3x3 matrix for two-dimensional transformations.
CSS transforms allows elements styled with CSS to be transformed in two-dimensional or three-dimensional space. This specification is the convergence of the CSS 2D transforms, CSS 3D transforms and SVG transforms specifications.
This specification replaces the former CSS 2D Transforms and CSS 3D Transforms specifications, as well as SVG Transforms.
The list of changes made to this specification is available.
This section is not normative.
The CSS visual formatting model describes a coordinate system within each element is positioned. Positions and sizes in this coordinate space can be thought of as being expressed in pixels, starting in the origin of point with positive values proceeding to the right and down.
This coordinate space can be modified with the 'transform' property. Using transform, elements can be translated, rotated and scaled in two or three dimensional space.
Additional properties make working with transforms easier, and allow the author to control how nested three-dimensional transforms interact.
Note that while some values of the 'transform' property allow an element to be transformed in a three-dimensional coordinate system, the elements themselves are not three-dimensional objects. Instead, they exist on a two-dimensional plane (a flat surface) and have no depth.
This module defines a set of CSS properties that affect the visual rendering of elements to which those properties are applied; these effects are applied after elements have been sized and positioned according to the Visual formatting model from [[!CSS21]]. Some values of these properties result in the creation of a containing block, and/or the creation of a stacking context.
Three-dimensional transforms can also affect the visual layering of elements, and thus override the back-to-front painting order described in Appendix E of [[!CSS21]].
Transforms affect the results of the Element Interface extensions getClientRects() and getBoundingClientRect(), which are specified in [[CSSOM-VIEW]].
Transforms affect the rendering of backgounds on elements with a value of ''fixed'' for the
'background-attachment' property,
which is specified in [[!CSS3BG]].
This specification follows the CSS property definition conventions from [[!CSS21]]. Value types not defined in this specification are defined in CSS Level 2 Revision 1 [[!CSS21]].
In addition to the property-specific values listed in their definitions, all properties defined in this specification also accept the inherit keyword as their property value. For readability it has not been repeated explicitly.
When used in this specification, terms have the meanings assigned in this section.
A bounding box is the object bounding box for all SVG elements without an associated CSS layout box and the border box for all other elements. The bounding box of a table is the border box of its table wrapper box, not its table box.
A transformable element is an element in one of these categories:
In general, a coordinate system defines locations and distances on the current canvas. The current local coordinate system (also user coordinate system) is the coordinate system that is currently active and which is used to define how coordinates and lengths are located and computed, respectively, on the current canvas.
A matrix computed from the values of the 'perspective' and 'perspective-origin' properties as described below.
A matrix that defines the mathematical mapping from one coordinate system into another. It is computed from the values of the 'transform' and 'transform-origin' properties as described below.
A matrix that defines the mapping from the local coordinate system into the viewport coordinate system.
A matrix computed for elements in a 3D rendering context, as described below.
A transform function that is equivalent to a identity 4x4 matrix (see Mathematical Description of Transform Functions). Examples for identity transform functions are ''translate(0)'', ''translate3d(0, 0, 0)'', ''translateX(0)'', ''translateY(0)'', ''translateZ(0)'', ''scale(1)'', ''scaleX(1)'', ''scaleY(1)'', ''scaleZ(1)'', ''rotate(0)'', ''rotate3d(1, 1, 1, 0)'', ''rotateX(0)'', ''rotateY(0)'', ''rotateZ(0)'', ''skew(0, 0)'', ''skewX(0)'', ''skewY(0)'', ''matrix(1, 0, 0, 1, 0, 0)'' and ''matrix3d(1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1)''. A special case is perspective: ''perspective(infinity)''. The value of m34 becomes infinitesimal small and the transform function is therefore assumed to be equal to the identity matrix.
A containing block hierarchy of one or more levels, instantiated by elements with a computed value for the 'transform-style' property of ''preserve-3d'', whose elements share a common three-dimensional coordinate system.
UAs may not always be able to render three-dimensional transforms and then just support a two-dimensional subset of this specification. In this case three-dimensional transforms and the properties 'transform-style', 'perspective', 'perspective-origin' and 'backface-visibility' must not be supported. Section 3D Transform Rendering does not apply. Matrix decomposing uses the technique taken from the "unmatrix" method in "Graphics Gems II, edited by Jim Arvo", simplified for the 2D case. Section Mathematical Description of Transform Functions is still effective but can be reduced by using a 3x3 transformation matrix where a equals m11, b equals m12, c equals m21, d equals m22, e equals m41 and f equals m42 (see A 2D 3x2 matrix with six parameter).
3x3 matrix for two-dimensional transformations.
Authors can easily provide a fallback if UAs do not provide support for three-dimensional transforms. The following example has two property definitions for 'transform'. The first one consists of two two-dimensional transform functions. The second one has a two-dimensional and a three-dimensional transform function.
div {
transform: scale(2) rotate(45deg);
transform: scale(2) rotate3d(0, 0, 1, 45deg);
}
With 3D support, the second definition will override the first one. Without 3D support, the second definition is invalid and a UA falls back to the first definition.
This section is normative.
Specifying a value other than ''none'' for the ''transform'' property establishes a new local coordinate system at the element that it is applied to. The mapping from where the element would have rendered into that local coordinate system is given by the element's transformation matrix. Transformations are cumulative. That is, elements establish their local coordinate system within the coordinate system of their parent. From the perspective of the user, an element effectively accumulates all the 'transform' properties of its ancestors as well as any local transform applied to it. The accumulation of these transforms defines a current transformation matrix (CTM) for the element.
The coordinate space is a coordinate system with two axes: the X axis increases horizontally to the right; the Y axis increases vertically downwards. Three-dimensional transform functions extend this coordinate space into three dimensions, adding a Z axis perpendicular to the plane of the screen, that increases towards the viewer.
Demonstration of the initial coordinate space.
The transformation matrix is computed from the 'transform' and 'transform-origin' properties as follows:
Transforms apply to transformable elements.
div {
transform: translate(100px, 100px);
}
This transform moves the element by 100 pixels in both the X and Y directions.
div {
height: 100px; width: 100px;
transform-origin: 50px 50px;
transform: rotate(45deg);
}
The 'transform-origin' property moves the point of origin by 50 pixels in both the X and Y directions. The transform rotates the element clockwise by 45° about the point of origin. After all transform functions were applied, the translation of the origin gets translated back by -50 pixels in both the X and Y directions.
div {
height: 100px; width: 100px;
transform: translate(80px, 80px) scale(1.5, 1.5) rotate(45deg);
}
This transform moves the element by 80 pixels in both the X and Y directions, then scales the element by 150%, then rotates it 45° clockwise about the Z axis. Note that the scale and rotation operate about the center of the element, since the element has the default transform-origin of ''50% 50%''.
