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<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01//EN"
"http://www.w3.org/TR/html4/strict.dtd">
<html lang=en>
<head>
<title>CSS 2D Transforms</title>
<link href=default.css rel=stylesheet type="text/css">
<style type="text/css">
.rhs { white-space: pre-wrap; }
code { font-size: inherit; }
#box-shadow-samples td { background: white; color: black; }
</style>
<link href="http://www.w3.org/StyleSheets/TR/W3C-ED.css" rel=stylesheet
type="text/css">
<body>
<div class=head> <!--begin-logo-->
<p><a href="http://www.w3.org/"><img alt=W3C height=48
src="http://www.w3.org/Icons/w3c_home" width=72></a> <!--end-logo-->
<h1>CSS 2D Transforms</h1>
<h2 class="no-num no-toc" id=longstatus-date>Editor's Draft 14 April 2011</h2>
<dl>
<dt>This version:
<dd> <a
href="http://www.w3.org/TR/2011/ED-css3-2d-transforms-20110414">http://dev.w3.org/csswg/css3-2d-transforms/</a>
<!--http://www.w3.org/TR/2011/WD-css3-2d-transforms-20110414-->
<dt>Latest version:
<dd><a
href="http://www.w3.org/TR/css3-2d-transforms">http://www.w3.org/TR/css3-2d-transforms</a>
<dt>Previous version:
<dd><a href="http://www.w3.org/TR/2009/WD-css3-2d-transforms-20090320/">
http://www.w3.org/TR/2009/WD-css3-2d-transforms-20090320/</a>
<dt id=editors-list>Editors:
<dd>Simon Fraser (<a href="http://www.apple.com/">Apple Inc</a>)
<simon.fraser @apple.com>
<dd>Dean Jackson (<a href="http://www.apple.com/">Apple Inc</a>) <dino
@apple.com>
<dd>David Hyatt (<a href="http://www.apple.com/">Apple Inc</a>) <hyatt
@apple.com>
<dd>Chris Marrin (<a href="http://www.apple.com/">Apple Inc</a>)
<cmarrin @apple.com>
<dd>Edward O'Connor (<a href="http://www.apple.com/">Apple Inc</a>)
<eoconnor @apple.com>
</dl>
<!--begin-copyright-->
<p class=copyright><a
href="http://www.w3.org/Consortium/Legal/ipr-notice#Copyright"
rel=license>Copyright</a> © 2011 <a
href="http://www.w3.org/"><acronym title="World Wide Web
Consortium">W3C</acronym></a><sup>®</sup> (<a
href="http://www.csail.mit.edu/"><acronym title="Massachusetts Institute
of Technology">MIT</acronym></a>, <a href="http://www.ercim.eu/"><acronym
title="European Research Consortium for Informatics and
Mathematics">ERCIM</acronym></a>, <a
href="http://www.keio.ac.jp/">Keio</a>), All Rights Reserved. W3C <a
href="http://www.w3.org/Consortium/Legal/ipr-notice#Legal_Disclaimer">liability</a>,
<a
href="http://www.w3.org/Consortium/Legal/ipr-notice#W3C_Trademarks">trademark</a>
and <a
href="http://www.w3.org/Consortium/Legal/copyright-documents">document
use</a> rules apply.</p>
<!--end-copyright-->
<hr title="Separator for header">
</div>
<h2 class="no-num no-toc" id=abstract>Abstract</h2>
<p>CSS 2D Transforms allows elements rendered by CSS to be transformed in
two-dimensional space.
<h2 class="no-num no-toc" id=status>Status of this document</h2>
<!--begin-status-->
<p>This is a public copy of the editors' draft. It is provided for
discussion only and may change at any moment. Its publication here does
not imply endorsement of its contents by W3C. Don't cite this document
other than as work in progress.
<p>The (<a
href="http://lists.w3.org/Archives/Public/www-style/">archived</a>) public
mailing list <a
href="mailto:www-style@w3.org?Subject=%5Bcss3-2d-transforms%5D%20PUT%20SUBJECT%20HERE">
www-style@w3.org</a> (see <a
href="http://www.w3.org/Mail/Request">instructions</a>) is preferred for
discussion of this specification. When sending e-mail, please put the text
“css3-2d-transforms” in the subject, preferably like this:
“[<!---->css3-2d-transforms<!---->] <em>…summary of
comment…</em>”
<p>This document was produced by the <a href="/Style/CSS/members">CSS
Working Group</a> (part of the <a href="/Style/">Style Activity</a>).
