reset to 0 on the "A", then it will
use the same font-size as the top-level math
root.
\displaystyle
, \textstyle
,
\scriptstyle
, and \scriptscriptstyle
correspond
to displaystyle
and scriptlevel
as
true
and 0
,
false
and 0
,
false
and 1
,
and false
and 2, respectively.
The attributes intent and arg are reserved as valid attributes.
This specification does not define any observable behavior that is
specific to the intent
and arg
attributes.
MathML can be mixed with HTML and SVG as described in the relevant specifications [[HTML]] [[SVG]].
When evaluating the SVG [^svg/requiredExtensions^] attribute, user agents must claim support for the language extension identified by the MathML namespace.
In this example, inline MathML and SVG elements are used inside
an HTML document. SVG elements <switch>
and
<foreignObject>
(with
proper <requiredExtensions>
) are used to
embed a MathML formula with a text fallback, inside a diagram.
HTML input
element is used within the
[^mtext^]
to include an interactive input field inside a mathematical
formula. See also
for an example of SVG and HTML inside an [^annotation-xml^]
element.
User agents must support various CSS features mentioned in this specification, including new ones described in . They must follow the computation rule for display: contents.
In this example, the MathML formula inherits the CSS color of its
parent and uses the font-family
specified via the
style attribute.
All documents containing MathML Core elements must include
CSS rules described in
as part of user-agent level style sheet defaults.
In particular, this adds !important
rules to force
writing mode
to horizontal-lr
on all MathML elements.
The float
property does
not create floating of elements whose parent's computed
display
value is
block math
or inline math
,
and does not take them out-of-flow.
The ::first-line and
::first-letter
pseudo-elements do not apply to elements whose computed
display
value is
block math
or inline math
, and such
elements do not contribute a first formatted line or first letter
to their ancestors.
The following CSS features are not supported and must be ignored:
white-space
is treated as nowrap
on all MathML elements.
align-content
, justify-content
,
align-self
, justify-self
have
no effects on MathML elements.
User agents supporting Web application APIs must ensure that they keep the visual rendering of MathML synchronized with the [[DOM]] tree, in particular perform necessary updates when MathML attributes are modified dynamically.
All the nodes representing MathML elements in the DOM must implement, and expose to scripts, the following MathMLElement interface.
The {{GlobalEventHandlers}} and
HTMLOrForeignElement
interfaces are defined in [[HTML]].
In the following example, a MathML formula is used to render the fraction "α over 2". When clicking the red α, it is changed into a blue β.
Because math fonts generally contain very tall glyphs such as big integrals, using typographic metrics is important to avoid excessive line spacing of text. As a consequence, user agents must take into account the USE_TYPO_METRICS flag from the OS/2 table [[OPEN-FONT-FORMAT]] when performing text layout.
MathML provides the ability for authors to allow for
interactivity in supporting interactive user agents
using the same concepts, approach and guidance to
Focus
as described in HTML, with modifications or
clarifications regarding application
for MathML as described in this section.
When an element is focused, all applicable CSS focus-related pseudo-classes as defined in Selectors Level 3 apply, as defined in that specification.
The contents of embedded [^math^] elements (including HTML elements inside token elements) contribute to the sequential focus order of the containing owner HTML document (combined sequential focus order).
The default [^math/display^] property is described in :
<math>
root,
it is equal to inline math
or block math
according to the value of the [^math/display^] attribute.
inline-table
,
table-row
and
table-cell
.
none
.
block math
.
In order to specify math layout in different writing modes, this specification uses concepts from [[CSS-WRITING-MODES-4]]:
horizontal-lr
and ltr
.
See ,
and
for examples of other
writing modes that are sometimes used for math layout.
Boxes used for MathML elements rely on several parameters in order to perform layout in a way that is compatible with CSS but also to take into account very accurate positions and spacing within math formulas:
Block metrics. The block size, first baseline set and last baseline set. The following baselines are defined for MathML boxes:
Given a MathML box, the following offsets are defined:
Here are examples of offsets obtained from line-relative metrics:
ltr
and
is the inline size of the box −
(line-left offset + inline size of
the child box) otherwise.
horizontal-lr
,
vertical-rl
or sideways-rl
and is the line-descent otherwise.
Each MathML element has an associated math content box, which is calculated as described in this chapter's layout algorithms using the following structure:
The following extra steps must be performed:
The box metrics and offsets of the padding box are obtained from the content box by taking into account the corresponding padding properties as described in CSS.
The baselines of the padding box are the same as the one of the content box.
If the content box has a top accent attachment then the padding box has the same property, increased by the inline-start padding. If the content box has an italic correction then the padding box has the same property, increased by the inline-end padding.
The box metrics and offsets of the border box are obtained from the padding box by taking into account the corresponding border-width property as described in CSS.
In general, the baselines of the border box are the same as the one of the padding box. However, if the line-over border is positive then the ink-over baseline is set to the line-over edge of the border box and if the line-under border is positive then the ink-under baseline is set to the line-under edge of the border box.
If the padding box has a top accent attachment then the border box has the same property, increased by the border-width of its inline-start egde. If the padding box has an italic correction then the border box has the same property, increased by the border-width of its inline-end egde.
The box metrics and offsets of the margin box are obtained from the border box by taking into account the corresponding margin properties as described in CSS.
The baselines of the margin box are the same as the one of the border box.
If the padding box has a top accent attachment then the margin box has the same property, increased by the inline-start margin. If the padding box has an italic correction then the margin box has the same property, increased by the inline-end margin.
During box layout, optional inline stretch size constraint and block stretch size constraint parameters may be used on embellished operators. The former indicates a target size that a [=embellished operator/core operator=] stretched along the inline axis should cover. The latter indicates an ink line-ascent and ink line-descent that a [=embellished operator/core operator=] stretched along the block axis should cover. Unless specified otherwise, these parameters are ignored during box layout and child boxes are laid out without any stretch size constraint.
An anonymous box is a box without any associated
element in the DOM tree and which is generated for layout purpose
only. The properties of anonymous boxes are inherited from the
enclosing non-anonymous box while non-inherited properties have
their initial value.
An anonymous <mrow> box is
an anonymous box with display
equal to
block math
and which is laid out as
described in section .
If a MathML element generates an anonymous <mrow> box then it wraps its children in an anonymous <mrow> box. I.e., its subtree in the visual formatting model is made of an anonymous <mrow> box which itself contains the boxes associated to the children of this MathML element.
In the following example, the [^math^] and
[^mrow^] elements are laid out as described in section
. In particular, the
<math>
element adds proper spacing around its
<mo>≠</mo>
child and the
<mrow>
element stretches its
<mo>|</mo>
children vertically.
The [^mtd^] element has
display: table-cell
and the
[^msqrt^] element displays a radical symbol around its
children. However, they also place their children in a way that
is similar to what is described in section
: the
<msqrt>
element adds proper spacing around its
<mo>+</mo>
child while the
<mtd>
element stretches its
<mo>
children vertically.
In order to make this possible,
each of these two elements
generates an anonymous <mrow> box.
MathML elements can overlap due to various spacing rules. They
can as well contain extra graphical items
(bars, radical symbol, etc).
A MathML element with computed style
display: block math
or display: inline math
generates a new stacking
context. The painting order
of in-flow children of such a MathML element
is exactly the same as block elements. The extra graphical
items are painted after text and background (right after
step 7.2.4 for display: inline math
and right after
step 7.2 for display: block math
).
Token elements in presentation markup are broadly intended to represent the smallest units of mathematical notation which carry meaning. Tokens are roughly analogous to words in text. However, because of the precise, symbolic nature of mathematical notation, the various categories and properties of token elements figure prominently in MathML markup. By contrast, in textual data, individual words rarely need to be marked up or styled specially.
<mtext>
The
mtext
element is used to represent arbitrary text
that should be rendered as itself. In general, the
<mtext>
element is intended to denote
commentary text.
The <mtext>
element accepts the attributes described
in .
In the following example, [^mtext^] is used to put conditional words in a definition:
<mtext>
If the element does not have its computed
[^math/display^] property equal to
block math
or inline math
then it is laid out according to the CSS specification where
the corresponding value is described.
Otherwise, the layout below is performed.
If the <mtext>
element contains only text
content without
forced line break
or
soft wrap opportunity
then, the anonymous child node generated for that text is
laid out as defined in the relevant CSS specification and:
<mtext>
element.
Otherwise, the mtext
element is laid out as a
block box
and corresponding min-content inline size,
max-content inline size,
inline size, block size,
first baseline set and last baseline set are
used for the math content box.
<mi>
The mi element represents a symbolic name or arbitrary text that should be rendered as an identifier. Identifiers can include variables, function names, and symbolic constants.
The <mi>
element accepts the attributes described
in as well as the following attribute:
The layout algorithm is the same as the [^mtext^] element. The user agent stylesheet must contain the following property in order to implement automatic italic via the text-transform value introduced in :
The
mathvariant
attribute,
if present, must be an
ASCII case-insensitive
match of normal
.
In that case, the user agent is expected to treat the attribute as a
presentational hint setting the element's
text-transform
property to none
. Otherwise it has no effects.
In [[MathML3]], the mathvariant
attribute was used
to define logical classes of token elements, each class providing
a collection of typographically-related symbolic tokens with
specific meaning within a given mathematical expression.
In MathML Core, this attribute is only used to cancel automatic italic of the [^mi^] element. For other use cases, the proper Mathematical Alphanumeric Symbols [[UNICODE]] should be used instead. See also section .
In the following example, [^mi^] is used to render
variables and function names. Note that per
the default
style text-transform: math-auto
has
no effect on the first <mi>
("cos" is made of three characters),
makes the second <mi>
render as math italic ("c" is made of a single
character U+0063 Latin Small Letter C which is
mapped to
U+1D450 Mathematical Italic Small C per the
italic table), has no effect
on the third <mi>
(overridden by
mathvariant="normal"
, setting
text-transform
to none) or on the fourth
<mi>
(no mapping defined for U+221E Infinity
in the italic table).
<mn>
The mn element represents a "numeric literal" or other data that should be rendered as a numeric literal. Generally speaking, a numeric literal is a sequence of digits, perhaps including a decimal point, representing an unsigned integer or real number.
The <mn>
element accepts the attributes described
in . Its layout algorithm is
the same as the
[^mtext^] element.
In the following example, [^mn^] is used to write a decimal number.
<mo>
The
mo
element represents an
operator or anything that should be rendered as an operator.
In general, the notational conventions for mathematical operators
are quite complicated, and therefore MathML provides a relatively
sophisticated mechanism for specifying the rendering behavior of an
<mo>
element.
As a consequence, in MathML the list of things that should "render as an operator" includes a number of notations that are not mathematical operators in the ordinary sense. Besides ordinary operators with infix, prefix, or postfix forms, these include fence characters such as braces, parentheses, and "absolute value" bars; separators such as comma and semicolon; and mathematical accents such as a bar or tilde over a symbol. This chapter uses the term "operator" to refer to operators in this broad sense.
The <mo>
element accepts the attributes described
in as well as the following
attributes:
This specification does not define any observable behavior that is specific to the fence and separator attributes.
Operators_fence
and Operators_separator
tables, or equivalently
the human-readable version
of the operator dictionary.
In the following example, the [^mo^] element
is used for the binary operator +. Default spacing is symmetric
around that operator. A tighter spacing is used if you rely
on the form
attribute to force it to be
treated as a prefix operator.
Spacing can also be specified explicitly using the
[^mo/lspace^] and
[^mo/rspace^] attributes.