Note that an identical rendering can be obtained by nesting elements with the equivalent transforms:
<div style="transform: translate(80px, 80px)">
<div style="transform: scale(1.5, 1.5)">
<div style="transform: rotate(45deg)"></div>
</div>
</div>
For elements whose layout is governed by the CSS box model, the transform property does not affect the flow of the content surrounding the transformed element. However, the extent of the overflow area takes into account transformed elements. This behavior is similar to what happens when elements are offset via relative positioning. Therefore, if the value of the 'overflow' property is ''scroll'' or ''auto'', scrollbars will appear as needed to see content that is transformed outside the visible area.
For elements whose layout is governed by the CSS box model, any value other than ''none'' 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.
Is this effect on position:fixed necessary? If so, 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. See Bug 16328.
Fixed backgrounds on the root element are affected by any transform specified for that element. For all other elements that are effected by a transform (i.e. have a transform applied to them, or to any of their ancestor elements), a value of ''fixed'' for the 'background-attachment' property is treated as if it had a value of ''scroll''. The computed value of 'background-attachment' is not affected.
If the root element is transformed, the transformation applies to the entire canvas, including any background specified for the root element. Since the background painting area for the root element is the entire canvas, which is infinite, the transformation might cause parts of the background that were originally off-screen to appear. For example, if the root element's background were repeating dots, and a transformation of ''scale(0.5)'' were specified on the root element, the dots would shrink to half their size, but there will be twice as many, so they still cover the whole viewport.
Normally, elements render as flat planes, and are rendered into the same plane as their containing block. Often this is the plane shared by the rest of the page. Two-dimensional transform functions can alter the appearance of an element, but that element is still rendered into the same plane as its containing block.
Three-dimensional transforms can result in transformation matrices with a non-zero Z component (where the Z axis projects out of the plane of the screen). This can result in an element rendering on a different plane than that of its containing block. This may affect the front-to-back rendering order of that element relative to other elements, as well as causing it to intersect with other elements. This behavior depends on whether the element is a member of a 3D rendering context, as described below.
This description does not exactly match what WebKit implements. Perhaps it should be changed to match current implementations? See Bug 19637.
This example shows the effect of three-dimensional transform applied to an element.
<style>
div {
height: 150px;
width: 150px;
}
.container {
border: 1px solid black;
}
.transformed {
transform: rotateY(50deg);
}
</style>
<div class="container">
<div class="transformed"></div>
</div>
The transform is a 50° rotation about the vertical, Y axis. Note how this makes the blue box appear narrower, but not three-dimensional.
The 'perspective' and 'perspective-origin' properties can be used to add a feeling of depth to a scene by making elements higher on the Z axis (closer to the viewer) appear larger, and those further away to appear smaller. The scaling is proportional to d/(d − Z) where d, the value of 'perspective', is the distance from the drawing plane to the the assumed position of the viewer's eye.
Diagrams showing how scaling depends on the 'perspective' property and Z position. In the top diagram, Z is half of d. In order to make it appear that the original circle (solid outline) appears at Z (dashed circle), the circle is scaled up by a factor of two, resulting in the light blue circle. In the bottom diagram, the circle is scaled down by a factor of one-third to make it appear behind the original position.
Normally the assumed position of the viewer's eye is centered on a drawing. This position can be moved if desired – for example, if a web page contains multiple drawings that should share a common perspective – by setting 'perspective-origin'.
Diagram showing the effect of moving the perspective origin upward.
The perspective matrix is computed as follows:
perspective(<length>)' transform function, where the length is provided by the value of the 'perspective' propertyThis example shows how perspective can be used to cause three-dimensional transforms to appear more realistic.
<style>
div {
height: 150px;
width: 150px;
}
.container {
perspective: 500px;
border: 1px solid black;
}
.transformed {
transform: rotateY(50deg);
}
</style>
<div class="container">
<div class="transformed"></div>
</div>
The inner element has the same transform as in the previous example, but its rendering is now influenced by the perspective property on its parent element. Perspective causes vertices that have positive Z coordinates (closer to the viewer) to be scaled up in X and Y, and those further away (negative Z coordinates) to be scaled down, giving an appearance of depth.
An element with a three-dimensional transform that is not contained in a 3D rendering context renders with the appropriate transform applied, but does not intersect with any other elements. The three-dimensional transform in this case can be considered just as a painting effect, like two-dimensional transforms. Similarly, the transform does not affect painting order. For example, a transform with a positive Z translation may make an element look larger, but does not cause that element to render in front of elements with no translation in Z.
An element with a three-dimensional transform that is contained in a 3D rendering context can visibly interact with other elements in that same 3D rendering context; the set of elements participating in the same 3D rendering context may obscure each other or intersect, based on their computed transforms. They are rendered as if they are all siblings, positioned in a common 3D coordinate space. The position of each element in that three-dimensional space is determined by accumulating the transformation matrices up from the element that establishes the 3D rendering context through each element that is a containing block for the given element, as described below.
<style>
div {
height: 150px;
width: 150px;
}
.container {
perspective: 500px;
border: 1px solid black;
}
.transformed {
transform: rotateY(50deg);
background-color: blue;
}
.child {
transform-origin: top left;
transform: rotateX(40deg);
background-color: lime;
}
</style>
<div class="container">
<div class="transformed">
<div class="child"></div>
</div>
</div>
This example shows how nested 3D transforms are rendered in the absence of ''transform-style: preserve-3d''. The blue div is transformed as in the previous example, with its rendering influenced by the perspective on its parent element. The lime element also has a 3D transform, which is a rotation about the X axis (anchored at the top, by virtue of the transform-origin). However, the lime element is being rendered into the plane of its parent because it is not a member of a 3D rendering context; the parent is "flattening".
Elements establish and participate in 3D rendering contexts as follows:
The final value of the transform used to render an element in a 3D rendering context is computed by accumulating an accumulated 3D transformation matrix as follows:
<style>
div {
height: 150px;
width: 150px;
}
.container {
perspective: 500px;
border: 1px solid black;
}
.transformed {
transform-style: preserve-3d;
transform: rotateY(50deg);
background-color: blue;
}
.child {
transform-origin: top left;
transform: rotateX(40deg);
background-color: lime;
}
</style>
This example is identical to the previous example, with the addition of ''transform-style: preserve-3d'' on the blue element. The blue element now establishes a 3D rendering context, of which the lime element is a member. Now both blue and lime elements share a common three-dimensional space, so the lime element renders as tilting out from its parent, influenced by the perspective on the container.