<p>This document was produced by a group operating under the <a
href="/Consortium/Patent-Policy-20040205/">5 February 2004 W3C Patent
Policy</a>. W3C maintains a <a href="/2004/01/pp-impl/32061/status"
rel=disclosure>public list of any patent disclosures</a> made in
connection with the deliverables of the group; that page also includes
instructions for disclosing a patent. An individual who has actual
knowledge of a patent which the individual believes contains <a
href="/Consortium/Patent-Policy-20040205/#def-essential">Essential
Claim(s)</a> must disclose the information in accordance with <a
href="/Consortium/Patent-Policy-20040205/#sec-Disclosure">section 6 of the
W3C Patent Policy</a>.</p>
<!--end-status-->
<p> The <a href=ChangeLog>list of changes made to this specification</a> is
available.
<h2 class="no-num no-toc" id=contents>Table of contents</h2>
<!--begin-toc-->
<ul class=toc>
<li><a href="#introduction"><span class=secno>1. </span>Introduction</a>
<li><a href="#transform-property"><span class=secno>2. </span> The <span
class=prop-name>‘<code
class=property>transform</code>’</span> Property </a>
<li><a href="#transform-origin-property"><span class=secno>3. </span> The
<span class=prop-name>‘<code
class=property>transform-origin</code>’</span> Property </a>
<li><a href="#transform-functions"><span class=secno>4. </span> The
Transformation Functions </a>
<li><a href="#transform-values"><span class=secno>5. </span> Transform
Values and Lists </a>
<li><a href="#animation"><span class=secno>6. </span> Transitions and
animations between transform values </a>
<li><a href="#matrix-decomposition"><span class=secno>7. </span> Matrix
decomposition for animation </a>
<ul class=toc>
<li><a href="#d-matrix-to-4x4-matrix-conversion"><span class=secno>7.1.
</span>2D matrix to 4x4 matrix conversion</a>
<li><a href="#unmatrix"><span class=secno>7.2. </span>Unmatrix</a>
<li><a href="#animating-the-components"><span class=secno>7.3.
</span>Animating the components</a>
<li><a href="#recomposing-the-matrix"><span class=secno>7.4.
</span>Recomposing the matrix</a>
</ul>
<li><a href="#dom-interfaces"><span class=secno>8. </span> DOM Interfaces
</a>
<ul class=toc>
<li><a href="#cssmatrix-interface"><span class=secno>8.1. </span>
CSSMatrix </a>
</ul>
<li><a href="#references"><span class=secno>9. </span>References</a>
<ul class=toc>
<li class=no-num><a href="#normative-references">Normative
references</a>
<li class=no-num><a href="#other-references">Other references</a>
</ul>
<li class=no-num><a href="#property-index">Property index</a>
<li class=no-num><a href="#index">Index</a>
</ul>
<!--end-toc-->
<h2 id=introduction><span class=secno>1. </span>Introduction</h2>
<p><em>This section is not normative.</em>
<p> The CSS <a href="http://www.w3.org/TR/REC-CSS2/visuren.html">visual
formatting model</a> describes a coordinate system within which each
element is positioned. Positions and sizes in this coordinate space can be
thought of as being expressed in pixels, starting in the upper left corner
of the parent with positive values proceeding to the right and down.
<p> This coordinate space can be modified with the <span
class=prop-name>‘<a href="#effects"><code
class=property>transform</code></a>’</span> property. Using
transform, elements can be translated, rotated and scaled in two
dimensional space. The coordinate space behaves as described in the <a
href="http://www.w3.org/TR/SVG/coords.html#EstablishingANewUserSpace">coordinate
system transformations</a> section of the SVG 1.1 specification. This is a
coordinate system with two axes: the X axis increases horizontally to the
right; the Y axis increases vertically downwards.
<p> Transforms apply to block-level and atomic inline-level elements, but
do not apply to elements which may be split into multiple inline-level
boxes.
<p> Specifying a value other than ‘<code
class=property>none</code>’ for the <span class=prop-name>‘<a
href="#effects"><code class=property>transform</code></a>’</span>
property establishes a new <em>local coordinate system</em> at the element
that it is applied to. Transformations are cumulative. That is, elements
establish their local coordinate system within the coordinate system of
their parent. In this way, a <span class=prop-name>‘<a
href="#effects"><code class=property>transform</code></a>’</span>
property effectively accumulates all the <span class=prop-name>‘<a
href="#effects"><code class=property>transform</code></a>’</span>
properties of its ancestors. The accumulation of these transforms defines
a <em>current transformation matrix (CTM)</em> for the element.