Another use case is for big operators such as summation. When displaystyle is true, such an operator is drawn larger but one can change that with the [^mo/largeop^] attribute. When displaystyle is false, underscripts are actually rendered as subscripts but one can change that with the [^mo/movablelimits^] attribute.
Operators are also used for stretchy symbols such as fences, accents, arrows etc. In the following example, the vertical arrow stretches to the height of the [^mspace^] element. One can override default stretch behavior with the [^mo/stretchy^] attribute e.g. to force an unstretched arrow. The [^mo/symmetric^] attribute allows to indicate whether the operator should stretch symmetrically above and below the math axis (fraction bar). Finally the [^mo/minsize^] and [^mo/maxsize^] attributes add additional constraints over the stretch size.
Note that the default properties of operators are dictionary-based, as explained in . For example a binary operator typically has default symmetric spacing around it while a fence is generally stretchy by default.
A MathML Core element is an embellished operator if it is:
The core operator of an embellished operator
is the <mo>
element defined recursively as
follows:
The stretch axis of an embellished operator
is inline if its
[=embellished operator/core operator=] contains only text content
made of a single character c
, and that character has
inline intrinsic stretch axis.
Otherwise, the stretch axis of the embellished operator
is block.
The same definitions apply for boxes in the visual formatting model where an anonymous <mrow> box is treated as a grouping element.
The form
property of an embellished operator is either
infix
, prefix
or
postfix
.
The corresponding form attribute on the
[^mo^] element, if present, must be an
ASCII case-insensitive
match to one of these values.
The algorithm for determining the form
of an embellished operator is as follows:
form
attribute is present and valid
on the [=embellished operator/core operator=], then its
ASCII lowercased value
is used.
prefix
.
postfix
.
postfix
.
infix
.
The
stretchy
,
symmetric
,
largeop
,
movablelimits
properties of an embellished operator are
either false
or true
. In the latter
case, it
is said that the embellished operator has the
property.
The corresponding stretchy, symmetric, largeop, movablelimits attributes on the
[^mo^] element, if present, must be a
boolean.
The
lspace
,
rspace
,
minsize
properties of an embellished operator are
<length-percentage>.
The maxsize
property
of an embellished operator is either a
<length-percentage> or ∞.
The
lspace,
rspace,
minsize and
maxsize attributes on the
[^mo^] element, if present,
must be a <length-percentage>.
The algorithm for determining the properties of an embellished operator is as follows:
stretchy
,
symmetric
,
largeop
,
movablelimits
,
lspace
,
rspace
,
maxsize
or
minsize
attribute is present and valid
on the [=embellished operator/core operator=], then the
ASCII lowercased value
of this property is used.form
of an embellished operator.Content
, then set Category
to the result of the
algorithm to determine the category of an operator
(Content, Form)
where Form
is the form
calculated at the previous step.
Category
is Default
and
the form
of embellished operator was not explicitly specified
as an attribute on its [=embellished operator/core operator=]:
Category
to the result of the
algorithm to determine the category of an operator
(Content, Form)
where Form
is
infix
.Category
is Default
, then
run the algorithm again with Form
set to
postfix
.Category
is Default
, then
run the algorithm again with Form
set to
prefix
.Category
.
When used during layout, the values of [=embellished operator/stretchy=], [=embellished operator/symmetric=], [=embellished operator/largeop=], [=embellished operator/movablelimits=], [=embellished operator/lspace=], [=embellished operator/rspace=], [=embellished operator/minsize=] are obtained by the algorithm for determining the properties of an embellished operator with the following extra resolutions:
lspace
,
rspace
are interpreted
relative to the value read from the dictionary
or to the fallback value above.
minsize
and maxsize
are described in
.
lspace
, rspace
,
minsize
and maxsize
rely on the
font style of the [=embellished operator/core operator=], not the one of the
embellished operator.
If the <mo>
element does not have its computed
[^math/display^] property equal to
block math
or inline math
then it is laid out according to the CSS specification where
the corresponding value is described.
Otherwise, the layout below is performed.
The text of the operator must only be painted if the
visibility of
the <mo>
element is visible
.
In that case, it must be painted with the
color
of the <mo>
element.
Operators are laid out as follows:
<mo>
element is not
made
of a single character c
then fall back to the
layout algorithm of .
c
in the inline direction
with the
first available font
then fall back to the
layout algorithm of .
T_{inline}
then
fall back to the
layout algorithm of .
T_{inline}
.
T_{inline}
and
at position determined by the previous box metrics.
c
in the block direction
with the
first available font
then fall back to the
layout algorithm of .
(U_{ascent}, U_{descent})
then
fall back to the
layout algorithm of .
T_{ascent}
and
T_{descent}
to
S_{ascent}
and
S_{descent}
respectively:
S_{ascent}
=
max(
U_{ascent}
− AxisHeight,
U_{descent}
+ AxisHeight
) + AxisHeight
S_{descent}
=
max(
U_{ascent}
− AxisHeight,
U_{descent}
+ AxisHeight
) − AxisHeight
U_{ascent}
and
U_{descent}
respectively.
T_{ascent}
− AxisHeight = T_{descent}
+ AxisHeight means that
an operator stretching exactly
T_{ascent}
above the baseline
and T_{descent}
below the
baseline would actually stretch symmetrically above
and below the math axis.
S_{ascent}
and
S_{descent}
are the minimal
values, that are respectively not less than
U_{ascent}
and
U_{descent}
, which satisfy
this property.
minsize
and maxsize
be the [=embellished operator/minsize=] and [=embellished operator/maxsize=] properties on the
operator. Percentage values are interpreted relative
to the height of the glyph for c
.
Let T
=
T_{ascent}
+
T_{descent}
be the target size.
If minsize
< 0 then set minsize
to 0.
If maxsize
< minsize
then
set maxsize
to minsize
.
With 0 ≤ minsize
≤ maxsize
:
T
≤ 0 then set
T_{ascent}
to
minsize
/ 2 + AxisHeight and
then set T_{descent}
to minsize
−
T_{ascent}
.
T
< minsize
then set T_{ascent}
to
max(0, (T_{ascent}
− AxisHeight) × minsize
/ T
+ AxisHeight) and
T_{descent}
to minsize
−
T_{ascent}
.
maxsize
< T
then set T_{ascent}
to
max(0, (T_{ascent}
− AxisHeight) × maxsize
/ T
+ AxisHeight) and
T_{descent}
to maxsize
−
T_{ascent}
.
maxsize
is value ∞ is
interpreted above as being larger than any other size,
i.e.
minsize ≤ maxsize
is always true while
maxsize < minsize
and
maxsize < T
are always false.
minsize
≤ T
≤ maxsize
holds.
Additionnally, if the target values correspond to symmetric stretching with respect to the math axis then property
T_{ascent}
− AxisHeight = T_{descent}
+ AxisHeight is preserved.
T_{ascent}
+
T_{descent}
.
The inline size of the math content is the width of
the stretchy glyph. The stretchy glyph is shifted
towards the line-under by a value Δ so that its
center aligns with the center of the target:
the ink ascent of the math content is
the ascent of the stretchy glyph − Δ
and the ink descent of the math content is
the descent of the stretchy glyph + Δ.
These centers have coordinates "½(ascent − descent)"
so Δ = [(ascent of stretchy glyph − descent of stretchy glyph) − (T_{ascent}
− T_{descent}
)] / 2.
T_{ascent}
+
T_{descent}
and at position determined by the previous box metrics
shifted by Δ towards the line-over.
math-style
on
the <mo>
element is normal
,
then:
Use the
MathVariants
table to try and find a glyph of height at least
DisplayOperatorMinHeight.
If none is found, fall back to the
largest non-base glyph. If none is found, fall back to
the layout algorithm of .
If the algorithm to shape a stretchy glyph has been used for one of the step above, then the italic correction of the math content is set to the value returned by that algorithm.
<mspace>
The mspace empty element represents a blank space of any desired size, as set by its attributes.
The <mspace>
element accepts the attributes described
in as well as the following
attributes:
The width, height, depth, if present, must have a value that is a valid <length-percentage>.
width
attribute is present, valid and not a percentage then
that attribute is used as a
presentational hint
setting the element's
width
property to the corresponding value.
height
attribute is absent, invalid or a percentage then the requested
line-ascent is 0
.
Otherwise the requested line-ascent is the resolved
value of the height
attribute, clamping
negative values to 0
.
height
and depth
attributes
are present, valid and not a percentage then they are used as a
presentational hint
setting the element's
height
property to the concatenation of the strings
"calc(
", the height
attribute value,
" +
", the depth
attribute value,
and ")
".
If only one of these attributes is
present, valid and not a percentage then it is treated as a
presentational hint
setting the element's
height
property to the corresponding value.
In the following example, [^mspace^] is used to force spacing within the formula (a 1px blue border is added to easily visualize the space):
If the <mspace>
element does not have its
computed
[^math/display^] property equal to
block math
or inline math
then it is laid out according to the CSS specification where
the corresponding value is described.
Otherwise,
the <mspace>
element is laid out as shown on
.
The min-content inline size,
max-content inline size and inline size of the math
content are equal to the resolved value of the
width property.
The block size of the math content is equal to the resolved
value of the height property.
The line-ascent of the math content is equal to the
requested line-ascent determined above.
A number of MathML presentation elements are "space-like" in the sense that they typically render as whitespace, and do not affect the mathematical meaning of the expressions in which they appear. As a consequence, these elements often function in somewhat exceptional ways in other MathML expressions.
A MathML Core element is a space-like element if it is:
The same definitions apply for boxes in the visual formatting model where an anonymous <mrow> box is treated as a grouping element.
<mphantom>
element is
primarily intended as an aid in aligning expressions, operators
adjacent to an <mphantom>
should behave
as if they were adjacent to the contents of the
<mphantom>
, rather than to an equivalently
sized area of whitespace.
<ms>
ms element is used to represent "string literals" in expressions meant to be interpreted by computer algebra systems or other systems containing "programming languages".
The <ms>
element accepts the attributes described
in . Its layout algorithm is
the same as the [^mtext^] element.
In the following example, [^ms^] is used to write a literal string of characters:
lquote
and
rquote
attributes to respectively specify the strings
to use as opening and closing quotes. These are no longer supported
and the quotes must instead be specified as part of the text of the
<ms>
element. One can add CSS rules to legacy
documents in order to preserve visual rendering. For example,
in left-to-right direction:
Besides tokens there are several families of MathML presentation elements. One family of elements deals with various "scripting" notations, such as subscript and superscript. Another family is concerned with matrices and tables. The remainder of the elements, discussed in this section, describe other basic notations such as fractions and radicals, or deal with general functions such as setting style properties and error handling.
<mrow>
The
mrow
element is used to group together any number of sub-expressions, usually
consisting of one or more <mo>
elements acting as
"operators" on one or more other expressions that are their "operands".
In the following example, [^mrow^] is used to group a sum "1 + 2/3" as a fraction numerator (first child of [^mfrac^]) and to construct a fenced expression (first child of [^msup^]) that is raised to the power of 5. Note that [^mrow^] alone does not add visual fences around its grouped content, one has to explicitly specify them using the [^mo^] element.
Within the [^mrow^] elements, one can see that vertical alignment of children (according to the alphabetic baseline or the mathematical baseline) is properly performed, fences are vertically stretched and spacing around the binary + operator automatically calculated.
The <mrow>
element accepts the attributes described
in . An <mrow>
element with in-flow children
child_{1}, child_{2}, …, child_{N}
is laid out as shown on . The child boxes
are put in a row one after the other with all their
alphabetic baselines
aligned.