Elements in the same 3D rendering context may intersect with each other. User agents must render intersection by subdividing the planes of intersecting elements as described by Newell's algorithm.
Untransformed elements in a 3D rendering context render on the Z=0 plane, yet may still intersect with transformed elements.
Within a 3D rendering context, the rendering order of non-intersecting elements is based on their position on the Z axis after the application of the accumulated transform. Elements at the same Z position render in stacking context order.
<style>
.container {
background-color: rgba(0, 0, 0, 0.3);
transform-style: preserve-3d;
perspective: 500px;
}
.container > div {
position: absolute;
left: 0;
}
.container > :first-child {
transform: rotateY(45deg);
background-color: orange;
top: 10px;
height: 135px;
}
.container > :last-child {
transform: translateZ(40px);
background-color: rgba(0, 0, 255, 0.75);
top: 50px;
height: 100px;
}
</style>
<div class="container">
<div></div>
<div></div>
</div>
This example shows show elements in a 3D rendering context can intersect. The container element establishes a 3D rendering context for itself and its two children. The children intersect with eachother, and the orange element also intersects with the container.
Using three-dimensional transforms, it's possible to transform an element such that its reverse side is towards the viewer. 3D-transformed elements show the same content on both sides, so the reverse side looks like a mirror-image of the front side (as if the element were projected onto a sheet of glass). Normally, elements whose reverse side is towards the viewer remain visible. However, the 'backface-visibility' property allows the author to make an element invisible when its reverse side is towards the viewer. This behavior is "live"; if an element with ''backface-visibility: hidden'' were animating, such that its front and reverse sides were alternately visible, then it would only be visible when the front side were towards the viewer.
This wording needs clarification; backface-visibility works per-3D rendering context, not relative to the root.
This is a first pass at an attempt to precisely specify how exactly to transform elements using the provided matrices. It might not be ideal, and implementer feedback is encouraged. See bug 15605.
The accumulated 3D transformation matrix is a 4×4 matrix, while the objects to be transformed are two-dimensional boxes. To transform each corner (a, b) of a box, the matrix must first be applied to (a, b, 0, 1), which will result in a four-dimensional point (x, y, z, w). This is transformed back to a three-dimensional point (x′, y′, z′) as follows:
If w > 0, (x′, y′, z′) = (x/w, y/w, z/w).
If w = 0, (x′, y′, z′) = (x ⋅ n, y ⋅ n, z ⋅ n). n is an implementation-dependent value that should be chosen so that x′ or y′ is much larger than the viewport size, if possible. For example, (5px, 22px, 0px, 0) might become (5000px, 22000px, 0px), with n = 1000, but this value of n would be too small for (0.1px, 0.05px, 0px, 0). This specification does not define the value of n exactly. Conceptually, (x′, y′, z′) is infinitely far in the direction (x, y, z).
If w < 0 for all four corners of the transformed box, the box is not rendered.
If w < 0 for one to three corners of the transformed box, the box must be replaced by a polygon that has any parts with w < 0 cut out. This will in general be a polygon with three to five vertices, of which exactly two will have w = 0 and the rest w > 0. These vertices are then transformed to three-dimensional points using the rules just stated. Conceptually, a point with w < 0 is "behind" the viewer, so should not be visible.
<style>
.transformed {
height: 100px;
width: 100px;
background: lime;
transform: perspective(50px) translateZ(100px);
}
</style>
All of the box's corners have z-coordinates greater than the perspective. This means that the box is behind the viewer and will not display. Mathematically, the point (x, y) first becomes (x, y, 0, 1), then is translated to (x, y, 100, 1), and then applying the perspective results in (x, y, 100, −1). The w-coordinate is negative, so it does not display. An implementation that doesn't handle the w < 0 case separately might incorrectly display this point as (−x, −y, −100), dividing by −1 and mirroring the box.
<style>
.transformed {
height: 100px;
width: 100px;
background: radial-gradient(yellow, blue);
transform: perspective(50px) translateZ(50px);
}
</style>
Here, the box is translated upward so that it sits at the same place the viewer is looking from. This is like bringing the box closer and closer to one's eye until it fills the entire field of vision. Since the default transform-origin is at the center of the box, which is yellow, the screen will be filled with yellow.
Mathematically, the point (x, y) first becomes (x, y, 0, 1), then is translated to (x, y, 50, 1), then becomes (x, y, 50, 0) after applying perspective. Relative to the transform-origin at the center, the upper-left corner was (−50, −50), so it becomes (−50, −50, 50, 0). This is transformed to something very far to the upper left, such as (−5000, −5000, 5000). Likewise the other corners are sent very far away. The radial gradient is stretched over the whole box, now enormous, so the part that's visible without scrolling should be the color of the middle pixel: yellow. However, since the box is not actually infinite, the user can still scroll to the edges to see the blue parts.
WebKit doesn't render this box unless the translateZ() is < 50px.
<style>
.transformed {
height: 50px;
width: 50px;
background: lime;
border: 25px solid blue;
transform-origin: left;
transform: perspective(50px) rotateY(-45deg);
}
</style>
The box will be rotated toward the viewer, with the left edge staying fixed while the right edge swings closer. The right edge will be at about z = 70.7px, which is closer than the perspective of 50px. Therefore, the rightmost edge will vanish ("behind" the viewer), and the visible part will stretch out infinitely far to the right.
Mathematically, the top right vertex of the box was originally (100, −50), relative to the transform-origin. It is first expanded to (100, −50, 0, 1). After applying the transform specified, this will get mapped to about (70.71, −50, 70.71, −0.4142). This has w = −0.4142 < 0, so we need to slice away the part of the box with w < 0. This results in the new top-right vertex being (50, −50, 50, 0). This is then mapped to some faraway point in the same direction, such as (5000, −5000, 5000), which is up and to the right from the transform-origin. Something similar is done to the lower right corner, which gets mapped far down and to the right. The resulting box stretches far past the edge of the screen.
Again, the rendered box is still finite, so the user can scroll to see the whole thing if he or she chooses. However, the right part has been chopped off. No matter how far the user scrolls, the rightmost 30px or so of the original box will not be visible. The blue border was only 25px wide, so it will be visible on the left, top, and bottom, but not the right.
The same basic procedure would apply if one or three vertices had w < 0. However, in that case the result of truncating the w < 0 part would be a triangle or pentagon instead of a quadrilateral.