<p> The transform property does not affect the flow of the content
surrounding the transformed element. However, the value of the overflow
area takes into account transformed elements. This behavior is similar to
what happens when elements are translated via relative positioning.
Therefore, if the value of the <span class=prop-name>‘<code
class=property>overflow</code>’</span> property is <span
class=prop-value>‘<code class=property>scroll</code>’</span>
or <span class=prop-value>‘<code
class=property>auto</code>’</span>, scrollbars will appear as needed
to see content that is transformed outside the visible area.
<p> Any value other than ‘<code class=property>none</code>’ for
the transform results in the creation of both a stacking context and a
containing block. The object acts as a containing block for fixed
positioned descendants.
<div class=todo> Need to go into more detail here about why fixed
positioned objects should do this, i.e., that it's much harder to
implement otherwise.</div>
<div class=issue> There are two roles for transformations in layout: (1)
transformations that adjust the position of the affected content without
changing the normal layout of that content (much like relative
positioning) and (2) transformation of the content prior to layout that
affects the layout of that content. See <a
href="http://lists.w3.org/Archives/Public/www-style/2007Oct/0209">http://lists.w3.org/Archives/Public/www-style/2007Oct/0209</a>
for examples of both cases. The "transform" property (as defined in this
document) is equally useful for both roles. This document is focused on
satisfying the first role. There is, however, an architectural question
that arises because there needs to be a way to distinguish which role an
author of a stylesheet wants. The key question is which is the default
behavior/role for the "transform" property and how is the other
behavior/role indicated by a stylesheet author. If you have an opinion on
this topic, please send feedback.</div>
<div class=issue> What do fixed backgrounds do in transforms? They should
probably ignore the transform completely, since - even transformed - the
object should be acting as "porthole" through which the fixed background
can be viewed in its original form.</div>
<div class=issue> This property should also be applicable to SVG elements.</div>
<div class=issue> We also need to specify that SVG transforms *do* combine
with this transform, e.g., if a <foreignObject> is inside
transformed SVG and then defines a transform of its own. This means we may
potentially have to examine the current SVG transform and combine with it
to set the correct transform.</div>
<!-- ======================================================================================================= -->
<h2 id=transform-property><span class=secno>2. </span> The <span
class=prop-name>‘<a href="#effects"><code
class=property>transform</code></a>’</span> Property</h2>
<p> A two-dimensional transformation is applied to an element through the
<span class=prop-name>‘<a href="#effects"><code
class=property>transform</code></a>’</span> property. This property
contains a list of <a href="#transform-functions">transform functions</a>.
The final transformation value for an element is obtained by performing a
matrix concatenation of each entry in the list. The set of transform
functions is similar to those allowed by SVG.
<table class=propdef>
<tbody>
<tr>
<td> <em>Name:</em>
<td> <dfn id=effects>transform</dfn>
<tr>
<td> <em>Value:</em>
<td> none | <transform-function> [ <transform-function> ]*
<tr>
<td> <em>Initial:</em>
<td> none
<tr>
<td> <em>Applies to:</em>
<td> block-level and atomic inline-level elements
<tr>
<td> <em>Inherited:</em>
<td> no
<tr>
<td> <em>Percentages:</em>
<td> refer to the size of the element's border box
<tr>
<td> <em>Media:</em>
<td> visual
<tr>
<td> <em>Computed value:</em>
<td> Same as specified value.
</table>
<!-- ======================================================================================================= -->
<h2 id=transform-origin-property><span class=secno>3. </span> The <span
class=prop-name>‘<a href="#transform-origin"><code
class=property>transform-origin</code></a>’</span> Property</h2>
<p> The <span class=prop-name>‘<a href="#transform-origin"><code
class=property>transform-origin</code></a>’</span> property
establishes the origin of transformation for an element. This property is
applied by first translating the element by the negated value of the
property, then applying the element's transform, then translating by the
property value. This effectively moves the desired transformation origin
of the element to (0,0) in the local coordinate system, then applies the
element's transform, then moves the element back to its original position.
<p>If only one value is specified, the second value is assumed to be
‘<code class=property>center</code>’. If two values are given
and at least one value is not a keyword, then the first value represents
the horizontal position (or offset) and the second represents the vertical
position (or offset). <var><percentage></var> and
<var><length></var> values here represent an offset of the transform
origin from the top left corner of the element's border box.