The algorithm for stretching operators along the block axis consists in the following steps:
L_{ToStretch}
containing
embellished operators with
a [=embellished operator/stretchy=] property and block [=embellished operator/stretch axis=];
and a second list L_{NotToStretch}
.
L_{NotToStretch}
.
If L_{ToStretch}
is empty then stop.
If L_{NotToStretch}
is empty, perform
layout with block stretch size constraint
(0, 0)
for
all the items of L_{ToStretch}
.
U_{ascent}
and U_{descent}
as respectively the maximum
ink ascent and maximum ink descent of the margin boxes of
in-flow children that
have been laid out in the previous step.
L_{ToStretch}
with
block stretch size constraint
(U_{ascent}, U_{descent})
.
<mrow>
If the box is not an anonymous <mrow> box
and the associated element does not have its computed
[^math/display^] property equal to
block math
or inline math
then it is laid out according to the CSS specification where
the corresponding value is described.
Otherwise, the layout below is performed.
A child box is slanted if it is not an embellished operator and has nonzero italic correction.
The min-content inline size (respectively max-content inline size) are calculated using the following algorithm:
add-space
to true if
the box corresponds to a [^math^] element
or is not an
embellished operator; and to false otherwise.
inline-offset
to 0.previous-italic-correction
to 0.inline-offset
by
previous-italic-correction
.
add-space
is true then
increment inline-offset
by
its [=embellished operator/lspace=] property.
inline-offset
by
the min-content inline size
(respectively max-content inline size) of
the child's margin box.
previous-italic-correction
to
its italic correction. Otherwise set it to 0.
add-space
is true then
increment inline-offset
by
its [=embellished operator/rspace=] property.
inline-offset
by
previous-italic-correction
.
inline-offset
.
The in-flow children are laid out using the algorithm for stretching operators along the block axis.
The inline size of the math content is calculated like the min-content inline size and max-content inline size of the math content, using the inline size of the in-flow children's margin boxes instead.
The ink line-ascent (respectively line-ascent) of the math content is the maximum of the ink line-ascents (respectively line-ascents) of all the in-flow children's margin boxes. Similarly, the ink line-descent (respectively line-descent) of the math content is the maximum of the ink line-descents (respectively ink line-ascents) of all the in-flow children's margin boxes.
The in-flow children are positioned using the following algorithm:
add-space
to true if
the box corresponds to a [^math^] element
or is not an
embellished operator; and to false otherwise.
inline-offset
to 0.previous-italic-correction
to 0.inline-offset
by
previous-italic-correction
.
add-space
is true then
increment inline-offset
by
its [=embellished operator/lspace=] property.
inline-offset
and its block offset such
that the alphabetic baseline of the child is aligned with the alphabetic baseline.
inline-offset
by
the inline size of the child's margin box.
previous-italic-correction
to
its italic correction. Otherwise set it to 0.
add-space
is true then
increment inline-offset
by
its [=embellished operator/rspace=] property.
The italic correction of the math content is set to the italic
correction of the last in-flow child, which is
the final value of previous-italic-correction
.
<mfrac>
The mfrac element is used for fractions. It can also be used to mark up fraction-like objects such as binomial coefficients and Legendre symbols.
If the <mfrac>
element does not have its computed
[^math/display^] property equal to block math
or inline math
then it is laid out according to the CSS specification where
the corresponding value is described.
Otherwise, the layout below is performed.
The <mfrac>
element accepts the attributes described
in as well as the
following attribute:
The linethickness attribute indicates the fraction line thickness to use for the fraction bar. If present, it must have a value that is a valid <length-percentage>. If the attribute is absent or has an invalid value, FractionRuleThickness is used as the default value. A percentage is interpreted relative to that default value. A negative value is interpreted as 0.
The following example contains four fractions with different [^mfrac/linethickness^] values. The bars are always aligned with the middle of plus and minus signs. The numerator and denominator are horizontally centered. The fractions that are not in displaystyle use smaller gaps and font-size.
The <mfrac>
element sets
displaystyle
to false
,
or if it was already false
increments
scriptlevel
by 1, within its children.
It sets math-shift to
compact
within its second child.
To avoid visual confusion between the fraction bar and another
adjacent items (e.g. minus sign or another fraction's bar),
a default 1-pixel space is added around the element.
The user agent stylesheet
must contain the following rules:
If the <mfrac>
element
has less or more than two in-flow children, its layout algorithm
is the same as the [^mrow^] element.
Otherwise, the first in-flow child is called
numerator, the second in-flow child is called
denominator and the layout algorithm is explained below.
<mfrac>
element has two children
that are in-flow. Hence the CSS rules basically perform
scriptlevel
, displaystyle
and math-shift
changes for the numerator and
denominator.
If the fraction line thickness is nonzero, the
<mfrac>
element is laid out as shown on .
The fraction bar must only be painted if the
visibility of
the <mfrac>
element is visible
.
In that case, the fraction bar must be painted with the
color
of the <mfrac>
element.
The min-content inline size (respectively max-content inline size) of content is the maximum between the min-content inline size (respectively max-content inline size) of the numerator's margin box and the min-content inline size (respectively max-content inline size) of the denominator's margin box.
If there is an inline stretch size constraint or a block stretch size constraint then the numerator is also laid out with the same stretch size constraint, otherwise it is laid out without any stretch size constraint. The denominator is always laid out without any stretch size constraint.
The inline size of the math content is the maximum between the inline size of the numerator's margin box and the inline size of the denominator's margin box.
NumeratorShift
is the maximum between:
compact
(respectively normal
).
compact
(respectively normal
) +
the ink line-descent of the numerator's margin box.
DenominatorShift
is the maximum between:
compact
(respectively normal
).
compact
(respectively normal
) +
the ink line-ascent of the denominator's margin box −
the AxisHeight.
The line-ascent of the math content is the maximum between:
Numerator Shift
+
the line-ascent of the numerator's margin box.
Denominator Shift
+
the line-ascent of the denominator's margin box
The line-descent of the math content is the maximum between:
Numerator Shift
+ the line-descent of the numerator's margin box.
Denominator Shift
+ the line-descent of the denominator's margin box.
The inline offset of the numerator (respectively denominator) is half the inline size of the math content − half the inline size of the numerator's margin box (respectively denominator's margin box).
The alphabetic baseline of the numerator (respectively denominator)
is shifted away from the alphabetic baseline by a distance of
NumeratorShift
(respectively
DenominatorShift
)
towards the line-over (respectively line-under).
The math content box is placed within the content box so that their block-start edges are aligned and the middles of these edges are at the same position.
The inline size of the fraction bar is the inline size of the content box and its inline-start edge is the aligned with the one the content box. The center of the fraction bar is shifted away from the alphabetic baseline of the math content box by a distance of AxisHeight towards the line-over. Its block size is the fraction line thickness.
If the fraction line thickness is zero,
the <mfrac>
element is instead laid out as
shown on .
The min-content inline size, max-content inline size and inline size of the math content are calculated the same as in .
If there is an inline stretch size constraint or a block stretch size constraint then the numerator is also laid out with the same stretch size constraint and otherwise it is laid out without any stretch size constraint. The denominator is always laid out without any stretch size constraint.
If the math-style is compact
then
TopShift
and
BottomShift
are respectively
set to StackTopShiftUp and StackBottomShiftDown.
Otherwise math-style is normal
and
they are respectively set to StackTopDisplayStyleShiftUp
and StackBottomDisplayStyleShiftDown.
The Gap
is defined to be
(BottomShift
−
the ink line-ascent of the denominator's margin box) +
(TopShift
−
the ink line-descent of the numerator's margin box).
If math-style is compact
then GapMin
is StackGapMin,
otherwise math-style is normal
and it is StackDisplayStyleGapMin.
If Δ = GapMin
− Gap
is positive then
TopShift
and BottomShift
are respectively increased by Δ/2 and Δ − Δ/2.
The line-ascent of the math content is the maximum between:
TopShift
+
the line-ascent of the numerator's margin box.
BottomShift
+ the line-ascent of the denominator's margin box.
The line-descent of the math content is the maximum between:
TopShift
+ the line-descent of the numerator's margin box.
BottomShift
+ the line-descent of the denominator's margin box.
The inline offsets of the numerator and denominator are calculated the same as in .
The alphabetic baseline of the numerator (respectively denominator) is
shifted away from the alphabetic baseline by a distance of
TopShift
(respectively −
BottomShift
) towards the
line-over (respectively line-under).
The math content box is placed within the content box so that their block-start edges are aligned and the middles of these edges are at the same position.
<msqrt>
, <mroot>
The radical elements construct an expression with a root symbol √ with a line over the content. The msqrt element is used for square roots, while the mroot element is used to draw radicals with indices, e.g. a cube root.
The <msqrt>
and <mroot>
elements accept the attributes described
in .
The following example contains a square root written with [^msqrt^] and a cube root written with [^mroot^]. Note that [^msqrt^] has several children and the square root applies to all of them. [^mroot^] has exactly two children: it is a root of index the second child (the number 3), applied to the first child (the square root). Also note these elements only change the font-size within the [^mroot^] index, but it is scaled down more than within the numerator and denumerator of the fraction.
The <msqrt>
and <mroot>
elements sets math-shift to
compact
.
The <mroot>
element
increments scriptlevel
by 2, and sets displaystyle
to "false" in all
but its first child.
The user agent stylesheet
must contain the following rule in order to implement that behavior:
If the <msqrt>
or <mroot>
element do not have their computed
[^math/display^] property equal to block math
or inline math
then they are laid out according to the CSS specification where
the corresponding value is described.
Otherwise, the layout below is performed.
If the <mroot>
has less or more than two
in-flow children,
its layout algorithm
is the same as the [^mrow^] element.
Otherwise, the first in-flow child is called
mroot base and
the second in-flow child is called
mroot index
and its layout algorithm is explained below.
<mroot>
element has two children
that are in-flow. Hence the CSS rules basically perform
scriptlevel
and displaystyle
changes for the index.
The <msqrt>
element
generates an anonymous <mrow> box
called the msqrt base.
The radical symbol must only be painted if the
visibility of
the <msqrt>
or <mroot>
element is visible
.
In that case, the radical symbol must be painted with the
color
of that element.
The radical glyph is the glyph obtained for the character U+221A SQUARE ROOT.
The radical gap is given by
RadicalVerticalGap
if the math-style is compact
and
RadicalDisplayStyleVerticalGap
if the math-style is normal
.
The radical target size for the stretchy radical glyph is the sum of RadicalRuleThickness, radical gap and the ink height of the base.
The box metrics of the radical glyph and painting of the surd are given by the algorithm to shape a stretchy glyph to block dimension the target size for the radical glyph.
The <msqrt>
element is laid out as shown on
.
The min-content inline size (respectively max-content inline size) of the math content is the sum of the preferred inline size of a glyph stretched along the block axis for the radical glyph and of the min-content inline size (respectively max-content inline size) of the msqrt base's margin box.
The inline size of the math content is the sum of the advance width of the box metrics of the radical glyph and of the inline size of the msqrt base's margin's box.
The line-ascent of the math content is the maximum between:
The line-descent of the math content is the maximum between:
The inline size of the overbar is the inline size of the msqrt base's margin's box. The inline offsets of the msqrt base and overbar are also the same and equal to the width of the box metrics of the radical glyph.
The alphabetic baseline of the msqrt base is aligned with the alphabetic baseline. The block size of the overbar is RadicalRuleThickness. Its vertical center is shifted away from the alphabetic baseline by a distance towards the line-over equal to the line-ascent of the math content, minus the RadicalExtraAscender, minus half the RadicalRuleThickness.
Finally, the painting of the surd is performed:
The <mroot>
element is laid out as shown on
.