A transformation is applied to the coordinate system an element renders in through the 'transform' property. This property contains a list of transform functions. The final transformation value for a coordinate system is obtained by converting each function in the list to its corresponding matrix like defined in Mathematical Description of Transform Functions, then multiplying the matrices.
| Name: | transform |
| Value: | none | <transform-function> [ <transform-function> ]* |
| Initial: | none |
| Applies to: | transformable elements |
| Inherited: | no |
| Percentages: | refer to the size of bounding box |
| Media: | visual |
| Computed value: | As specified, but with relative lengths converted into absolute lengths. |
| Animatable: | as transform |
Any value other than ''none'' 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.
| Name: | transform-origin |
| Value: |
[ left | center | right | top | bottom | <percentage> | <length> ] | [ left | center | right | <percentage> | <length> ] [ top | center | bottom | <percentage> | <length> ] <length>? | [ center | [ left | right ] ] && [ center | [ top | bottom ] ] <length>? |
| Initial: | 50% 50% |
| Applies to: | transformable elements |
| Inherited: | no |
| Percentages: | refer to the size of bounding box |
| Media: | visual |
| Computed value: | For <length> the absolute value, otherwise a percentage |
| Animatable: | as simple list of length, percentage, or calc |
The default value for SVG elements without associated CSS layout box is ''0 0''.
The values of the 'transform' and 'transform-origin' properties are used to compute the transformation matrix, as described above.
If only one value is specified, the second value is assumed to be ''center''. If one or two values are specified, the third value is assumed to be ''0px''.
If two or more values are defined and either no value is a keyword, or the only used keyword is ''center'', then the first value represents the horizontal position (or offset) and the second represents the vertical position (or offset). A third value always represents the Z position (or offset) and must be of type <length>.
A percentage for the horizontal offset is relative to the width of the bounding box. A percentage for the vertical offset is relative to height of the bounding box. The value for the horizontal and vertical offset represent an offset from the top left corner of the bounding box.
A length value gives a fixed length as the offset. The value for the horizontal and vertical offset represent an offset from the top left corner of the bounding box.
For SVG elements without an associated CSS layout box the horizontal and vertical offset represent an offset from the point of origin of the element's local coordinate space.
The resolved value of 'transform-origin' is the used value (i.e., percentages are resolved to absolute lengths).
| Name: | transform-style |
| Value: | flat | preserve-3d |
| Initial: | flat |
| Applies to: | transformable elements |
| Inherited: | no |
| Percentages: | N/A |
| Media: | visual |
| Computed value: | Same as specified value. |
| Animatable: | no |
A value of ''preserve-3d'' for 'transform-style' establishes a stacking context.
The following CSS property values require the user agent to create a flattened representation of the descendant elements before they can be applied, and therefore override the behavior of ''transform-style: preserve-3d'':
The computed value of 'transform-style' is not affected.
The values of the 'transform' and 'transform-origin' properties are used to compute the transformation matrix, as described above.
| Name: | perspective |
| Value: | none | <length> |
| Initial: | none |
| Applies to: | transformable elements |
| Inherited: | no |
| Percentages: | N/A |
| Media: | visual |
| Computed value: | Absolute length or "none". |
| Animatable: | as length |
If the value is ''none'', no perspective transform is applied. Lengths must be positive.
The use of this property with any value other than ''none'' establishes a stacking context. It also establishes a containing block (somewhat similar to ''position: relative''), just like the 'transform' property does.
The values of the 'perspective' and 'perspective-origin' properties are used to compute the perspective matrix, as described above.
The 'perspective-origin' property establishes the origin for the perspective property. It effectively sets the X and Y position at which the viewer appears to be looking at the children of the element.
| Name: | perspective-origin |
| Value: |
[ left | center | right | top | bottom | <percentage> | <length> ] | [ left | center | right | <percentage> | <length> ] [ top | center | bottom | <percentage> | <length> ] | [ center | [ left | right ] ] && [ center | [ top | bottom ] ] |
| Initial: | 50% 50% |
| Applies to: | transformable elements |
| Inherited: | no |
| Percentages: | refer to the size of the bounding box |
| Media: | visual |
| Computed value: | For <length> the absolute value, otherwise a percentage. |
| Animatable: | as simple list of length, percentage, or calc |
The values of the 'perspective' and 'perspective-origin' properties are used to compute the perspective matrix, as described above.
If only one value is specified, the second value is assumed to be ''center''.
If at least one of the two values is not a keyword, then the first value represents the horizontal position (or offset) and the second represents the vertical position (or offset).
The values for 'perspective-origin' represent an offset of the perspective origin from the top left corner of the bounding box.
A percentage for the horizontal perspctive offset is relative to the width of the bounding box. A percentage for the vertical offset is relative to height of the bounding box. The value for the horizontal and vertical offset represent an offset from the top left corner of the bounding box.
A length value gives a fixed length as the offset. The value for the horizontal and vertical offset represent an offset from the top left corner of the bounding box.
The resolved value of 'perspective-origin' is the used value (i.e., percentages are resolved to absolute lengths).
The 'backface-visibility' property determines whether or not the "back" side of a transformed element is visible when facing the viewer. With an identity transform, the front side of an element faces the viewer. Applying a rotation about Y of 180 degrees (for instance) would cause the back side of the element to face the viewer.
Note that this property is useful when you place two elements back-to-back, as you would to create a playing card. Without this property, the front and back elements could switch places at times during an animation to flip the card. Another example is creating a box out of 6 elements, but where you want to see the inside faces of the box. This is useful when creating the backdrop for a 3 dimensional stage.
| Name: | backface-visibility |
| Value: | visible | hidden |
| Initial: | visible |
| Applies to: | transformable elements |
| Inherited: | no |
| Percentages: | N/A |
| Media: | visual |
| Computed value: | Same as specified value. |
| Animatable: | no |
The visibility of an element with ''backface-visibility: hidden'' is determined as follows:
The reasoning for this definition is as follows. Assume elements are rectangles in the x–y plane with infinitesimal thickness. The front of the untransformed element has coordinates like (x, y, ε), and the back is (x, y, −ε), for some very small ε. We want to know if after the transformation, the front of the element is closer to the viewer than the back (higher z-value) or further away. The z-coordinate of the front will be M13x + M23y + M33ε + M43, before accounting for perspective, and the back will be M13x + M23y − M33ε + M43. The first quantity is greater than the second if and only if M33 > 0. (If it equals zero, the front and back are equally close to the viewer. This probably means something like a 90-degree rotation, which makes the element invisible anyway, so we don't really care whether it vanishes.)
The SVG 1.1 specification did not specify the attributes 'transform', 'gradientTransform' or 'patternTransform' as presentation attributes. In order to improve the integration of SVG and HTML, this specification makes these SVG attributes presentation attributes and makes the 'transform' property one that applies to transformable elements in the SVG namespace.