<p>If three or four values are given, then each
<var><percentage></var> or<var><length></var> represents an
offset and must be preceded by a keyword, which specifies from which edge
the offset is given. For example, ‘<code class=css>transform-origin:
bottom 10px right 20px</code>’ represents a ‘<code
class=css>10px</code>’ vertical offset up from the bottom edge and a
‘<code class=css>20px</code>’ horizontal offset leftward from
the right edge. If three values are given, the missing offset is assumed
to be zero.
<p>Positive values represent an offset <em>inward</em> from the edge of the
border box. Negative values represent an offset <em>outward</em> from the
edge of the border box.
<table class=propdef>
<tbody>
<tr>
<td> <em>Name:</em>
<td> <dfn id=transform-origin>transform-origin</dfn>
<tr>
<td> <em>Value:</em>
<td> [ top | bottom ] |<br>
[ <percentage> | <length> | left | center | right ] [
<percentage> | <length> | top | center | bottom ]? |<br>
[ center | [ left | right ] [ <percentage> | <length> ]? ]
&& [ center | [ top | bottom ] [ <percentage> |
<length> ]? ]<br>
<tr>
<td> <em>Initial:</em>
<td> 50% 50%
<tr>
<td> <em>Applies to:</em>
<td> block-level and atomic inline-level elements
<tr>
<td> <em>Inherited:</em>
<td> no
<tr>
<td> <em>Percentages:</em>
<td> refer to the size of the element's border box
<tr>
<td> <em>Media:</em>
<td> visual
<tr>
<td> <em>Computed value:</em>
<td> For <length> the absolute value, otherwise a percentage
</table>
<!-- ======================================================================================================= -->
<h2 id=transform-functions><span class=secno>4. </span> The Transformation
Functions</h2>
<p> The value of the <a class=prop-name href="#effects">transform</a>
property is a list of <transform-functions> applied in the order
provided. The individual transform functions are separated by whitespace.
The set of allowed transform functions is given below. In this list the
type <translation-value> is defined as a <length> or
<percentage> value, and the <angle> type is defined by <a
href="http://www.w3.org/TR/css3-values/">CSS Values and Units.</a>
<dl>
<dt> <span class=prop-value>matrix(<number>, <number>,
<number>, <number>, <number>, <number>)</span>
<dd> specifies a 2D transformation in the form of a <a
href="http://www.w3.org/TR/SVG/coords.html#TransformMatrixDefined">transformation
matrix</a> of six values. <span
class=prop-value>matrix(a,b,c,d,e,f)</span> is equivalent to applying the
transformation matrix <strong>[a b c d e f]</strong>.
<dt> <span class=prop-value>translate(<translation-value>[,
<translation-value>])</span>
<dd> specifies a <a
href="http://www.w3.org/TR/SVG/coords.html#TranslationDefined">2D
translation</a> by the vector [tx, ty], where tx is the first
translation-value parameter and ty is the optional second
translation-value parameter. If <em><ty></em> is not provided, ty
has zero as a value.
<dt> <span class=prop-value>translateX(<translation-value>)</span>
<dd> specifies a <a
href="http://www.w3.org/TR/SVG/coords.html#TranslationDefined">translation</a>
by the given amount in the X direction.
<dt> <span class=prop-value>translateY(<translation-value>)</span>
<dd> specifies a <a
href="http://www.w3.org/TR/SVG/coords.html#TranslationDefined">translation</a>
by the given amount in the Y direction.
<dt> <span class=prop-value>scale(<number>[, <number>])</span>
<dd> specifies a <a
href="http://www.w3.org/TR/SVG/coords.html#ScalingDefined">2D scale</a>
operation by the [sx,sy] scaling vector described by the 2 parameters. If
the second parameter is not provided, it is takes a value equal to the
first.
<dt> <span class=prop-value>scaleX(<number>)</span>
<dd> specifies a scale operation using the [sx,1] scaling vector, where sx
is given as the parameter.
<dt> <span class=prop-value>scaleY(<number>)</span>
<dd> specifies a scale operation using the [1,sy] scaling vector, where sy
is given as the parameter.
<dt> <span class=prop-value>rotate(<angle>)</span>
<dd> specifies a <a
href="http://www.w3.org/TR/SVG/coords.html#RotationDefined">2D
rotation</a> by the angle specified in the parameter about the origin of
the element, as defined by the <a
href="#transform-origin"><em>transform-origin</em></a> property.