The mroot index is first ignored and the mroot base
and
radical glyph are laid out as
shown on figure
using the same algorithm as in
in order to produce a margin box B (represented in green).
The min-content inline size (respectively max-content inline size) of the math content is the sum of max(0, RadicalKernBeforeDegree), the mroot index's min-content inline size (respectively max-content inline size) of the mroot index's margin box, max(−min-content inline size, RadicalKernAfterDegree) (respectively max(−max-content inline size of the mroot index's margin box, RadicalKernAfterDegree)) and of the min-content inline size (respectively max-content inline size) of B.
Using the same clamping, AdjustedRadicalKernBeforeDegree and AdjustedRadicalKernAfterDegree are respectively defined as max(0, RadicalKernBeforeDegree) and is max(−inline size of the index's margin box, RadicalKernAfterDegree).
The inline size of the math content is the sum of AdjustedRadicalKernBeforeDegree, the inline size of the index's margin box, AdjustedRadicalKernAfterDegree and of the inline size of B.
The line-ascent of the math content is the maximum between:
The line-descent of the math content is the maximum between:
The inline offset of the index is AdjustedRadicalKernBeforeDegree. The inline-offset of the mroot base is the same + the inline size of the index's margin box.
The alphabetic baseline of B is aligned with the alphabetic baseline. The alphabetic baseline of the index is shifted away from the line-under edge by a distance of RadicalDegreeBottomRaisePercent × the block size of B + the line-descent of the index's margin box.
<mstyle>
Historically, the mstyle element was introduced to make style changes that affect the rendering of its contents.
The <mstyle>
element accepts the attributes described in
. Its layout algorithm is the
same as the [^mrow^] element.
<mstyle>
is implemented for compatibility with full MathML. Authors whose only target is MathML Core are encouraged to use CSS for styling.
In the following example, [^mstyle^] is used to set the scriptlevel and displaystyle. Observe this is respectively affecting the font-size and placement of subscripts of their descendants. In MathML Core, one could just have used [^mrow^] elements instead.
<merror>
The merror element displays its contents as an ”error message”. The intent of this element is to provide a standard way for programs that generate MathML from other input to report syntax errors in their input.
In the following example, [^merror^] is used to indicate a parsing error for some LaTeX-like input:
The <merror>
element accepts the attributes described in
. Its layout algorithm is the
same as the [^mrow^] element.
The user agent stylesheet
must contain the following rule in order to visually highlight the error
message:
<mpadded>
The
mpadded
element renders the same as its in-flow child content, but with the
size and relative positioning point of its
content modified according to <mpadded>
’s attributes.
The <mpadded>
element accepts the attributes described
in as well as the following
attributes:
The width, height, depth, lspace and voffset if present, must have a value that is a valid <length-percentage>.
In the following example, [^mpadded^] is used to tweak spacing around a fraction (a blue background is used to visualize it). Without attributes, it behaves like an [^mrow^] but the attributes allow to specify the size of the box (width, height, depth) and position of the fraction within that box (lspace and voffset).
The [^mpadded^] element generates an anonymous <mrow> box called the mpadded inner box with parameters called inner inline size, inner line-ascent and inner line-descent.
The requested <mpadded>
parameters are determined as follows:
width
attribute is present, valid and not a percentage then
that attribute is used as a
presentational hint
setting the element's
width
property to the corresponding value.
height
attribute is absent, invalid or a percentage then the requested
height is the inner line-ascent.
Otherwise the requested height is the resolved
value of the height
attribute, clamping
negative values to 0
.
depth
attribute is absent, invalid or a percentage then the requested
depth is the inner line-ascent.
Otherwise the requested depth is the resolved
value of the depth
attribute, clamping
negative values to 0
.
lspace
attribute is absent, invalid or a percentage then the requested
lspace is 0. Otherwise the requested lspace is the resolved
value of the lspace
attribute, clamping
negative values to 0
.
voffset
attribute is absent, invalid or a percentage then the requested
voffset is 0. Otherwise the requested voffset is the resolved
value of the voffset
attribute.
voffset
values are not clamped to
0
.
<mpadded>
If the <mpadded>
element does not have its
computed
[^math/display^] property equal to block math
or inline math
then it is laid out according to the CSS specification where
the corresponding value is described.
Otherwise, it is laid out as shown on
.
The min-content inline size (respectively max-content inline size) of the math content is the requested width calculated in but using the min-content inline size (respectively max-content inline size) of the mpadded inner box instead of the "inner inline size".
The inline size of the math content is the requested width calculated in .
The line-ascent of the math content is the requested height. The line-descent of the math content is the requested depth.
The mpadded inner box is placed so that its alphabetic baseline is shifted away from the alphabetic baseline by the requested voffset towards the line-over.
<mphantom>
Historically, the mphantom element was introduced to render its content invisibly, but with the same metrics size and other dimensions, including alphabetic baseline position that its contents would have if they were rendered normally.
In the following example, [^mphantom^] is used to ensure alignment of corresponding parts of the numerator and denominator of a fraction:
The <mphantom>
element accepts the attributes described
in . Its layout algorithm is
the same as the [^mrow^] element.
The user agent stylesheet
must contain the following rule in order to hide the content:
<mphantom>
is implemented for compatibility with full MathML. Authors whose only target is MathML Core are encouraged to use CSS for styling.
The elements described in this section position one or more scripts around a base. Attaching various kinds of scripts and embellishments to symbols is a very common notational device in mathematics. For purely visual layout, a single general-purpose element could suffice for positioning scripts and embellishments in any of the traditional script locations around a given base. However, in order to capture the abstract structure of common notation better, MathML provides several more specialized scripting elements.
In addition to sub-/superscript elements, MathML has overscript and underscript elements that place scripts above and below the base. These elements can be used to place limits on large operators, or for placing accents and lines above or below the base.
<msub>
, <msup>
, <msubsup>
The msub, msup and msubsup elements are used to attach subscript and superscript to a MathML expression. They accept the attributes described in .
The following example shows basic use of subscripts and superscripts. The font-size is automatically scaled down within the scripts.
If the
<msub>
,
<msup>
or
<msubsup>
elements do not have their
computed
[^math/display^] property equal to block math
or inline math
then they are laid out according to the CSS specification where
the corresponding value is described.
Otherwise, the layout below is performed.
<msub>
,
<msup>
, <msubsup>
If the <msub>
element
has less or more than two in-flow children, its layout algorithm
is the same as the [^mrow^] element.
Otherwise, the first in-flow child is called the
msub base, the second in-flow child is called the
msub subscript and the layout algorithm is explained
in .
If the <msup>
element
has less or more than two in-flow children, its layout algorithm
is the same as the [^mrow^] element.
Otherwise, the first in-flow child is called the
msup base, the second in-flow child is called the
msup superscript and the layout algorithm is explained
in .
If the <msubsup>
element
has less or more than three in-flow children, its layout algorithm
is the same as the [^mrow^] element.
Otherwise, the first in-flow child is called the
msubsup base, the second in-flow child
is called the msubsup subscript,
its third in-flow child is called
the msubsup superscript and the layout algorithm is explained
in .
The <msub>
element is laid out as shown on
.
LargeOpItalicCorrection
is the italic correction of the msub base
if it is an embellished operator with
the [=embellished operator/largeop=] property and 0 otherwise.
The
min-content inline size (respectively max-content inline size) of the math content is the
min-content inline size (respectively max-content inline size) of the msub base's margin box −
LargeOpItalicCorrection
+
min-content inline size (respectively max-content inline size) of
the msub subscript's margin box + SpaceAfterScript.
If there is an inline stretch size constraint or a block stretch size constraint then the msub base is also laid out with the same stretch size constraint and otherwise it is laid out without any stretch size constraint. The scripts are always laid out without any stretch size constraint.
The inline size of the math content
is the inline size of the msub base's margin box −
LargeOpItalicCorrection
+
the inline size of
the msub subscript's margin box + SpaceAfterScript.
SubShift
is the maximum between:
The line-ascent of the math content is the maximum between:
SubShift
.The line-descent of the math content is the maximum between:
SubShift
.
The inline offset of the msub base is 0 and the inline offset of the
msub subscript is the inline size of the msub base's margin box −
LargeOpItalicCorrection
.
The msub base is placed so that its alphabetic baseline
matches the alphabetic baseline. The msub subscript is placed so that its alphabetic baseline
is shifted away from the alphabetic baseline by SubShift
towards the line-under.
The <msup>
element is laid out as shown on
.
ItalicCorrection
is the italic correction of the msup base
if it is not an embellished operator with
the [=embellished operator/largeop=] property and 0 otherwise.
The
min-content inline size (respectively max-content inline size)
of the math content
is the
min-content inline size (respectively max-content inline size) of
the msup base's margin box +
ItalicCorrection
+
the min-content inline size (respectively max-content inline size) of
the msup superscript's margin box + SpaceAfterScript.
If there is an inline stretch size constraint or a block stretch size constraint then the msup base is also laid out with the same stretch size constraint and otherwise it is laid out without any stretch size constraint. The scripts are always laid out without any stretch size constraint.
The inline size of the math content
is the inline size of the msup base's margin box +
ItalicCorrection
+
the inline size of
the msup superscript's margin box + SpaceAfterScript.
SuperShift
is the maximum between:
compact
, or
SuperscriptShiftUp otherwise.The line-ascent of the math content is the maximum between:
SuperShift
.The line-descent of the math content is the maximum between:
SuperShift
.
The inline offset of the msup base is 0 and the inline offset of
msup superscript is the inline size of the msup base's margin box +
ItalicCorrection
.
The msup base is placed so that its alphabetic baseline
matches the alphabetic baseline. The msup superscript is placed so that its
alphabetic baseline
is shifted away from the alphabetic baseline by SuperShift
towards the line-over.
The <msubsup>
element is laid out as shown on
.
LargeOpItalicCorrection
and SubShift
are set as in .
ItalicCorrection
and SuperShift
are set as in .
The min-content inline size (respectively max-content inline size and inline size) of the math content is the maximum between the min-content inline size (respectively max-content inline size and inline size) of the math content calculated in and .
If there is an inline stretch size constraint or a block stretch size constraint then the msubsup base is also laid out with the same stretch size constraint and otherwise it is laid out without any stretch size constraint. The scripts are always laid out without any stretch size constraint.
If there is an inline stretch size constraint or a block stretch size constraint then the msubsup base is also laid out with the same stretch size constraint and otherwise it is laid out without any stretch size constraint. The scripts are always laid out without any stretch size constraint.
SubSuperGap
is the gap between the two scripts
along the block axis and is defined by
(SubShift
− the ink line-ascent of the msubsup subscript's
margin box) +
(SuperShift
− the ink line-descent of the
msubsup superscript's margin box).
If SubSuperGap
is not at least
SubSuperscriptGapMin then the following steps are
performed to ensure that the condition holds:
SuperShift
− the ink line-descent of the
msubsup superscript's margin box).
If Δ > 0 then set Δ to the minimum between Δ set
SubSuperscriptGapMin − SubSuperGap
and
increase SuperShift
(and so
SubSuperGap
too) by Δ.
SubSuperGap
.
If Δ > 0 then
increase SubscriptShift
(and so
SubSuperGap
too) by Δ.
The ink line-ascent (respectively line-ascent, ink line-descent,
line-descent) of the math content
is set to the maximum
of the
ink line-ascent (respectively line-ascent, ink line-descent,
line-descent) of the math content
calculated in
and
but using the adjusted values SubShift
and
SuperShift
above.
The inline offset and block offset of the msubsup base and scripts are performed the same as described in and .
Even when the msubsup subscript (respectively msubsup superscript) is an empty
box, <msubsup>
does not generally render the same as
(respectively )
because of the additional constraint on
SubSuperGap
.
Moreover, positioning the empty msubsup subscript
(respectively msubsup superscript)
may also change the total size.
In order to keep the algorithm simple, no attempt is made to handle empty scripts in a special way.
<munder>
, <mover>
, <munderover>
The munder, mover and munderover elements are used to attach accents or limits placed under or over a MathML expression.
The <munderover>
element accepts the attribute
described in as well as the
following attributes:
Similarly, the <mover>
element
(respectively <munder>
element) accepts the
attribute described in
as well as the [^mover/accent^]
attribute (respectively the
[^mover/accentunder^] attribute).
accent,
accentunder
attributes, if present, must have values that are booleans.
If these attributes are absent or invalid, they are treated as
equal to false
.
User agents must implement them as described in
.
The following example shows basic use of under- and overscripts. The font-size is automatically scaled down within the scripts, unless they are meant to be accents.
If the
<munder>
,
<mover>
or
<munderover>
elements do not have their
computed
[^math/display^] property equal to block math
or inline math
then they are laid out according to the CSS specification where
the corresponding value is described.
Otherwise, the layout below is performed.
<munder>
,
<mover>
, <munderover>
If the <munder>
element
has less or more than two in-flow children, its layout algorithm
is the same as the [^mrow^] element.
Otherwise, the first in-flow child is called the
munder base and the second in-flow child is called the
munder underscript.
If the <mover>
element
has less or more than two in-flow children, its layout algorithm
is the same as the [^mrow^] element.
Otherwise, the first in-flow child is called the
mover base and the second in-flow child is called the
mover overscript.
If the <munderover>
element
has less or more than three in-flow children, its layout algorithm
is the same as the [^mrow^] element.
Otherwise, the first in-flow child is called the
munderover base, the second in-flow child
is called the munderover underscript
and its third in-flow child is called
the munderover overscript.
If the
<munder>
, <mover>
or
<munderover>
elements have a computed
math-style property equal to compact
and their base is an embellished operator with the
[=embellished operator/movablelimits=] property, then
their layout algorithms are respectively
the same as the ones described for
<msub>
, <msup>
and
<msubsup>
in
,
and
.
Otherwise, the
<munder>
, <mover>
and
<munderover>
layout algorithms are respectively
described in
,
and
.
The algorithm for stretching operators along the inline axis is as follows.
L_{ToStretch}
containing
embellished operators with
a [=embellished operator/stretchy=] property and inline [=embellished operator/stretch axis=];
and a second list L_{NotToStretch}
.
L_{NotToStretch}
.
If L_{ToStretch}
is empty then stop.
If L_{NotToStretch}
is empty, perform
layout with inline stretch size constraint 0 for
all the items of L_{ToStretch}
.
T
to
the maximum inline size of the
margin boxes of child boxes that have been laid out in the
previous step.
L_{ToStretch}
with inline stretch size constraint T
.
The <munder>
element is laid out as shown on
.
LargeOpItalicCorrection
is the italic correction of the munder base
if it is an embellished operator with
the [=embellished operator/largeop=] property and 0 otherwise.
The min-content inline size (respectively max-content inline size) of the math content are calculated like the inline size of the math content below but replacing the inline sizes of the munder base's margin box and munder underscript's margin box with the min-content inline size (respectively max-content inline size) of the munder base's margin box and munder underscript's margin box.
The in-flow children are laid out using the algorithm for stretching operators along the inline axis.
The inline size of the math content is calculated by determining the absolute difference between:
LargeOpItalicCorrection
.LargeOpItalicCorrection
.
If m is the minimum calculated in the second item above then the
inline offset
of the munder base is −m − half the inline size of the base's margin box.
The inline offset of the munder underscript is
−m − half the inline size of the munder underscript's margin box −
half LargeOpItalicCorrection
.
Parameters
UnderShift
and UnderExtraDescender
are determined by considering three cases in the following order:
The munder base is an
embellished operator with the
[=embellished operator/largeop=] property.
UnderShift
is the maximum of
UnderExtraDescender
is 0.
The munder base is an
embellished operator with the
[=embellished operator/stretchy=] property
and [=embellished operator/stretch axis=] inline.
UnderShift
is the maximum of:
UnderExtraDescender
is 0.
UnderShift
is equal to UnderbarVerticalGap
if the [^munder/accentunder^] attribute is not an
ASCII case-insensitive match to true
and to zero otherwise.
UnderExtraAscender
is
UnderbarExtraDescender.
The line-ascent of the math content is the maximum between:
UnderShift
.The line-descent of the math content is the maximum between:
UnderShift
+ UnderExtraAscender
.
The alphabetic baseline of the munder base is aligned with the alphabetic baseline.
The alphabetic baseline of the munder underscript is shifted away from the alphabetic baseline
and towards the line-under by a distance equal to
the ink line-descent of the munder base's margin box
+ UnderShift
.
The math content box is placed within the content box so that their block-start edges are aligned and the middles of these edges are at the same position.
The <mover>
element is laid out as shown on
.
LargeOpItalicCorrection
is the italic correction of the mover base
if it is an embellished operator with
the [=embellished operator/largeop=] property and 0 otherwise.
The min-content inline size (respectively max-content inline size) of the math content are calculated like the inline size of the math content below but replacing the inline sizes of the mover base's margin box and mover overscript's margin box with the min-content inline size (respectively max-content inline size) of the mover base's margin box and mover overscript's margin box.
The in-flow children are laid out using the algorithm for stretching operators along the inline axis.
The TopAccentAttachment
is the
top accent attachment of the mover overscript or
half the inline size of the mover overscript's margin box
if it is undefined.
The inline size of the math content is calculated by applying the algorithm for stretching operators along the inline axis for layout and determining the absolute difference between:
TopAccentAttachment
+
half LargeOpItalicCorrection
.TopAccentAttachment
+
half LargeOpItalicCorrection
.
If m is the minimum calculated in the second item above then the
inline offset
of the mover base is −m − half the inline size of the base's margin.
The inline offset of the mover overscript is
−m − half the inline size of the mover overscript's margin box +
half LargeOpItalicCorrection
.
Parameters
OverShift
and OverExtraDescender
are determined by considering three cases in the following order:
The mover base is an
embellished operator with the
[=embellished operator/largeop=] property.
OverShift
is the maximum of
OverExtraAscender
is 0.
The mover base is an
embellished operator with the
[=embellished operator/stretchy=] property and
[=embellished operator/stretch axis=] inline.
OverShift
is the maximum of:
OverExtraDescender
is 0.
Otherwise, OverShift
is equal to
true
.
OverExtraAscender
is OverbarExtraAscender.
The line-ascent of the math content is the maximum between:
OverShift
+ OverExtraAscender
.The line-descent of the math content is the maximum between:
OverShift
.
The alphabetic baseline of the mover base is aligned with the alphabetic baseline.
The alphabetic baseline of the mover overscript is shifted away from the alphabetic baseline
and towards the line-over by a distance equal to
the ink line-ascent of the base + OverShift
.
The math content box is placed within the content box so that their block-start edges are aligned and the middles of these edges are at the same position.
The general layout of <munderover>
is shown on
. The
LargeOpItalicCorrection
,
UnderShift
,
UnderExtraDescender
,
OverShift
,
OverExtraDescender
parameters
are calculated the same as in
and
.
The min-content inline size, max-content inline size and inline size of the math content are calculated as an absolute difference between a maximum inline offset and minimum inline offset. These extrema are calculated by taking the extremum value of the corresponding extrema calculated in and . The inline offsets of the munderover base, munderover underscript and munderover overscript are calculated as in these sections but using the new minimum m (minimum of the corresponding minima).
Like in these sections, the in-flow children are laid out using the algorithm for stretching operators along the inline axis.
The line-ascent and line-descent of the math content are also calculated by taking the extremum value of the extrema calculated in and .
Finally, the alphabetic baselines of the munderover base, munderover underscript and munderover overscript are calculated as in sections and .
The math content box is placed within the content box so that their block-start edges are aligned and the middles of these edges are at the same position.
When the underscript (respectively overscript) is an empty box, the base and overscript (respectively underscript) are laid out similarly to (respectively ) but the position of the empty underscript (respectively overscript) may add extra space. In order to keep the algorithm simple, no attempt is made to handle empty scripts in a special way.
<mmultiscripts>
Presubscripts and tensor notations are represented by the mmultiscripts element. The mprescripts element is used as a separator between the postscripts and prescripts. These two elements accept the attributes described in .
The following example shows basic use of prescripts and postscripts, involving a [^mprescripts^]. Empty [^mrow^] elements are used at positions where no scripts are rendered. The font-size is automatically scaled down within the scripts.
If the
<mmultiscripts>
or
<mprescripts>
elements do not have their
computed
[^math/display^] property equal to block math
or inline math
then they are laid out according to the CSS specification where
the corresponding value is described.
Otherwise, the layout below is performed.
The
<mprescripts>
element is laid out as an [^mrow^]
element.
A valid <mmultiscripts>
element contains the
following in-flow children:
If an <mmultiscripts>
element is not valid then
it is laid out the same as the
[^mrow^] element.
Otherwise the layout algorithm is performed as in
.
The <mmultiscripts>
element is laid out as
shown on .
For each subscript/superscript pair of
mmultiscripts postscripts,
the ItalicCorrection
LargeOpItalicCorrection
are defined as
in
and .
The min-content inline size (respectively max-content inline size) of the math content is calculated the same as the inline size of the math content below, but replacing "inline size" with "min-content inline size" (respectively "max-content inline size") for the mmultiscripts base's margin box and scripts' margin boxes.
If there is an inline stretch size constraint or a block stretch size constraint the mmultiscripts base is also laid out with the same stretch size constraint. Otherwise it is laid out without any stretch size constraint. The other elements are always laid out without any stretch size constraint.
The inline size of the math content is calculated with the following algorithm:
inline-offset
to 0.
For each subscript/superscript pair of
mmultiscripts prescripts, increment
inline-offset
by SpaceAfterScript + the
maximum of
inline-offset
by the inline size of the
mmultiscripts base's margin box and
set inline-size
to inline-offset
.
For each subscript/superscript pair of
mmultiscripts postscripts, modify
inline-size
to be at least:
LargeOpItalicCorrection
.
ItalicCorrection
.
Increment inline-offset
to the maximum of:
Increment inline-offset
by
SpaceAfterScript.
inline-size
.
SubShift
(respectively SuperShift
)
is calculated by taking the maximum of all subshifts
(respectively supershifts) of each
subscript/superscript pair as described in
.
The line-ascent of the math content is calculated
by taking the maximum of all the line-ascent
of each subscript/superscript pair as described in
but using the SubShift
and
SuperShift
values calculated above.
The line-descent of the math content is calculated
by taking the maximum of all the line-descent
of each subscript/superscript pair as described in
but using the SubShift
and
SuperShift
values calculated above.
Finally, the placement of the in-flow children is performed using the following algorithm:
inline-offset
to 0.For each subscript/superscript pair of mmultiscripts prescripts:
inline-offset
by
SpaceAfterScript.
pair-inline-size
to the maximum of
inline-offset
+ pair-inline-size
− the inline size of the subscript's margin box.
inline-offset
+ pair-inline-size
− the inline size of the superscript's margin box.
SubShift
(respectively SuperShift
)
towards the line-under (respectively line-over).
inline-offset
by
pair-inline-size
.
<mprescripts>
boxes
at inline offsets
inline-offset
and with their alphabetic baselines
aligned with the alphabetic baseline.
For each subscript/superscript pair of mmultiscripts postscripts:
pair-inline-size
to the maximum of
inline-offset
− LargeOpItalicCorrection
.
inline-offset
+ ItalicCorrection
.
SubShift
(respectively SuperShift
)
towards the line-under (respectively line-over).
inline-offset
by
pair-inline-size
.
inline-offset
by
SpaceAfterScript.
An <mmultiscripts>
with only one
subscript/superscript pair of
mmultiscripts postscripts is laid out the same as a
<msubsup>
with the same in-flow children.
However, as
noticed for
<msubsup>
,
if additionally the subscript (respectively superscript) is an
empty box then it is not necessarily laid out the same as an
<msub>
(respectively <msup>
) element.
In order to keep the algorithm simple, no attempt is made to
handle empty scripts in a special
way.
For all scripted elements, the rule of thumb is to set
displaystyle
to false
and
to increment scriptlevel
in all child
elements but the first one.
However, an [^mover^] (respectively
[^munderover^])
element with an [^mover/accent^]
attribute that is an
ASCII case-insensitive
match to true
does not increment scriptlevel within
its second child (respectively third child). Similarly,
[^mover^] and
[^munderover^] elements
with an [^mover/accentunder^]
attribute that is an
ASCII case-insensitive
match to true
do not increment scriptlevel within
their second child.
<mmultiscripts>
sets
math-shift
to
compact
on its children at even position if they are
before an [^mprescripts^], and on those at odd position
if they are after
an [^mprescripts^].
The <msub>
and <msubsup>
elements set math-shift
to
compact
on their second child.
[^mover^] and
[^munderover^]
elements with an [^mover/accent^]
attribute that is an
ASCII case-insensitive
match to true
also set math-shift
to
compact
within their first child.
The must contain the following style in order to implement this behavior:
<mprescripts>
is empty.
Hence the CSS rules essentially perform automatic displaystyle
and
scriptlevel
changes for the scripts; and
math-shift
changes for
subscripts and sometimes the base.
Matrices, arrays and other table-like mathematical notation are marked up using [^mtable^] [^mtr^] [^mtd^] elements. These elements are similar to the [^table^], [^tr^] and [^td^] elements of [[HTML]].
The following example shows how tabular layout allows to write a matrix. Note that it is vertically centered with the fraction bar and the middle of the equal sign.
<mtable>
The mtable is laid out as an
inline-table
and sets
displaystyle
to false
. The
user agent stylesheet must contain
the following rules in order to implement these properties:
The mtable
element is as a CSS
table
and the
min-content inline size, max-content inline size,
inline size, block size,
first baseline set and last baseline set
sets are determined
accordingly.
The center of the table is aligned with the math axis.
<mtr>
The mtr is laid out as
table-row
. The
user agent stylesheet must contain
the following rules in order to implement that behavior:
<mtd>
The mtd is laid out as
a table-cell
with content centered in the cell and
a default padding. The
user agent stylesheet must contain
the following rules:
The <mtd>
accepts the attributes described
in as well as the following attributes:
The columnspan
(respectively
rowspan
) attribute has the same
syntax and semantics as the
[^td/colspan^]
(respectively
[^td/rowspan^])
attribute on the <td>
element from [[HTML]].
In particular, the parsing of these attributes is handled as
described in the
algorithm for processing rows, always reading "colspan
" as
"columnspan
".
columnspan
and is preserved for backward
compatibility reasons.
The <mtd>
element
generates an anonymous <mrow> box.
Historically, the maction element provides a mechanism for binding actions to expressions.
The <maction>
element accepts the attributes described
in as well as the following
attributes:
This specification does not define any observable behavior that is specific to the actiontype and selection attributes.
The following example shows the "toggle" action type from [[MathML3]] where the renderer alternately displays the selected subexpression, starting from "one third" and cycling through them when there is a click on the selected subexpression ("one quarter", "one half", "one third", etc). This is not part of MathML Core but can be implemented using JavaScript and CSS polyfills. The default behavior is just to render the first child.
The layout algorithm of the <maction>
element
is the same as the <mrow>
element.
The user agent stylesheet
must contain the following rules in order to hide all but
its first child element,
which is the default behavior for the legacy actiontype
values:
<maction>
is implemented for compatibility with full MathML. Authors whose only target is MathML Core are encouraged to use other HTML, CSS and JavaScript mechanisms to implement custom actions. They may
rely on maction attributes defined in [[MathML3]].
The
semantics
element is the container element that associates
annotations with a MathML expression. Typically, the
<semantics>
element has as its first child element
a MathML expression to be annotated while subsequent child elements
represent
text annotations within an annotation
element, or more complex markup annotations within
an annotation-xml element.
The following example shows how the fraction "one half" can be annotated with a textual annotation (LaTeX) or an XML annotation (content MathML), which are not intended to be rendered by the user agent. This fraction is also annotated with equivalent SVG and HTML markup.
The <semantics>
element accepts the attributes
described in . Its layout algorithm
is the same as the [^mrow^] element.
The user agent stylesheet
must contain the following rule in order to only render the annotated
MathML expression:
The <annotation-xml>
and
<annotation>
element accepts the attributes
described in as well as the
following attribute:
This specification does not define any observable behavior that is specific to the encoding attribute.
The layout algorithm of the <annotation-xml>
and <annotation>
element is the same as the [^mtext^] element.
/* Hide the annotated child. */ semantics > :first-child { display: none; } /* Show all text annotations. */ semantics > annotation { display: inline; } /* Show all HTML annotations. */ semantics > annotation-xml[encoding="text/html" i], semantics > annotation-xml[encoding="application/xhtml+xml" i] { display: inline-block; }
display: block math
and display: inline math
valueThe display
property
from
is extended with a new inner display type:
Name: | display |
---|---|
New values: | <display-outside> || [ <display-inside> | math ] |
For elements that are not MathML elements, if the specified
value of display
is block math
or
inline math
then the computed value is
block flow
and inline flow
respectively.
For the [^mtable^] element
the computed value is block table
and
inline table
respectively.
For the [^mtr^] element, the computed value
is table-row
.
For the [^mtd^] element, the computed value
is table-cell
.
MathML elements with a
computed display
value equal to
block math
or inline math
control box generation and layout according to their tag name, as
described in the relevant sections.
Unknown MathML elements
behave the same as the [^mrow^] element.
display: block math
and
display: inline math
values provide a default
layout for MathML elements while at the same time allowing
to override it with either native display values or
custom values.
This allows authors or polyfills to define their own custom notations
to tweak or extend MathML Core.
In the following example, the default layout of the MathML [^mrow^] element is overridden to render its content as a grid.
text-transform
valueThe text-transform property from is extended with a new value:
Name: | text-transform |
---|---|
New value: | math-auto |
On text nodes containing a single character, if the computed value
is math-auto
and the character is present in the
"Original" column of
then it is converted to the corresponding character from the "italic"
column.
A common style convention is to render
identifiers with multiple letters (e.g. the function name "exp")
with normal style and identifiers with a single letter
(e.g. the variable "n") with italic style. The
math-auto
property is intended to implement this
default behavior, which can be overridden by authors if necessary.
Note that mathematical fonts are designed with a special kind
of italic glyphs located at the
Unicode positions of
, which differ from the shaping
obtained via italic font style. Compare this
mathematical formula
rendered with the Latin Modern Math font using
font-style: italic
(left) and
text-transform: math-auto
(right):
math-style
propertyName: | math-style |
---|---|
Value: | normal | compact |
Initial: | normal |
Applies to: | All elements |
Inherited: | yes |
Percentages: | n/a |
Computed value: | specified keyword |
Canonical order: | n/a |
Animation type: | not animatable |
Media: | visual |
When math-style
is compact
,
the math layout on descendants tries to minimize the
logical height by
applying the following rules:
math
and
the computed value of math-depth
is
auto-add
(default for [^mfrac^])
as described in .The following example shows a
mathematical formula rendered with
its [^math^] root styled with
math-style: compact
(left) and
math-style: normal
(right).
In the former case, the font-size is automatically scaled down
within the fractions and the summation limits are rendered as
subscript and superscript of the ∑. In the latter case, the ∑ is
drawn bigger than normal text and
vertical gaps within fractions (even relative to current
font-size) are larger.
These two math-style
values typically correspond to
mathematical expressions in inline and display
mode respectively [[TeXBook]].
A mathematical formula in display mode
may automatically switch to inline mode within some subformulas
(e.g. scripts, matrix elements, numerators and denominators, etc)
and it is sometimes desirable to override this default behavior.
The math-style property allows to easily implement these
features for MathML in the
user agent stylesheet
and with the displaystyle attribute; and also exposes
them to polyfills.
math-shift
propertyName: | math-shift |
---|---|
Value: | normal | compact |
Initial: | normal |
Applies to: | All elements |
Inherited: | yes |
Percentages: | n/a |
Computed value: | specified keyword |
Canonical order: | n/a |
Animation type: | not animatable |
Media: | visual |
If the value of math-shift
is compact
, the math layout on descendants will use the
superscriptShiftUpCramped parameter to place superscript.
If the value of math-shift
is normal
, the math
will use the superscriptShiftUp parameter instead.
This property is used for positioning superscript during the layout of MathML scripted elements. See § , and .
In the following example, the two "x squared" are rendered with
compact math-style and the same font-size
.
However, the one within the square root is rendered with
compact math-shift
while
the other one is rendered with
normal math-shift
, leading
to subtle different shift of the superscript "2".
Per [[TeXBook]], a mathematical formula uses normal style by default but may switch to compact style ("cramped" in TeX's terminology) within some subformulas (e.g. radicals, fraction denominators, etc). The math-shift property allows to easily implement these rules for MathML in the user agent stylesheet. Page authors or developers of polyfills may also benefit from having access to this property to tweak or refine the default implementation.
math-depth
property
A new math-depth property is introduced to describe a notion
of "depth" for each element of a mathematical formula, with respect to
the top-level container of that formula. Concretely, this is used to
determine the computed value of the
font-size
property when its specified value is math
.
Name: | math-depth |
---|---|
Value: | auto-add | add(<integer>) | <integer> |
Initial: | 0 |
Applies to: | All elements |
Inherited: | yes |
Percentages: | n/a |
Computed value: | an integer, see below |
Canonical order: | n/a |
Animation type: | not animatable |
Media: | visual |
The computed value of the math-depth value is determined as follows:
auto-add
and
the inherited value of math-style
is compact
then the computed value of
math-depth of the element is its inherited value plus one.
add(<integer>)
then the computed value
of math-depth of the element is its inherited value plus
the specified integer.
<integer>
then the computed value
of math-depth of the element is the specified integer.
If the specified value of
font-size
is math
then the
computed value of
font-size
is obtained by multiplying the inherited value of
font-size
by a nonzero scale factor calculated by the
following procedure:
InvertScaleFactor
to true.InvertScaleFactor
to false.InvertScaleFactor
is false and 1/S otherwise.The following example shows a mathematical formula with normal math-style rendered with the Latin Modern Math font. When entering subexpressions like scripts or fractions, the font-size is automatically scaled down according to the values of MATH table contained in that font. Note that font-size is scaled down when entering the superscripts but even faster when entering a root's prescript. Also it is scaled down when entering the inner fraction but not when entering the outer one, due to automatic change of math-style in fractions.
These rules from [[TeXBook]] are subtle and it's worth having a
separate math-depth
mechanism to express and
handle them. They can be implemented in MathML using the
user agent stylesheet.
Page authors or developers of polyfills may also benefit from
having access to this property to tweak or refine the default
implementation. In particular, the scriptlevel attribute
from MathML provides a way to perform math-depth
changes.
MATH
table
This chapter describes features provided by MATH
table
of an OpenType font [[OPEN-FONT-FORMAT]]. Throughout this chapter,
a C-like notation
Table.Subtable1[index].Subtable2.Parameter
is used to
denote OpenType parameters.
Such parameters may not be available (e.g. if the font lacks one of the
subtable, has an invalid offset, etc) and so fallback options are
provided.
OpenType values expressed in design units (perhaps indirectly via a
MathValueRecord
entry) are scaled to appropriate values
for layout purpose, taking into account
head.unitsPerEm
, CSS
font-size
or zoom level.
MathConstants
)These are global layout constants for the first available font:
post.underlineThickness
or
Default fallback constant if the constant is not available.
MATH.MathConstants.scriptPercentScaleDown / 100
or
0.71 if MATH.MathConstants.scriptPercentScaleDown
is
null or not available.
MATH.MathConstants.scriptScriptPercentScaleDown / 100
or
0.5041 if
MATH.MathConstants.scriptScriptPercentScaleDown
is
null or not available.
MATH.MathConstants.displayOperatorMinHeight
or
Default fallback constant
if the constant is not available.MATH.MathConstants.axisHeight
or half
OS/2.sxHeight
if the constant is not available.MATH.MathConstants.accentBaseHeight
or OS/2.sxHeight
if the constant is not available.MATH.MathConstants.subscriptShiftDown
or OS/2.ySubscriptYOffset
if the constant is not available.MATH.MathConstants.subscriptTopMax
or ⅘ × OS/2.sxHeight
if the constant is not available.MATH.MathConstants.subscriptBaselineDropMin
or
Default fallback constant if the constant is not available.MATH.MathConstants.superscriptShiftUp
or OS/2.ySuperscriptYOffset
if the constant is not available.MATH.MathConstants.superscriptShiftUpCramped
or
Default fallback constant if the constant is not available.MATH.MathConstants.superscriptBottomMin
or ¼ × OS/2.sxHeight
if the constant is not available.MATH.MathConstants.superscriptBaselineDropMax
or
Default fallback constant if the constant is not available.MATH.MathConstants.subSuperscriptGapMin
or 4 × default rule thickness if the constant is not available.MATH.MathConstants.superscriptBottomMaxWithSubscript
or ⅘ × OS/2.sxHeight
if the constant is not available.MATH.MathConstants.spaceAfterScript
or 1/24em if the constant is not available.MATH.MathConstants.upperLimitGapMin
or
Default fallback constant if the constant is not available.MATH.MathConstants.upperLimitBaselineRiseMin
or Default fallback constant if the constant is not available.MATH.MathConstants.lowerLimitGapMin
or Default fallback constant if the constant is not available.MATH.MathConstants.lowerLimitBaselineDropMin
or Default fallback constant if the constant is not available.MATH.MathConstants.stackTopShiftUp
or Default fallback constant if the constant is not available.MATH.MathConstants.stackTopDisplayStyleShiftUp
or Default fallback constant if the constant is not available.MATH.MathConstants.stackBottomShiftDown
or Default fallback constant if the constant is not available.MATH.MathConstants.stackBottomDisplayStyleShiftDown
or Default fallback constant if the constant is not available.MATH.MathConstants.stackGapMin
or 3 × default rule thickness if the constant is not available.MATH.MathConstants.stackDisplayStyleGapMin
or 7 × default rule thickness if the constant is not available.MATH.MathConstants.stretchStackTopShiftUp
or Default fallback constant if the constant is not available.MATH.MathConstants.stretchStackBottomShiftDown
or Default fallback constant if the constant is not available.MATH.MathConstants.stretchStackGapAboveMin
or Default fallback constant if the constant is not available.MATH.MathConstants.stretchStackGapBelowMin
or Default fallback constant if the constant is not available.MATH.MathConstants.fractionNumeratorShiftUp
or Default fallback constant if the constant is not available.MATH.MathConstants.fractionNumeratorDisplayStyleShiftUp
or Default fallback constant if the constant is not available.MATH.MathConstants.fractionDenominatorShiftDown
or Default fallback constant if the constant is not available.MATH.MathConstants.fractionDenominatorDisplayStyleShiftDown
or Default fallback constant if the constant is not available.MATH.MathConstants.fractionNumeratorGapMin
or default rule thickness if the constant is not available.MATH.MathConstants.fractionNumDisplayStyleGapMin
or 3 × default rule thickness if the constant is not available.MATH.MathConstants.fractionRuleThickness
or default rule thickness if the constant is not available.MATH.MathConstants.fractionDenominatorGapMin
or default rule thickness if the constant is not available.MATH.MathConstants.fractionDenomDisplayStyleGapMin
or 3 × default rule thickness if the constant is not available.MATH.MathConstants.overbarVerticalGap
or 3 × default rule thickness if the constant is not available.MATH.MathConstants.overbarExtraAscender
or default rule thickness if the constant is not available.MATH.MathConstants.underbarVerticalGap
or 3 × default rule thickness if the constant is not available.MATH.MathConstants.underbarExtraDescender
or default rule thickness if the constant is not available.MATH.MathConstants.radicalVerticalGap
or 1¼ × default rule thickness if the constant is not available.MATH.MathConstants.radicalDisplayStyleVerticalGap
or default rule thickness + ¼ OS/2.sxHeight
if the constant is not available.MATH.MathConstants.radicalRuleThickness
or default rule thickness if the constant is not available.MATH.MathConstants.radicalExtraAscender
or default rule thickness if the constant is not available.MATH.MathConstants.radicalKernBeforeDegree
or 5/18em if the constant is not available.MATH.MathConstants.radicalKernAfterDegree
or −10/18em if the constant is not available.MATH.MathConstants.radicalDegreeBottomRaisePercent / 100.0
or 0.6 if the constant is not available.MathGlyphInfo
)These are per-glyph tables for the first available font:
MATH.MathGlyphInfo.MathItalicsCorrectionInfo
of italics correction values. Use the corresponding value in
MATH.MathGlyphInfo.MathItalicsCorrectionInfo.italicsCorrection
if there is one for the requested glyph or
0
otherwise.
MATH.MathGlyphInfo.MathTopAccentAttachment
of positioning top math accents along the inline axis.
Use the corresponding value in
MATH.MathGlyphInfo.MathTopAccentAttachment.topAccentAttachment
if there is one for the requested glyph or
half the advance width of the glyph otherwise.
MathVariants
)
This section describes how to handle stretchy glyphs of arbitrary
size using the MATH.MathVariants
table.
GlyphAssembly
tableThis section is based on [[?OPEN-TYPE-MATH-IN-HARFBUZZ]]. For convenience, the following definitions are used:
MATH.MathVariant.minConnectorOverlap
.
GlyphPartRecord
is an extender
if and only if
GlyphPartRecord.partFlags
has the
fExtender
flag set.
GlyphAssembly
is horizontal
if it is obtained from
MathVariant.horizGlyphConstructionOffsets
.
Otherwise it is vertical (and obtained from
MathVariant.vertGlyphConstructionOffsets
).
GlyphAssembly
table,
N_{Ext} (respectively
N_{NonExt}) is the number of extenders
(respectively non-extenders) in
GlyphAssembly.partRecords
.
GlyphAssembly
table,
S_{Ext} (respectively
S_{NonExt}) is the sum of
GlyphPartRecord.fullAdvance
for all extenders (respectively non-extenders) in
GlyphAssembly.partRecords
.
User agents must treat the GlyphAssembly
as invalid
if the following conditions are not satisfied:
GlyphPartRecord
in GlyphAssembly.partRecords
,
the values of
GlyphPartRecord.startConnectorLength
and
GlyphPartRecord.endConnectorLength
must be at least o_{min}.
Otherwise, it is not possible to satisfy the condition of
MathVariant.minConnectorOverlap
.
In this specification, a glyph assembly is built by repeating each extender r times and using the same overlap value o between each glyph. The number of glyphs in such an assembly is AssemblyGlyphCount(r) = N_{NonExt} + r N_{Ext} while the stretch size is AssembySize(o, r) = S_{NonExt} + r S_{Ext} − o (AssemblyGlyphCount(r) − 1).
r_{min} is the minimal number of repetitions needed to obtain an assembly of size at least T, i.e. the minimal r such that AssembySize(o_{min}, r) ≥ T. It is defined as the maximum between 0 and the ceiling of ((T − S_{NonExt} + o_{min} (N_{NonExt} − 1)) / S_{Ext,NonOverlapping}).
o_{max,theorical} = (AssembySize(0, r_{min}) − T) / (AssemblyGlyphCount(r_{min}) − 1) is the theorical overlap obtained by splitting evenly the extra size of an assembly built with null overlap.
o_{max} is the maximum overlap possible to build an assembly of size at least T by repeating each extender r_{min} times. If AssemblyGlyphCount(r_{min}) ≤ 1, then the actual overlap value is irrelevant. Otherwise, o_{max} is defined to be the minimum of:
GlyphPartRecord.startConnectorLength
for all
the entries in
GlyphAssembly.partRecords
, excluding the
last one if it is not an extender.
GlyphPartRecord.endConnectorLength
for all
the entries in
GlyphAssembly.partRecords
, excluding the
first one if it is not an extender.
The glyph assembly stretch size for a target size T is AssembySize(o_{max}, r_{min}).
The glyph assembly width, glyph assembly ascent and glyph assembly descent are defined as follows:
GlyphAssembly
is vertical,
the width is the maximum advance width of the glyphs of ID
GlyphPartRecord.glyphID
for all the
GlyphPartRecord
in
GlyphAssembly.partRecords
,
the ascent is the
glyph assembly stretch size
for a given target size T
and the descent is 0.
GlyphAssembly
is horizontal,
the width is glyph assembly stretch size
for a given target size T
while
the ascent (respectively descent) is the
maximum ascent (respectively descent) of the glyphs of ID
GlyphPartRecord.glyphID
for all the
GlyphPartRecord
in
GlyphAssembly.partRecords
.
The glyph assembly height is the sum of the glyph assembly ascent and glyph assembly descent.
T
.
The shaping of the glyph assembly is performed with the following algorithm:
(x, y)
to (0, 0)
,
RepetitionCounter
to 0 and
PartIndex
to -1.
RepetitionCounter
is 0:
PartIndex
.PartIndex
is
GlyphAssembly.partCount
then stop.Part
to
GlyphAssembly.partRecords[PartIndex]
.
Set RepetitionCounter
to
r_{min} if
Part
is an extender and to 1 otherwise.
Part.glyphID
so that its (left, baseline) coordinates
are at position (x, y)
.
Set x
to
x + Part.fullAdvance −
o_{max}.
Part.glyphID
so that its (left, bottom) coordinates
are at position (x, y)
.
Set y
to
y − Part.fullAdvance +
o_{max}.
RepetitionCounter
.The preferred inline size of a glyph stretched along the block axis is calculated using the following algorithm:
S
to the glyph's advance width.
MathGlyphConstruction
table
in the MathVariants.vertGlyphConstructionOffsets
table for the given glyph:
MathGlyphVariantRecord
in
MathGlyphConstruction.mathGlyphVariantRecord
,
ensure that S
is at least
the advance width of the glyph of id
MathGlyphVariantRecord.variantGlyph
.
GlyphAssembly
subtable,
then ensure
that S
is at least the
glyph assembly width.
S
.
The algorithm to shape a stretchy glyph to inline
(respectively block) dimension T
is the following:
MathGlyphConstruction
table
in the MathVariants.horizGlyphConstructionOffsets
table (respectively
MathVariants.vertGlyphConstructionOffsets
table)
for the given glyph then exit with failure.
T
then use normal shaping and bounding box for that glyph,
the MathItalicsCorrectionInfo for that glyph as
italic correction and exit with success.
MathGlyphVariantRecord
in
MathGlyphConstruction.mathGlyphVariantRecord
.
If one MathGlyphVariantRecord.advanceMeasurement
is at least T
then use
normal shaping and bounding box
for MathGlyphVariantRecord.variantGlyph
,
the MathItalicsCorrectionInfo for that glyph as
italic correction and exit with success.
GlyphAssembly
subtable
then use the bounding box given by
glyph assembly width,
glyph assembly height,
glyph assembly ascent,
glyph assembly descent, the value
GlyphAssembly.italicsCorrection
as italic
correction, perform shaping of the glyph assembly and
exit with success.
T
, then choose last one that was tried and exit
with success.
@namespace url(http://www.w3.org/1998/Math/MathML); /* Universal rules */ /* The <math> element */ /* <mrow>-like elements */ /* Token elements */ /* Tables */ /* Fractions */ /* Other rules for scriptlevel, displaystyle and math-shift */
The algorithm to set the properties of an operator from its category is as follows:
minsize
to 100%
.maxsize
to ∞
.lspace
and rspace
to the
value specified in the corresponding column.stretchy
,
symmetric
, largeop
,
movablelimits
, set that property to true
if it is listed in the last column or to false
otherwise.The algorithm to determine the category of an operator
(Content
, Form
) is as folllows:
Content
as an UTF-16 string does not have length
or 1 or 2 then exit with category Default
.
Content
is a single character in the
range U+0320–U+03FF
then exit with category Default
. Otherwise,
if it has two characters:
Content
is the surrogate pairs corresponding
to
U+1EEF0 ARABIC MATHEMATICAL OPERATOR MEEM WITH HAH WITH TATWEEL
or U+1EEF1 ARABIC MATHEMATICAL OPERATOR HAH WITH DAL and
Form
is postfix
, exit with category
I
.Content
with the first character and move to step
3.Content
is listed in
Operators_2_ascii_chars
then
replace Content
with the
Unicode character
"U+0320 plus the index of Content
in
Operators_2_ascii_chars
" and move to step
3.
Default
.Form
is infix and Content
corresponds
to one of U+007C VERTICAL LINE or U+223C TILDE OPERATOR then exit
with category ForceDefault
. If the category of
(Content
, Form
)
provided by table
has N/A encoding in table
(namely if it has category L
or M
), then
exit with that category.
Otherwise:
Key
to Content
if it is in
range U+0000–U+03FF; or to Content
− 0x1C00
if it is in range U+2000–U+2BFF. Otherwise, exit with
category Default
.
Key
according to whether Form
is infix
, prefix
,
postfix
respectively.
Key
is at most 0x2FFF.Entry
in table
such that Entry
% 0x4000 is equal to
Key
. If one is found then return the category
corresponding to encoding Entry
/ 0x1000 in
.
Otherwise, return category Default
.
The intrinsic stretch axis a Unicode character
c
is inline if it belongs to the list below.
Otherwise, the intrinsic stretch axis of c
is
block.
The following dictionary provides a human-readable version
of . Please refer to
for explanation about
how to use this dictionary and how to
determine the values Content
and Form
indexing together
the dictionary.
The values for [=embellished operator/rspace=] and [=embellished operator/lspace=] are indicated
in the corresponding columns.
The values of
[=embellished operator/stretchy=],
[=embellished operator/symmetric=],
[=embellished operator/largeop=],
[=embellished operator/movablelimits=]
are true
if they are listed in the "properties" column.
The following table gives mappings between spacing and non spacing characters when used in MathML accent constructs.
The following table provides fallback that user agents may use for
stretching a given base character when the font does not
provide a MATH.MathVariants
table.
The algorithms of
work the same except with some adjustments:
MathVariants.horizGlyphConstructionOffsets[]
item;
if it is vertical it corresponds to
a MathVariants.vertGlyphConstructionOffsets[]
item.
MathGlyphConstruction.mathGlyphVariantRecord
is
always empty.
MathVariants.minConnectorOverlap
,
GlyphPartRecord.startConnectorLength
and
GlyphPartRecord.endConnectorLength
are treated as 0.
MathGlyphConstruction.GlyphAssembly.partRecords
is built
from each table row as follows:
The following tables enumerate the mathematical alphanumeric symbols with form bold, italic, fraktur, monospace, double-struck etc that are available in Unicode. For each of them, the character in its normal form is provided as well as the difference between the code points of the transformed and original characters.
It is sometimes needed to distinguish between
Chancery and Roundhand style for MATHEMATICAL SCRIPT characters.
These are notably used in LaTeX for the
\mathcal
and \mathscr
commands.
One way to do that is to rely on
Chapter 23.4 Variation Selectors of
Unicode which describes a way to
specify selection of particular glyph variants [[UNICODE]].
Indeed, the
StandardizedVariants.txt
file from the
Unicode Character Database indicates that variant selectors
U+FE00 and U+FE01 can be used on capital script to specify
Chancery and Roundhand respectively.
Alternatively, some
mathematical fonts rely on salt
or
ssXY
properties from [[OPEN-FONT-FORMAT]]
to provide both styles. Page authors may use the
font-variant-alternates property with corresponding OpenType font features
to access these glyphs.
bold-script
mappingsbold-italic
mappingstailed
mappingsbold
mappingsfraktur
mappingsscript
mappingsmonospace
mappingsinitial
mappingssans-serif
mappingsdouble-struck
mappingslooped
mappingsstretched
mappingsitalic
mappingsbold-fraktur
mappingssans-serif-bold-italic
mappingssans-serif-italic
mappingsbold-sans-serif
mappingsMathML Core is based on MathML3. See the appendix E of [[MathML3]] for the people that contributed to that specification.
MathML Core was initially developed by the MathML Community Group, and then by the Math Working Group. Working Group or Community Group members who regularly participated in MathML Core meetings during the development of this specification: Brian Kardell, Bruce Miller, Daniel Marques, David Carlisle, David Farmer, Deyan Ginev, Frédéric Wang, Louis Mahler, Moritz Schubotz, Murray Sargent, Neil Soiffer, Patrick Ion, Rob Buis, Steve Noble and Sam Dooley.
In addition, we would like to extend special thanks to Brian Kardell, Neil Soiffer and Rob Buis for help with the editing.
Many thanks also to the following people for their help with the test suite: Brian Kardell, Frédéric Wang, Neil Soiffer and Rob Buis. Several tests are also based on MathML tests from browser repositories and we are grateful to the Mozilla and WebKit contributors.
We would like to thank the people who, through their input and feedback on public communication channels, have helped us with the creation of this specification: André Greiner-Petter, Anne van Kesteren, Boris Zbarsky, Brian Smith, Elika Etemad, Emilio Cobos Álvarez, ExE Boss, Ian Kilpatrick, Koji Ishii, L. David Baron, Michael Kohlhase, Michael Smith, Ryosuke Niwa, Sergey Malkin, Tab Atkins Jr., Viktor Yaffle and frankvel.
This specification adds script execution mechanisms via the MathML event handler attributes described in . UAs may decide to prevent execution of scripts specified in these attributes, following the same security restrictions as those applying to HTML or SVG elements.
In [[MathML3]], it was possible to make any element linkable
via href
or xlink:href
attributes, with
an URL pointing to an untrusted resource or even
javascript:
execution. These attributes are not
available in MathML Core. However, as described in
it is possible to embed
HTML or SVG content inside MathML, including HTML or SVG links.
In [[MathML3]], it was possible to use the
[^maction^] element with
the actiontype
value set to "statusline"
in order to override the text of the browser statusline. In particular,
an attacker could use this
to hide the URL text of an untrusted link e.g.
This feature is not available in MathML Core, where the [^maction^] element essentially behaves like an [^mrow^] container with extra style.
An attacker can try to hang the UA by inserting very large
stretchy operators, effectively making the algorithm
shaping of the glyph assembly deal with a huge amount of
glyphs. UAs may work around this issue
by limiting r_{min} and
GlyphAssembly.partCount
to
maximum values.
As described in CSS Fonts Module, an attacker can try to rely on malformed or malicious fonts to exploit potential security faults in browser implementations. Because the OpenType MATH table is used extensively in this specification, UAs should ensure their font sanitization mechanisms are able to deal with that table.
Finally, in order to reduce attack surface, some UAs expose runtime options to disable part of the web platform. Disabling MathML layout can essentially be achieved by forcing elements in the DOM tree to be put in the HTML namespace and disabling .
As explained in ,
MathML can be embedded into an SVG image via the
<foreignObject>
element which can thus be used in a
[^canvas^]
element.
UA may decide to implement any measure to prevent potential
information leakage
such as tainting the canvas and returning a
{{"SecurityError"}}
when one tries to access the canvas' content via JavaScript APIs.
In the following example, the canvas image is set to the image of
some MathML content with an HTML link to https://example.org/
.
It should not be possible for an attacker to determine whether that
link was visited by reading pixels via context.{{CanvasImageData/getImageData()}}
.
For more about links in MathML, see
.
This specification describes layout of DOM
elements which may involve system
fonts. Like for HTML/CSS layout,
it is thus possible to use JavaScript APIs
(e.g.
context.{{CanvasImageData/getImageData()}}
on content embedded in a canvas context, or even just
{{Element/getBoundingClientRect()}})
to measure box sizes and positions and infer data from system fonts.
By combining miscellaneous tests on such fonts and
comparing measurements against results of well-known fonts, an attacker
can try and determine the default fonts of the user.
The following
HTML+CSS+JavaScript document relies on a Web font with exotic metrics
to try and determine whether A Well Known System Font
is available by default.
The following HTML+CSS+JavaScript document tries to determine whether the UI serif font provides Asian glyphs:
The following
HTML+CSS document contains the same text rendered with
text-decoration-thickness set to from-font
and 1em
(here
100 pixels)
respectively. By comparing the heights of the two underlines,
one can calculate a good approximation of the
underlineThickness
value from the PostScript Table
[[OPEN-FONT-FORMAT]].
This specification relies on information from
to render MathML content. One
can get good approximation of most
layout parameters from MathConstants
and
MathGlyphInfo
using measurement
techniques similar to what is described above for
HTML+CSS+JavaScript document. The use of the MathVariants
table for MathML rendering can also be observed by putting stretchy
operators of different sizes inside a canvas
context.
Although none of these parameters taken individually are personal, implementing this specification increases the set of exposed font information that can be used by an attacker to implement fingerprinting techniques. Typically, they could help determine available and preferred math fonts for a user.
Conformance requirements are expressed with a combination of descriptive assertions and RFC 2119 terminology. The key words “MUST”, “MUST NOT”, “REQUIRED”, “SHALL”, “SHALL NOT”, “SHOULD”, “SHOULD NOT”, “RECOMMENDED”, “MAY”, and “OPTIONAL” in the normative parts of this document are to be interpreted as described in RFC 2119. However, for readability, these words do not appear in all uppercase letters in this specification.
All of the text of this specification is normative except sections explicitly marked as non-normative, examples, and notes. [[RFC2119]]
Examples in this specification are introduced with the words
“for example” or are set apart from the normative text with
class="example"
, like this:
This is an example of an informative example.
Informative notes begin with the word “Note” and are set apart from
the normative text with class="note"
, like this:
Note, this is an informative note.
Advisements are normative sections styled to evoke special attention
and are set apart from other normative text with
<strong class="advisement">
, like this:
UAs MUST provide an accessible alternative.