This specification will also introduce the new presentation attributes 'transform-origin', 'perspective', 'perspective-origin', 'transform-style' and 'backface-visibility'.
Values on new introduced presentation attributes get parsed following the syntax rules on SVG Data Types [[SVG11]].
Since the previously named SVG attributes become presentation attributes, their participation in the CSS cascade is determined by the specificity of presentation attributes, as explained in the SVG specification.
This example shows the combination of the 'transform' style property and the 'transform' presentation attribute.
<svg xmlns="http://www.w3.org/2000/svg">
<style>
.container {
transform: translate(100px, 100px);
}
</style>
<g class="container" transform="translate(200 200)">
<rect width="100" height="100" fill="blue" />
</g>
</svg>
Because of the participation to the CSS cascade, the 'transform' style property overrides the 'transform' presentation attribute. Therefore the container gets translated by ''100px'' in both the horizontal and the vertical directions, instead of ''200px''.
To provide backwards compatibility, the syntax of the 'transform' presentation attribute differs from the syntax of the 'transform' style property as shown in the example above. However, the syntax used for the 'transform' style property can be used for a 'transform' presentation attribute value. Authors are advised to follow the rules of CSS Values and Units Module. Therefore an author should write ''transform="translate(200px, 200px)"'' instead of ''transform="translate (200 200)"'' because the second example with the spaces before the ''('', the missing comma between the arguments and the values without the explicit unit notation would be valid for the attribute only.
The value for the 'transform' attribute consists of a transform list with zero or more transform functions using functional notation. If the transform list consists of more than one transform function, these functions are separated by optional whitespace, an optional comma ('','') and optional whitespace. The transform list can have optional whitespace characters before and after the list.
The syntax starts with the name of the function followed by optional whitespace characters followed by a left parenthesis followed by optional whitespace followed by the argument(s) to the notation followed by optional whitespace followed by a right parenthesis. If a function takes more than one argument, the arguments are either separated by a comma ('','') with optional whitespace characters before and after the comma, or by one or more whitespace characters.
Arguments on all new introduced presentation attributes consist of data types in the sense of CSS Values and Units Module. The definitions of data types in CSS Values and Units Module are enhanced as follows:
A translation-value or length can be a <number> without an unit identifier. In this case the number gets interpreted as "user unit". A user unit in the the initial coordinate system is equivalent to the parent environment's notion of a pixel unit.
An angle can be a <number> without an unit identifier. In this case the number gets interpreted as a value in degrees.
SVG supports scientific notations for numbers. Therefore a number gets parsed like described in SVG Basic data types for SVG attributes.
SVG specifies the attributes 'gradientTransform' and 'patternTransform'. This specification makes both attributes presentation attributes. Both attributes use the same syntax as the SVG 'transform' attribute. This specification does not introduce corresponding CSS style properties. Both, the 'gradientTransform' and the 'patternTransform' attribute, are presentation attributes for the 'transform' property.
For backwards compatibility with existing SVG content, this specification supports all transform functions defined by The 'transform' attribute in [[SVG11]]. Therefore the two-dimensional transform function ''rotate(<angle>)'' is extended as follows:
rotate(<angle>[, <translation-value>,
<translation-value>])
User agents are just required to support the two optional arguments for translation on elements in the SVG namespace.
This specification explicitly requires three-dimensional transform functions to apply to the container elements: 'a element', 'g element', 'svg element', all graphics elements, all graphics referencing elements and the SVG 'foreignObject element' element.
Three-dimensional transform functions and the properties 'perspective', 'perspective-origin', 'transform-style' and 'backface-visibility' can not be used for the elements: 'clipPath element', 'linearGradient element', 'radialGradient element' and 'pattern element'. If a transform list includes a three-dimensional transform function, the complete transform list must be ignored. The values of every previously named property must be ignored. Transformable elements that are contained by one of these elements can have three-dimensional transform functions. The 'clipPath element', 'mask element', 'pattern element' elements require the user agent to create a flattened representation of the descendant elements before they can be applied, and therefore override the behavior of ''transform-style: preserve-3d''.
If the 'vector-effect' property is set to ''non-scaling-stroke'' and an object is within a 3D rendering context the property has no affect on stroking the object.
For the 'pattern element', 'linearGradient element', 'radialGradient element' and 'clipPath element' elements the 'transform', 'patternTransform', 'gradientTransform' presentation attributes represents values in the current user coordinate system in place at the time when these elements are referenced (i.e., the user coordinate system for the element referencing the 'pattern' element via a 'fill' or 'stroke' property). Percentage values are relative to the bounding box of the referencing element.
In particualar the 'patternUnit', 'gradientUnit' and 'maskUnit' attributes don't affect the user coordinate system used for transformations [[SVG11]].
For all other transformable elements the 'transform' presentation attribute represents values in the current user coordinate system of the parent. All percentage values of the 'transform' presentation attribute are relative to the element's bounding box.
The 'transform-origin' property on the pattern in the following example specifies a ''50%'' translation of the origin in the horizontal and vertical dimension. The 'transform' property specifies a translation as well, but in absolute lengths.
<svg xmlns="http://www.w3.org/2000/svg">
<style>
pattern {
transform: rotate(45deg);
transform-origin: 50% 50%;
}
</style>
<defs>
<pattern id="pattern-1">
<rect id="rect1" width="100" height="100" fill="blue" />
</pattern>
</defs>
<rect width="200" height="200" fill="url(#pattern-1)" />
</svg>
An SVG 'pattern element' element doesn't have a bounding box. The bounding box of the referencing 'rect element' element is used instead to solve the relative values of the 'transform-origin' property. Therefore the point of origin will get translated by 100 pixels temporarily to rotate the user space of the 'pattern element' elements content.
The SVG specification defines the 'SVGAnimatedTransformList' interface in the SVG DOM to provide access to the animated and the base value of the SVG 'transform', 'gradientTransform' and 'patternTransform' attributes. To ensure backwards compatibility, this API must still be supported by user agents.
The 'transform' property contributes to the CSS cascade. According to SVG 1.1 user agents conceptually insert a new author style sheet for presentation attributes, which is the first in the author style sheet collection. 'baseVal' gives the author the possibility to access and modify the values of the SVG 'transform' attribute. To provide the necessary backwards compatibility to the SVG DOM, 'baseVal' must reflect the values of this author style sheet. All modifications to SVG DOM objects of 'baseVal' must affect this author style sheet immediately.