<dt> <span class=prop-value>skewX(<angle>)</span>
<dd> specifies a <a
href="http://www.w3.org/TR/SVG/coords.html#SkewXDefined">skew
transformation along the X axis</a> by the given angle.
<dt> <span class=prop-value>skewY(<angle>)</span>
<dd> specifies a <a
href="http://www.w3.org/TR/SVG/coords.html#SkewYDefined">skew
transformation along the Y axis</a> by the given angle.
</dl>
<h2 id=transform-values><span class=secno>5. </span> Transform Values and
Lists</h2>
<p> The <translation-value> values are defined as [<percentage>
| <length>]. All other value types are described <a
href="http://www.w3.org/TR/REC-CSS2/syndata.html#values">as CSS types</a>.
If a list of transforms is provided, then the net effect is as if each
transform had been specified separately in the order provided. For
example,
<pre>
<div style="transform:translate(-10px,-20px) scale(2) rotate(45deg) translate(5px,10px)"/>
</pre>
<p> is functionally equivalent to:
<pre>
<div style="transform:translate(-10px,-20px)">
<div style="transform:scale(2)">
<div style="transform:rotate(45deg)">
<div style="transform:translate(5px,10px)">
</div>
</div>
</div>
</div>
</pre>
<div class=example>
<pre>
div {
transform: translate(100px, 100px);
}
</pre>
Move the element by 100 pixels in both the X and Y directions.
<div class=figure> <img alt="The 100px translation in X and Y"
src=transform1.png></div>
</div>
<div class=example>
<pre>
div {
height: 100px; width: 100px;
transform: translate(80px, 80px) scale(1.5, 1.5) rotate(45deg);
}
</pre>
Move the element by 80 pixels in both the X and Y directions, then scale
the element by 150%, then rotate it 45 degrees clockwise about the Z axis.
Note that the scale and rotate operate about the center of the element,
since the element has the default transform-origin of 50% 50%.
<div class=figure> <img alt="The transform specified above"
src="compound_transform.png"></div>
</div>
<!-- ======================================================================================================= -->
<h2 id=animation><span class=secno>6. </span> Transitions and animations
between transform values</h2>
<p> When animating or transitioning the value of a transform property the
rules described below are applied. The ‘<code
class=property>from</code>’ transform is the transform at the start
of the transition or current keyframe. The ‘<code
class=property>end</code>’ transform is the transform at the end of
the transition or current keyframe.
<ul>
<li> If the ‘<code class=property>from</code>’ and
‘<code class=property>to</code>’ transforms are both single
functions of the same type:
<ul>
<li> For translate, translateX, translateY, scale, scaleX, scaleY,
rotate, skew, skewX and skewY functions:
<ul>
<li> the individual components of the function are interpolated
numerically.
</ul>
<li> For matrix:
<ul>
<li> the matrix is decomposed using <a
href="http://tog.acm.org/GraphicsGems/gemsii/unmatrix.c">the method
described by unmatrix</a> into separate translation, scale, rotation
and skew matrices, then each decomposed matrix is interpolated
numerically, and finally combined in order to produce a resulting 3x2
matrix.
</ul>
</ul>
<li> If both the ‘<code class=property>from</code>’ and
‘<code class=property>to</code>’ transforms are "none":
<ul>
<li> There is no interpolation necessary
</ul>
<li> If one of the ‘<code class=property>from</code>’ or
‘<code class=property>to</code>’ transforms is "none":
<ul>
<li> The ‘<code class=property>none</code>’ is replaced by
an equivalent identity function list for the corresponding transform
function list.
<p> For example, if the ‘<code class=property>from</code>’
transform is "scale(2)" and the ‘<code
class=property>to</code>’ transform is "none" then the value
"scale(1)" will be used as the ‘<code
class=property>to</code>’ value, and animation will proceed
using the rule above. Similarly, if the ‘<code
class=property>from</code>’ transform is "none" and the
‘<code class=property>to</code>’ transform is "scale(2)
rotate(50deg)" then the animation will execute as if the ‘<code
class=property>from</code>’ value is "scale(1) rotate(0)".</p>
<p> The identity functions are translate(0), translateX(0),
translateY(0), scale(1), scaleX(1), scaleY(1), rotate(0), skewX(0),
skewY(0) and matrix(1, 0, 0, 1, 0, 0).</p>
</ul>
<li> If both the ‘<code class=property>from</code>’ and
‘<code class=property>to</code>’ transforms have the same
number of transform functions and corresponding functions in each
transform list are of the same type:
<ul>
<li> Each transform function is animated with its corresponding
destination function in isolation using the rules described above. The
individual values are then applied as a list to produce resulting
transform value.