'animVal' represents the computed style of the 'transform' property. Therefore it includes all applied CSS3 Transitions, CSS3 Animations or SVG Animations if any of those are underway. The computed style and SVG DOM objects of 'animVal' can not be modified.
The attribute 'type' of 'SVGTransform' must return 'SVG_TRANSFORM_UNKNOWN' for Transform Functions or unit types that are not supported by this interface. If a two-dimensional transform function is not supported, the attribute 'matrix' must return a 3x2 'SVGMatrix' with the corresponding values as described in the section Mathematical Description of Transform Functions.
With this specification, the 'animate element' element and the 'set element' element can animate the data type <transform-list>.
The animation effect is post-multiplied to the underlying value for additive 'animate element' animations (see below) instead of added to the underlying value, due to the specific behavior of <transform-list> animations.
From-to, from-by and by animations are defined in SMIL to be equivalent to a corresponding values animation. However, to animations are a mixture of additive and non-additive behavior [[SMIL3]].
To animations on 'animate element' provide specific functionality to get a smooth change from the underlying value to the 'to' attribute value, which conflicts mathematically with the requirement for additive transform animations to be post-multiplied. As a consequence, the behavior of to animations for 'animate element' is undefined. Authors are suggested to use from-to, from-by, by or values animations to achieve any desired transform animation.
The value 'paced' is undefined for the attribute 'calcMode' on 'animate element' for animations of the data type <transform-list>. If specified, UAs may choose the value 'linear' instead. Future versions of this specification may define how paced animations can be performed on <transform-list>.
Note: The following paragraphs extend Elements, attributes and properties that can be animated [[SVG11]].
The introduce presentation attributes 'transform', 'transform-origin', 'perspective', 'perspective-origin', 'transform-style' and 'backface-visibility' are animatable. 'transform-style' and 'backface-visibility' are non-additive.
With this specification the SVG basic data type <transform-list> is equivalent to a list of <transform-function>s. <transform-list> is animatable and additive. The data type can be animated using the SVG 'animate element' element and the SVG 'set element' element. SVG animations must run the same animation steps as described in section Transitions and Animations between Transform Values.
The set of animatable data types gets extended by <translation-value>. The new data type is animatable and additive.
| Data type | Additive? | 'animate element' | 'set element' | 'animateColor element' | 'animateTransform' | Notes |
|---|---|---|---|---|---|---|
| <transform-list> | yes | yes | yes | no | yes | Additive for 'animateTransform element' means that a transformation is post-multiplied to the base set of transformations. |
| <translation-value> | yes | yes | yes | no | no |
Some animations require a neutral element for addition. For transform functions this is a scalar or a list of scalars of 0. Examples of neutral elements for transform functions are ''translate(0)'', ''translate3d(0, 0, 0)'', ''translateX(0)'', ''translateY(0)'', ''translateZ(0)'', ''scale(0)'', ''scaleX(0)'', ''scaleY(0)'', ''scaleZ(0)'', ''rotate(0)'', ''rotate3d(vx, vy, vz, 0)'' (where v is a context dependent vector), ''rotateX(0)'', ''rotateY(0)'', ''rotateZ(0)'', ''skew(0, 0)'', ''skewX(0)'', ''skewY(0)'', ''matrix(0, 0, 0, 0, 0, 0)'', ''matrix3d(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)'' and ''perspective(0)''.
Animations to or from the neutral element of additions ''matrix'', ''matrix3d'' and ''perspective'' fall back to discrete animations (See Interpolation of Matrices).
A by animation with a by value vb is equivalent to the same animation with a values list with 2 values, the neutral element for addition for the domain of the target attribute (denoted 0) and vb, and ''additive="sum"''. [[SMIL3]]
<rect width="100" height="100">
<animateTransform attributeName="transform" attributeType="XML"
type="scale" by="1" dur="5s" fill="freeze"/>
</rect>
The neutral element for addition when performing a by animation with ''type="scale"'' is the value 0. Thus, performing the animation of the example above causes the rectangle to be invisible at time 0s (since the animated transform list value is ''scale(0)''), and be scaled back to its original size at time 5s (since the animated transform list value is ''scale(1)'').
SVG 1.1 Animation defines the 'attributeName' attribute to specify the name of the target attribute. For the presentation attributes 'gradientTransform' and 'patternTransform' it will also be possible to use the value 'transform'. The same 'transform' property will get animated.
In this example the gradient transformation of the linear gradient gets animated.
<linearGradient gradientTransform="scale(2)">
<animate attributeName="gradientTransform" from="scale(2)" to="scale(4)"
dur="3s" additive="sum"/>
<animate attributeName="transform" from="translate(0, 0)" to="translate(100px, 100px)"
dur="3s" additive="sum"/>
</linearGradient>
The 'linearGradient' element specifies the 'gradientTransform' presentation attribute. The two 'animate' elements address the target attribute 'gradientTransform' and 'transform'. Even so all animations apply to the same gradient transformation by taking the value of the 'gradientTransform' presentation attribute, applying the scaling of the first animation and applying the translation of the second animation one after the other.
The value of the 'transform' property is a list of <transform-functions>. The set of allowed transform functions is given below. For <transform-functions> the type <translation-value> is defined as a <length> or <percentage> value, and the <angle> type is defined by CSS Values and Units Module. Wherever <angle> is used in this specification, a <number> that is equal to zero is also allowed, which is treated the same as an angle of zero degrees.
matrix(<number>, <number>, <number>, <number>, <number>, <number>)
translate(<translation-value>[, <translation-value>])
translateX(<translation-value>)
translateY(<translation-value>)
scale(<number>[, <number>])
scaleX(<number>)
scaleY(<number>)
rotate(<angle>)
skew(<angle>[, <angle>])
Note that the behavior of ''skew'' is different from mutliplying ''skewX'' with ''skewY''. Implementations must support this function for compatibility with legacy content.
skewX(<angle>)
skewY(<angle>)
matrix3d(<number>, <number>, <number>, <number>, <number>, <number>, <number>, <number>, <number>, <number>, <number>, <number>, <number>, <number>, <number>, <number>)
translate3d(<translation-value>, <translation-value>, <length>)
translateZ(<length>)
scale3d(<number>, <number>, <number>)
scaleZ(<number>)
rotate3d(<number>, <number>, <number>, <angle>)
Note that the rotation is clockwise as one looks from the end of the vector toward the origin.
rotateX(<angle>)
rotate3d(1, 0, 0, <angle>).
rotateY(<angle>)
rotate3d(0, 1, 0, <angle>).
rotateZ(<angle>)
rotate3d(0, 0, 1, <angle>),
which is also the same as rotate(<angle>).
perspective(<length>)
If a list of <transform-functions> is provided, then the net effect is as if each transform function had been specified separately in the order provided. For example,
<div style="transform:translate(-10px,-20px) scale(2) rotate(45deg) translate(5px,10px)"/>
is functionally equivalent to:
<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>
That is, in the absence of other styling that affects position and dimensions, a nested set of transforms is equivalent to a single list of transform functions, applied from the outside in. The resulting transform is the matrix multiplication of the list of transforms.