</ul>
<li> Otherwise:
<ul>
<li> The transform function lists are each converted into the equivalent
matrix value and animation proceeds using the rule for a single
function above.
</ul>
</ul>
<p> In some cases, an animation might cause a transformation matrix to be
singular or non-invertible. For example, an animation in which scale moves
from 1 to -1. At the time when the matrix is in such a state, the
transformed element is not rendered.
<h2 id=matrix-decomposition><span class=secno>7. </span> Matrix
decomposition for animation</h2>
<p> When interpolating between 2 matrices, each is decomposed into the
corresponding translation, rotation, scale, skew, and perspective values.
Not all matrices can be accurately described by these values. Those that
can't are decomposed into the most accurate representation possible, using
the technique below. This technique is taken from The "unmatrix" method in
"Graphics Gems II, edited by Jim Arvo". The pseudocode below works on a
4x4 homogeneous matrix. A 3x2 2D matrix is therefore first converted to
4x4 homogeneous form.
<h3 id=d-matrix-to-4x4-matrix-conversion><span class=secno>7.1. </span>2D
matrix to 4x4 matrix conversion</h3>
<div class=issue>
<p> The pseudocode here is written with column-major matrices, as would be
used in C/C++, but the SVG spec illustrates matrixes using row-major
notation. Change this pseudocode to avoid confusion?</p>
<p> It's also unfortunate that we have to include math for 4x4 matrices
into the 2D transforms spec. Find a way to avoid that?</p>
</div>
<pre>
Input: matrix2d ; a 3x2 transformation matrix
; rotation matrix in m11,m12,m21,m22
; x,y translation in m31,m32
Output: matrix ; a 4x4 homgeneous matrix
matrix[0][0] = matrix2d[0][0]
matrix[0][1] = matrix2d[0][1]
matrix[0][2] = 0
matrix[0][3] = 0
matrix[1][0] = matrix2d[1][0]
matrix[1][1] = matrix2d[1][1]
matrix[1][2] = 0
matrix[1][3] = 0
matrix[2][0] = 0
matrix[2][1] = 0
matrix[2][2] = 1
matrix[2][3] = 0
matrix[3][0] = matrix2d[2][0]
matrix[3][1] = matrix2d[2][1]
matrix[3][2] = 0
matrix[3][3] = 1
return matrix
</pre>
<h3 id=unmatrix><span class=secno>7.2. </span>Unmatrix</h3>
<pre>
Input: matrix ; a 4x4 matrix
Output: translate ; a 3 component vector
rotate ; Euler angles, represented as a 3 component vector
scale ; a 3 component vector
skew ; skew factors XY,XZ,YZ represented as a 3 component vector
perspective ; a 4 component vector
Returns false if the matrix cannot be decomposed, true if it can
Supporting functions (point is a 3 component vector, matrix is a 4x4 matrix):
float determinant(matrix) returns the 4x4 determinant of the matrix
matrix inverse(matrix) returns the inverse of the passed matrix
matrix transpose(matrix) returns the transpose of the passed matrix
point multVecMatrix(point, matrix) multiplies the passed point by the passed matrix
and returns the transformed point
float length(point) returns the length of the passed vector
point normalize(point) normalizes the length of the passed point to 1
float dot(point, point) returns the dot product of the passed points
float cos(float) returns the cosine of the passed angle in radians
float asin(float) returns the arcsine in radians of the passed value
float atan2(float y, float x) returns the principal value of the arc tangent of
y/x, using the signs of both arguments to determine
the quadrant of the return value
Decomposition also makes use of the following function:
point combine(point a, point b, float ascl, float bscl)
result[0] = (ascl * a[0]) + (bscl * b[0])
result[1] = (ascl * a[1]) + (bscl * b[1])
result[2] = (ascl * a[2]) + (bscl * b[2])
return result
// Normalize the matrix.
if (matrix[3][3] == 0)
return false
for (i = 0; i < 4; i++)
for (j = 0; j < 4; j++)
matrix[i][j] /= matrix[3][3]
// perspectiveMatrix is used to solve for perspective, but it also provides
// an easy way to test for singularity of the upper 3x3 component.
perspectiveMatrix = matrix
for (i = 0; i < 3; i++)
perspectiveMatrix[i][3] = 0
perspectiveMatrix[3][3] = 1
if (determinant(perspectiveMatrix) == 0)
return false
// First, isolate perspective.
if (matrix[0][3] != 0 || matrix[1][3] != 0 || matrix[2][3] != 0)
// rightHandSide is the right hand side of the equation.
rightHandSide[0] = matrix[0][3];
rightHandSide[1] = matrix[1][3];
rightHandSide[2] = matrix[2][3];
rightHandSide[3] = matrix[3][3];
// Solve the equation by inverting perspectiveMatrix and multiplying
// rightHandSide by the inverse.
inversePerspectiveMatrix = inverse(perspectiveMatrix)
transposedInversePerspectiveMatrix = transposeMatrix4(inversePerspectiveMatrix)
perspective = multVecMatrix(rightHandSide, transposedInversePerspectiveMatrix)
// Clear the perspective partition
matrix[0][3] = matrix[1][3] = matrix[2][3] = 0
matrix[3][3] = 1
else
// No perspective.
perspective[0] = perspective[1] = perspective[2] = 0
perspective[3] = 1
// Next take care of translation
translate[0] = matrix[3][0]
matrix[3][0] = 0
translate[1] = matrix[3][1]
matrix[3][1] = 0
translate[2] = matrix[3][2]
matrix[3][2] = 0
// Now get scale and shear. 'row' is a 3 element array of 3 component vectors
for (i = 0; i < 3; i++)
row[i][0] = matrix[i][0]
row[i][1] = matrix[i][1]
row[i][2] = matrix[i][2]
// Compute X scale factor and normalize first row.
scale[0] = length(row[0])
row[0] = normalize(row[0])
// Compute XY shear factor and make 2nd row orthogonal to 1st.
skew[0] = dot(row[0], row[1])
row[1] = combine(row[1], row[0], 1.0, -skew[0])
// Now, compute Y scale and normalize 2nd row.
scale[1] = length(row[1])
row[1] = normalize(row[1])
skew[0] /= scale[1];
// Compute XZ and YZ shears, make 3rd row orthogonal
skew[1] = dot(row[0], row[2])
row[2] = combine(row[2], row[0], 1.0, -skew[1])
skew[2] = dot(row[1], row[2])
row[2] = combine(row[2], row[1], 1.0, -skew[2])
// Next, get Z scale and normalize 3rd row.
scale[2] = length(row[2])
row[2] = normalize(row[2])
skew[1] /= scale[2]
skew[2] /= scale[2]
// At this point, the matrix (in rows) is orthonormal.
// Check for a coordinate system flip. If the determinant
// is -1, then negate the matrix and the scaling factors.
pdum3 = cross(row[1], row[2])
if (dot(row[0], pdum3) < 0)
for (i = 0; i < 3; i++) {
scale[0] *= -1;
row[i][0] *= -1
row[i][1] *= -1
row[i][2] *= -1
// Now, get the rotations out
rotate[1] = asin(-row[0][2]);
if (cos(rotate[1]) != 0)
rotate[0] = atan2(row[1][2], row[2][2]);
rotate[2] = atan2(row[0][1], row[0][0]);
else
rotate[0] = atan2(-row[2][0], row[1][1]);
rotate[2] = 0;
return true;</pre>
<h3 id=animating-the-components><span class=secno>7.3. </span>Animating the
components</h3>
<p> Once decomposed, each component of each returned value of the source
matrix is linearly interpolated with the corresponding component of the
destination matrix. For instance, the translate[0], translate[1] and
translate[2] values are interpolated numerically, and the result is used
to set the translation of the animating element.
<h3 id=recomposing-the-matrix><span class=secno>7.4. </span>Recomposing the
matrix</h3>
<p><em>This section is not normative.</em>
<p> After interpolation the resulting values are used to position the
element. One way to use these values is to recompose them into a 4x4
matrix. This can be done using the Transformation Functions of the <a
href="#effects"><em>transform</em></a> property. The following JavaScript
example produces a string for this purpose. The values passed in are the
output of the Unmatrix function above:
<pre>
function compose(translate, rotate, scale, skew, perspective, elementID)
{
var s = "matrix3d(1,0,0,0, 0,1,0,0, 0,0,1,0, " +
perspective[0] + ", " + perspective[1] + ", " +
perspective[2] + ", " + perspective[3] + ")";
s += " translate3d(" + translate[0] + ", " + translate[1] + ", " + translate[2] + ")";
s += " rotateX(" + rotate[0] + ")";
s += " rotateY(" + rotate[0] + ")";
s += " rotateZ(" + rotate[0] + ")";
s += " matrix3d(1,0,0,0, 0,1,0,0, 0," + skew[2] + ",1,0, 0,0,0,1)";
s += " matrix3d(1,0,0,0, 0,1,0,0, " + skew[1] + ",0,1,0, 0,0,0,1)";
s += " matrix3d(1,0,0,0, " + skew[0] + ",1,0,0, 0,0,1,0, 0,0,0,1)";
s += " scale3d(" + scale[0] + ", " + scale[1] + ", " + scale[2] + ")";
document.getElementById(elementID).style.transform = s;
}</pre>
<h2 id=dom-interfaces><span class=secno>8. </span> DOM Interfaces</h2>
<p> This section describes the interfaces and functionality added to the
DOM to support runtime access to the functionality described above.
<h3 id=cssmatrix-interface><span class=secno>8.1. </span> CSSMatrix</h3>
<dl>
<dt> <b>Interface <i><a id=DOM-CSSMatrix
name=DOM-CSSMatrix>CSSMatrix</a></i></b>
<dd>
<p> The <code>CSSMatrix</code> interface represents a 4x4 homogeneous
matrix.</p>
<div class=issue> We actually describe a 3x2 matrix here, similar to <a
href="http://www.w3.org/TR/SVG/coords.html#InterfaceSVGMatrix">SVGMatrix</a>.
</div>
<dl>
<dt> <b>IDL Definition</b>
<dd>
<div class=idl-code>
<pre>
interface CSSMatrix {
attribute float a;
attribute float b;
attribute float c;
attribute float d;
attribute float e;
attribute float f;
void setMatrixValue(in DOMString string) raises(DOMException);
CSSMatrix multiply(in CSSMatrix secondMatrix);
CSSMatrix inverse() raises(DOMException);
CSSMatrix translate(in float x, in float y);
CSSMatrix scale(in float scaleX, in float scaleY);
CSSMatrix skewX(in float angle);
CSSMatrix skewY(in float angle);
CSSMatrix rotate(in float angle);
};</pre>
</div>
<br>
</dd>
<!-- IDL -->
<dt> <b>Attributes</b>
<dd>
<dl>
<dt> <code class=attribute-name><a id=DOM-CSSMatrix-matrix
name=DOM-CSSMatrix-matrix>a-f</a></code> of type <code>float</code>
<dd> Each of these attributes represents one of the values in the 3x2
matrix.<br>
</dl>
</dd>
<!-- Attributes -->
<dt> <b>Methods</b>
<dd>
<dl><!-- ===================================================== -->
<dt> <code class=method-name><a id=DOM-CSSMatrix-setMatrixValue
name=DOM-CSSMatrix-setMatrixValue>setMatrixValue</a></code>
<dd>
<div class=method> The <code>setMatrixValue</code> method replaces
the existing matrix with one computed from parsing the passed string
as though it had been assigned to the transform property in a CSS
style rule.
<div class=parameters> <b>Parameters</b>
<div class=paramtable>
<dl>
<dt> <code class=parameter-name>string</code> of type
<code>DOMString</code>
<dd> The string to parse.<br>
</dl>
</div>
</div>
<!-- parameters -->
<div class=return-value> <b>No Return Value</b></div>
<div> <b>Exceptions</b>
<div class=returnvalue>
<dl>
<dt> <code>DOMException SYNTAX_ERR</code>
<dd> Thrown when the provided string can not be parsed into a
CSSMatrix.
</dl>
</div>
</div>
</div>
</dd>
<!-- setMatrixValue -->
<!-- ===================================================== -->
<dt> <code class=method-name><a id=DOM-CSSMatrix-multiply
name=DOM-CSSMatrix-multiply>multiply</a></code>
<dd>
<div class=method> The <code>multiply</code> method returns a new
CSSMatrix which is the result of this matrix multiplied by the
passed matrix, with the passed matrix to the right. This matrix is
not modified.
<div class=parameters> <b>Parameters</b>
<div class=paramtable>
<dl>
<dt> <code class=parameter-name>secondMatrix</code> of type
<code>CSSMatrix</code>
<dd> The matrix to multiply.<br>
</dl>
</div>
</div>
<!-- parameters -->
<div class=return-value> <b>Return Value</b>
<div class=returnvalue>
<dl>
<dt> <code>CSSMatrix</code>
<dd> The result matrix.<br>
</dl>
</div>
</div>
<div> <b>No Exceptions</b></div>
</div>
</dd>
<!-- multiply() -->
<!-- ===================================================== -->