If a transform function causes the current transformation matrix (CTM) of an object to be non-invertible, the object and its content do not get displayed.
The object in the following example gets scaled by 0.
<style>
.box {
transform: scale(0);
}
</style>
<div class="box">
Not visible
</div>
The scaling causes a non-invertible CTM for the coordinate space of the div box. Therefore neither the div box, nor the text in it get displayed.
When animating or transitioning transforms, the transform function lists must be interpolated. For interpolation between one transform from-transform and a second transforms to-transform, the rules described below are applied.
For example, if from-transform is ''scale(2)'' and to-transform is ''none'' then the value ''scale(1)'' will be used for to-transform and animation will proceed using the next rule. Similarly, if from-transform is ''none'' and to-transform is ''scale(2) rotate(50deg)'' then the animation will execute as if from-transform is ''scale(1) rotate(0)''.
For example, if from-transform is ''scale(1) translate(0)'' and to-transform is ''translate(100px) scale(2)'' then ''scale(1)'' and ''translate(100px)'' as well as ''translate(0)'' and ''scale(2)'' don't share a common primitive and therefore can not get interpolated following this rule.
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.
Some transform functions can be represented by more generic transform functions. These transform functions are called derived transform functions, the generic transform functions primitives. Primitives for two-dimensional and three-dimensional transform functions are listed below.
Two-dimensional primitives with derived transform functions are:
translate(<translation-value>,
<translation-value>)
translateX(<translation-value>),
translateY(<translation-value>) and
translate(<translation-value>).
rotate(<angle>, <translation-value>,
<translation-value>)
rotate(<angle>) if
rotate with three arguments is supported.
scale(<number>, <number>)
scaleX(<number>), scaleY(<number>) and scale(<number>).
Three-dimensional primitives with derived transform functions are:
translate3d(<translation-value>,
<translation-value>, <length>)
translateX(<translation-value>), translateY(<translation-value>), translateZ(<number>) and translate(<translation-value>[,
<translation-value>]).
scale3d(<number>, <number>, <number>)
scaleX(<number>), scaleY(<number>), scaleZ(<number>) and scale(<number>[, <number>]).
rotate3d(<number>, <number>, <number>, <angle>)
rotate(<number>), rotateX(<number>), rotateY(<number>) and rotateZ(<number>).
For derived transform functions that have a two-dimensional primitive and a three-dimensional primitive, the context decides about the used primitive. See Interpolation of primitives and derived transform functions.
Two transform functions with the same name and the same number of arguments are interpolated numerically without a former conversion. The calculated value will be of the same transform function type with the same number of arguments. Special rules apply to ''rotate3d'', ''matrix'', ''matrix3d'' and ''perspective''.
The two transform functions ''translate(0)'' and ''translate(100px)'' are of the same type, have the same number of arguments and therefore can get interpolated numerically. ''translateX(100px)'' is not of the same type and ''translate(100px, 0)'' does not have the same number of arguments, therefore these transform functions can not get interpolated without a former conversion step.
Two different types of transform functions that share the same primitive, or transform functions of the same type with different number of arguments can be interpolated. Both transform functions need a former conversion to the common primitive first and get interpolated numerically afterwards. The computed value will be the primitive with the resulting interpolated arguments.
The following example describes a transition from ''translateX(100px)'' to
''translateY(100px)'' in 3 seconds on hovering over the div box. Both transform
functions derive from the same primitive translate(<translation-value>, <translation-value>)
and therefore can be interpolated.
div {
transform: translateX(100px);
}
div:hover {
transform: translateY(100px);
transition: transform 3s;
}
For the time of the transition both transform functions get transformed to the common primitive. ''translateX(100px)'' gets converted to ''translate(100px, 0)'' and ''translateY(100px)'' gets converted to ''translate(0, 100px)''. Both transform functions can then get interpolated numerically.
If both transform functions share a primitive in the two-dimensional space, both transform functions get converted to the two-dimensional primitive. If one or both transform functions are three-dimensional transform functions, the common three-dimensional primitive is used.
In this example a two-dimensional transform function gets animated to a
three-dimensional transform function. The common primitive is translate3d.
div {
transform: translateX(100px);
}
div:hover {
transform: translateZ(100px);
transition: transform 3s;
}
First 'translateX(100px)' gets converted to 'translate3d(100px, 0, 0)' and 'translateZ(100px)' to 'translate3d(0, 0, 100px)' respectively. Then both converted transform
functions get interpolated numerically.
The transform functions ''matrix'', ''matrix3d'' and ''perspective'' get converted into 4x4 matrices first and interpolated as defined in section Interpolation of Matrices afterwards.
For interpolations with the primitive ''rotate3d'', the direction vectors of the transform functions get normalized first. If the normalized vectors are equal, the rotation angle gets interpolated numerically. Otherwise the transform functions get converted into 4x4 matrices first and interpolated as defined in section Interpolation of Matrices afterwards.
When interpolating between two 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 pseudocode in Decomposing the Matrix. The resulting values get interpolated numerically and recomposed back to a matrix in a final step.
In the following example the element gets translated by 100 pixel in both the X and Y directions and rotated by 1170° on hovering. The initial transformation is 45°. With the usage of transition, an author might expect a animated, clockwise rotation by three and a quarter turns (1170°).
<style>
div {
transform: rotate(45deg);
}
div:hover {
transform: translate(100px, 100px) rotate(1215deg);
transition: transform 3s;
}
</style>
<div></div>
The number of transform functions on the source transform ''rotate(45deg)'' differs from the number of transform functions on the destination transform ''translate(100px, 100px) rotate(1125deg)''. According to the last rule of Interpolation of Transforms, both transforms must be interpolated by matrix interpolation. With converting the transformation functions to matrices, the information about the three turns gets lost and the element gets rotated by just a quarter turn (90°).
To achieve the three and a quarter turns for the example above, source and destination transforms must fulfill the third rule of Interpolation of Transforms. Source transform could look like ''translate(0, 0) rotate(45deg)'' for a linear interpolation of the transform functions.
If one of the matrices for interpolation is non-invertible, the used animation function must fallback to a discrete animation according to the rules of the respective animation specification.
The pseudocode below is based upon the "unmatrix" method in "Graphics Gems II, edited by Jim Arvo", but modified to use Quaternions instead of Euler angles to avoid the problem of Gimbal Locks.
The following pseudocode works on a 4x4 homogeneous matrix:
Input: matrix ; a 4x4 matrix
Output: translation ; 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
quaternion ; 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):
double 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
double length(point) returns the length of the passed vector
point normalize(point) normalizes the length of the passed point to 1
double dot(point, point) returns the dot product of the passed points
double sqrt(double) returns the root square of passed value
double max(double y, double x) returns the bigger value of the two passed values
Decomposition also makes use of the following function:
point combine(point a, point b, double ascl, double 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)
else
// No perspective.
perspective[0] = perspective[1] = perspective[2] = 0
perspective[3] = 1
// Next take care of translation
for (i = 0; i < 3; i++)
translate[i] = matrix[3][i]
// 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, orthogonalize 3rd row
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[i] *= -1;
row[i][0] *= -1
row[i][1] *= -1
row[i][2] *= -1
// Now, get the rotations out
quaternion[0] = 0.5 * sqrt(max(1 + row[0][0] - row[1][1] - row[2][2], 0))
quaternion[1] = 0.5 * sqrt(max(1 - row[0][0] + row[1][1] - row[2][2], 0))
quaternion[2] = 0.5 * sqrt(max(1 - row[0][0] - row[1][1] + row[2][2], 0))
quaternion[3] = 0.5 * sqrt(max(1 + row[0][0] + row[1][1] + row[2][2], 0))
if (row[2][1] > row[1][2])
quaternion[0] = -quaternion[0]
if (row[0][2] > row[2][0])
quaternion[1] = -quaternion[1]
if (row[1][0] > row[0][1])
quaternion[2] = -quaternion[2]
return true
Each component of the decomposed values translation, scale, skew and perspective of the source matrix get linearly interpolated with each corresponding component of the destination matrix.
For instance, translate[0] of the source matrix and
translate[0] of the destination matrix are interpolated numerically,
and the result is used to set the translation of the animating element.
Quaternions of the decomposed source matrix are interpolated with quaternions of the decomposed destination matrix using the spherical linear interpolation (Slerp) as described by the pseudocode below:
Input: quaternionA ; a 4 component vector
quaternionB ; a 4 component vector
t ; interpolation parameter with 0 <= t <= 1
Output: quaternionDst ; a 4 component vector
Supporting functions (vector is a 4 component vector):
double dot(vector, vector) returns the dot product of the passed vectors
vector multVector(vector, vector) multiplies the passed vectors
double sqrt(double) returns the root square of passed value
double max(double y, double x) returns the bigger value of the two passed values
double min(double y, double x) returns the smaller value of the two passed values
double cos(double) returns the cosines of passed value
double sin(double) returns the sine of passed value
double acos(double) returns the inverse cosine of passed value
product = dot(quaternionA, quaternionB)
// Clamp product to -1.0 <= product <= 1.0
product = max(product, 1.0)
product = min(product, -1.0)
if (product == 1.0)
quaternionDst = quaternionA
return
theta = acos(dot)
w = sin(t * theta) * 1 / sqrt(1 - product * product)
for (i = 0; i < 4; i++)
quaternionA[i] *= cos(t * theta) - product * w
quaternionB[i] *= w
quaternionDst[i] = quaternionA[i] + quaternionB[i]
return
After interpolation the resulting values are used to transform the elements user space. One way to use these values is to recompose them into a 4x4 matrix. This can be done following the pseudocode below:
Input: translation ; 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
quaternion ; a 4 component vector
Output: matrix ; a 4x4 matrix
Supporting functions (matrix is a 4x4 matrix):
matrix multiply(matrix a, matrix b) returns the 4x4 matrix product of a * b
// apply perspective
for (i = 0; i < 4; i++)
matrix[i][3] = perspective[i]
// apply translation
for (i = 0; i < 3; i++)
for (j = 0; j < 3; j++)
matrix[3][i] += translation[j] * matrix[j][i]
// apply rotation
x = quaternion[0]
y = quaternion[1]
z = quaternion[2]
w = quaternion[3]
// Construct a composite rotation matrix from the quaternion values
// rotationMatrix is a identity 4x4 matrix initially
rotationMatrix[0][0] = 1 - 2 * (y * y + z * z)
rotationMatrix[0][1] = 2 * (x * y - z * w)
rotationMatrix[0][2] = 2 * (x * z + y * w)
rotationMatrix[1][0] = 2 * (x * y + z * w)
rotationMatrix[1][1] = 1 - 2 * (x * x + z * z)
rotationMatrix[1][2] = 2 * (y * z - x * w)
rotationMatrix[2][0] = 2 * (x * z - y * w)
rotationMatrix[2][1] = 2 * (y * z + x * w)
rotationMatrix[2][2] = 1 - 2 * (x * x + y * y)
matrix = multiply(matrix, rotationMatrix)
// apply skew
// temp is a identity 4x4 matrix initially
if (skew[2])
temp[2][1] = skew[2]
matrix = multiply(matrix, temp)
if (skew[1])
temp[2][1] = 0
temp[2][0] = skew[1]
matrix = multiply(matrix, temp)
if (skew[0])
temp[2][0] = 0
temp[1][0] = skew[0]
matrix = multiply(matrix, temp)
// apply scale
for (i = 0; i < 3; i++)
for (j = 0; j < 3; j++)
matrix[i][j] *= scale[i]
return
Mathematically, all transform functions can be represented as 4x4 transformation matrices of the following form:
One translation unit on a matrix is equivalent to 1 pixel in the local coordinate system of the element.
A 2D 3x2 matrix with six parameters a, b, c, d, e and f is equivalent to the matrix:
A 2D translation with the parameters tx and ty is equivalent to a 3D translation where tz has zero as a value.
A 2D scaling with the parameters sx and sy is equivalent to a 3D scale where sz has one as a value.
A 2D rotation with the parameter alpha is equivalent to a 3D rotation with vector [0,0,1] and parameter alpha.
A 2D skew like transformation with the parameters alpha and beta is equivalent to the matrix:
A 2D skew transformation along the X axis with the parameter alpha is equivalent to the matrix:
A 2D skew transformation along the Y axis with the parameter beta is equivalent to the matrix:
A 3D translation with the parameters tx, ty and tz is equivalent to the matrix:
A 3D scaling with the parameters sx, sy and sz is equivalent to the matrix:
A 3D rotation with the vector [x,y,z] and the parameter alpha is equivalent to the matrix:
where:
A perspective projection matrix with the parameter d is equivalent to the matrix:
