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1. Introduction

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.

1.1. Module Interactions

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 [CSS2]. Some values of these properties result in the creation of a containing block, and/or the creation of a stacking context.

Transforms affect the rendering of backgrounds on elements with a value of fixed for the background-attachment property, which is specified in [CSS3BG].

Transforms affect the client rectangles returned by the Element Interface Extensions getClientRects() and getBoundingClientRect(), which are specified in [CSSOM-VIEW].

Transforms affect the computation of the scrollable overflow region as described by [CSS-OVERFLOW-3].

1.2. CSS Values

This specification follows the CSS property definition conventions from [CSS2]. Value types not defined in this specification are defined in CSS Values & Units [CSS-VALUES-3]. Other CSS modules may expand the definitions of these value types.

In addition to the property-specific values listed in their definitions, all properties defined in this specification also accept the CSS-wide keywords keywords as their property value. For readability they have not been repeated explicitly.

2. Terminology

When used in this specification, terms have the meanings assigned in this section.

transformable element

A transformable element is an element in one of these categories:

all elements whose layout is governed by the CSS box model except for non-replaced inline boxes, table-column boxes, and table-column-group boxes [CSS2],
all SVG paint server elements, the clipPath element and SVG renderable elements with the exception of any descendant element of text content elements [SVG2].

transformed element

An element with a computed value other than none for the transform property.

user coordinate system

local coordinate system

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. The current user coordinate system has its origin at the top-left of a reference box specified by the transform-box property. Percentage values are relative to the dimension of this reference box. One unit equals one CSS pixel.

transformation matrix

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.

current transformation matrix (CTM)

A matrix that defines the mapping from the local coordinate system into the viewport coordinate system.

2D matrix

A 3x2 transformation matrix, or a 4x4 matrix where the items m₃₁, m₃₂, m₁₃, m₂₃, m₄₃, m₁₄, m₂₄, m₃₄ are equal to 0 and m₃₃, m₄₄ are equal to 1.

identity transform function

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), translateX(0), translateY(0), scale(1), scaleX(1), scaleY(1), rotate(0), skew(0, 0), skewX(0), skewY(0) and matrix(1, 0, 0, 1, 0, 0).

post-multiply

post-multiplied

Term A post-multiplied by term B is equal to A · B.

pre-multiply

pre-multiplied

Term A pre-multiplied by term B is equal to B · A.

multiply

Multiply term A by term B is equal to A · B.

3. The Transform Rendering Model

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.

The transformation matrix is computed from the transform and transform-origin properties as follows:

Start with the identity matrix.
Translate by the computed X and Y of transform-origin
Multiply by each of the transform functions in transform property from left to right
Translate by the negated computed X and Y values of transform-origin

An element has a transform property that is not none.

div {
  transform-origin: 0 0;
  transform: translate(-10px, -20px) scale(2) rotate(45deg);
}

The transform-origin property is set to 0 0 and can be omitted. The transformation matrix TM gets computed by post-multying the <translate()>, <scale()> and <rotate()> <transform-function>s.

$TM = \begin{bmatrix} 1 & 0 & 0 & -10 \\ 0 & 1 & 0 & -20 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \cdot \begin{bmatrix} 2 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \cdot \begin{bmatrix} cos(45) & -sin(45) & 0 & 0 \\ sin(45) & cos(45) & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}$

Transforms apply to transformable elements.

The coordinate space is a coordinate system with two axes: the X axis increases horizontally to the right; the Y axis increases vertically downwards.

Transformations are cumulative. That is, elements establish their local coordinate system within the coordinate system of their parent.

To map a point p_local with the coordinate pair x_local and y_local from the local coordinate system of an element into the parent’s coordinate system, post-multiply the transformation matrix TM of the element by p_local. The result is the mapped point p_parent with the coordinate pair x_parent and y_parent in the parent’s local coordinate system.

$\begin{bmatrix} x_{parent} \\ y_{parent} \\ 0 \\ 1 \end{bmatrix} = TM \cdot \begin{bmatrix} x_{local} \\ y_{local} \\ 0 \\ 1 \end{bmatrix}$

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 current transformation matrix is computed by post-multiplying all transformation matrices starting from the viewport coordinate system and ending with the transformation matrix of an element.

This example has multiple, nested elements in an SVG document. Some elements get transformed by a transformation matrix.

<svg xmlns="http://www.w3.org/2000/svg">
  <g transform="translate(-10, 20)">
    <g transform="scale(2)">
      <rect width="200" height="200" transform="rotate(45)"/>
    </g>
  </g>
</svg>

translate(-10, 20) computes to the transformation matrix T1
scale(2) computes to the transformation matrix T2
rotate(45) computes to the transformation matrix T3

The CTM for the SVG rect element is the result of multiplying T1, T2 and T3 in order.

$CTM = \begin{bmatrix} 1 & 0 & 0 & -10 \\ 0 & 1 & 0 & -20 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \cdot \begin{bmatrix} 2 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \cdot \begin{bmatrix} cos(45) & -sin(45) & 0 & 0 \\ sin(45) & cos(45) & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}$

To map a point p_local with the coordinate pair x_local and y_local from the local coordinate system of the SVG rect element into the viewport coordinate system, post-multiply the current transformation matrix CTM of the element by p_local. The result is the mapped point p_viewport with the coordinate pair x_viewport and y_viewport in the viewport coordinate system.

$\begin{bmatrix} x_{viewport} \\ y_{viewport} \\ 0 \\ 1 \end{bmatrix} = CTM \cdot \begin{bmatrix} x_{local} \\ y_{local} \\ 0 \\ 1 \end{bmatrix}$

Note: Transformations do affect the visual rendering, but have no affect on the CSS layout other than affecting overflow. Transforms are also taken into account when computing client rectangles exposed via the Element Interface Extensions, namely getClientRects() and getBoundingClientRect(), which are specified in [CSSOM-VIEW].

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.

The point of origin gets translated temporary

div {
  height: 100px; width: 100px;
  transform: translate(80px, 80px) scale(1.5, 1.5) rotate(45deg);
}

The visual appareance is as if the div element gets translated by 80px to the bottom left direction, then scaled up by 150% and finally rotated by 45°.

Each <transform-function> can get represented by a corresponding 4x4 matrix. To map a point from the coordinate space of the div box to the coordinate space of the parent element, these transforms get multiplied in the reverse order:

The rotation matrix gets post-multiplied by the scale matrix.
The result of the previous multiplication is then post-multiplied by the translation matrix to create the accumulated transformation matrix.
Finally, the point to map gets pre-multiplied with the accumulated transformation matrix.

For more details see The Transform Function Lists.

Note: The 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. Specifically, transforms can extend (but do not shrink) the size of the overflow area, which is computed as the union of the bounds of the elements before and after the application of transforms.

For elements whose layout is governed by the CSS box model, any value other than none for the transform property results in the creation of a stacking context. Implementations must paint the layer it creates, within its parent stacking context, at the same stacking order that would be used if it were a positioned element with z-index: 0. If an element with a transform is positioned, the z-index property applies as described in [CSS2], except that auto is treated as 0 since a new stacking context is always created.

For elements whose layout is governed by the CSS box model, any value other than none for the transform property also causes the element to establish a containing block for all descendants. Its padding box will be used to layout for all of its absolute-position descendants, fixed-position descendants, and descendant fixed background attachments.

To demostrate the effect of containing block for all descendants on fixed-position descendants, the following code snippets should behave identically:

<style>
#container {
  width: 300px;
  height: 200px;
  border: 5px dashed black;
  padding: 5px;
  overflow: scroll;
}

#bloat {
  height: 1000px;
}

#child {
  right: 0;
  bottom: 0;
  width: 10%;
  height: 10%;
  background: green;
}
</style>

<div id="container" style="transform:translateX(5px);">
  <div id="bloat"></div>
  <div id="child" style="position:fixed;"></div>
</div>

versus

<div id="container" style="position:relative; z-index:0; left:5px;">
  <div id="bloat"></div>
  <div id="child" style="position:absolute;"></div>
</div>

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.

Note: 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.

4. The transform Property

A transformation is applied to the coordinate system an element renders into 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-list>
Initial:	none
Applies to:	transformable elements
Inherited:	no
Percentages:	refer to the size of reference box
Computed value:	as specified, but with lengths made absolute
Canonical order:	per grammar
Animation type:	transform list, see interpolation rules

Any computed value other than none for the transform affects containing block and stacking context, as described in §3 The Transform Rendering Model.

<transform-list> = <transform-function>+

4.1. Serialization of <transform-function>s

To serialize the <transform-function>s, serialize as per their individual grammars, in the order the grammars are written in, avoiding <calc()> expressions where possible, avoiding <calc()> transformations, omitting components when possible without changing the meaning, joining space-separated tokens with a single space, and following each serialized comma with a single space.

4.2. Serialization of the computed value of <transform-list>

A <transform-list> for the computed value is serialized to one <matrix()> function by the following algorithm:

Let transform be a 4x4 matrix initialized to the identity matrix. The elements m11, m22, m33 and m44 of transform must be set to 1 all other elements of transform must be set to 0.
Post-multiply all <transform-function>s in <transform-list> to transform.
Serialize transform to a <matrix()> function.

5. The transform-origin Property

Name:	transform-origin
Value:	[ left \| center \| right \| top \| bottom \| <length-percentage> ] \| [ left \| center \| right \| <length-percentage> ] [ top \| center \| bottom \| <length-percentage> ] <length>? \| [[ center \| left \| right ] && [ center \| top \| bottom ]] <length>?
Initial:	50% 50%
Applies to:	transformable elements
Inherited:	no
Percentages:	refer to the size of reference box
Computed value:	see background-position
Canonical order:	per grammar
Animation type:	by computed value

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>.

<length-percentage>: A percentage for the horizontal offset is relative to the width of the reference box. A percentage for the vertical offset is relative to the height of the reference box. The value for the horizontal and vertical offset represent an offset from the top left corner of the reference box.
<length>: 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 reference box.
top: Computes to 0% for the vertical position.
right: Computes to 100% for the horizontal position.
bottom: Computes to 100% for the vertical position.
left: Computes to 0% for the horizontal position.
center: Computes to 50% (left 50%) for the horizontal position if the horizontal position is not otherwise specified, or 50% (top 50%) for the vertical position if it is.

For SVG elements without associated CSS layout box the initial used value is 0 0 as if the user agent style sheet contained:

*:not(svg), *:not(foreignObject) > svg {
    transform-origin: 0 0;
}

The transform-origin property is a resolved value special case property like height. [CSSOM]

6. Transform reference box: the transform-box property

Name:	transform-box
Value:	content-box \| border-box \| fill-box \| stroke-box \| view-box
Initial:	view-box
Applies to:	transformable elements
Inherited:	no
Percentages:	N/A
Computed value:	specified keyword
Canonical order:	per grammar
Animation type:	discrete

All transformations defined by the transform and transform-origin property are relative to the position and dimensions of the reference box of the element. The reference box is specified by one of the following:

content-box

Uses the content box as reference box. The reference box of a table is the border box of its table wrapper box, not its table box.

border-box

Uses the border box as reference box. The reference box of a table is the border box of its table wrapper box, not its table box.

fill-box

Uses the object bounding box as reference box.

stroke-box

Uses the stroke bounding box as reference box.

view-box

Uses the nearest SVG viewport as reference box.

If a viewBox attribute is specified for the SVG viewport creating element:

The reference box is positioned at the origin of the coordinate system established by the viewBox attribute.
The dimension of the reference box is set to the width and height values of the viewBox attribute.

For the SVG pattern element, the reference box gets defined by the patternUnits attribute [SVG2].

For the SVG linearGradient and radialGradient elements, the reference box gets defined by the gradientUnits attribute [SVG2].

For the SVG clipPath element, the reference box gets defined by the clipPathUnits attribute [CSS-MASKING].

A reference box adds an additional offset to the origin specified by the transform-origin property.

For SVG elements without associated CSS layout box, the used value for content-box is fill-box and for border-box is stroke-box.

For elements with associated CSS layout box, the used value for fill-box is content-box and for stroke-box and view-box is border-box.

7. The SVG transform Attribute

7.1. SVG presentation attributes

The transform-origin CSS property is also a presentation attribute and extends the list of existing presentation attributes [SVG2].

SVG 2 defines the transform, patternTransform, gradientTransform attributes as presentation attributes, represented by the CSS transform property [SVG2].

The participation in the CSS cascade is determined by the specificity of presentation attributes in the SVG specification. According to SVG, user agents conceptually insert a new author style sheet for presentation attributes, which is the first in the author style sheet collection [SVG2].

This example shows the combination of the transform style property and the transform 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 attribute. Therefore the container gets translated by 100px in both the horizontal and the vertical directions, instead of 200px.

7.2. Syntax of the SVG transform attribute

For backwards compatibility reasons, the syntax of the transform, patternTransform, gradientTransform attributes differ from the syntax of the transform CSS property. For the attributes, there is no support for additional <transform-function>s defined for the CSS transform property. Specifically, <translateX()>, <translateY()>, <scaleX()>, <scaleY()> and <skew()> are not supported by the transform, patternTransform, gradientTransform attributes.

The following list uses the Backus-Naur Form (BNF) to define values for the transform, patternTransform and gradientTransform attributes followed by an informative rail road diagram. The following notation is used:

*: 0 or more
+: 1 or more
?: 0 or 1
(): grouping
|: separates alternatives
double quotes surround literals. Literals consists of letters [CSS-SYNTAX-3], left parenthesis and right parenthesis.
<number-token> defined by the CSS Syntax module [CSS-SYNTAX-3].

Note: The syntax reflects implemented behavior in user agents and differs from the syntax defined by SVG 1.1.

left parenthesis (

U+0028 LEFT PARENTHESIS

right parenthesis )

U+0029 RIGHT PARENTHESIS

comma

U+002C COMMA.

wsp

Either a U+000A LINE FEED, U+000D CARRIAGE RETURN, U+0009 CHARACTER TABULATION, or U+0020 SPACE.

comma-wsp

(wsp+ comma? wsp*) | (comma wsp*)

translate

"translate" wsp* "(" wsp* number ( comma-wsp? number )? wsp* ")"

scale

"scale" wsp* "(" wsp* number ( comma-wsp? number )? wsp* ")"

rotate

"rotate" wsp* "(" wsp* number ( comma-wsp? number comma-wsp? number )? wsp* ")"

skewX

"skewY" wsp* "(" wsp* number wsp* ")"

skewY

"skewY" wsp* "(" wsp* number wsp* ")"

matrix

"matrix" wsp* "(" wsp*
    number comma-wsp?
    number comma-wsp?
    number comma-wsp?
    number comma-wsp?
    number comma-wsp?
    number wsp* ")"

transform

matrix
| translate
| scale
| rotate
| skewX
| skewY

transforms

transform
| transform comma-wsp transforms

transform-list

wsp* transforms? wsp*

7.3. SVG transform functions

SVG transform functions of the transform, patternTransform, gradientTransform attributes defined by the syntax above are mapped to CSS <transform-function>s as follows:

SVG transform function	CSS <transform-function>	Additional notes
translate	<translate()>	Number values interpreted as CSS <length> types with px units.
scale	<scale()>
rotate	<rotate()>	Only single value version. Number value interpreted as CSS <angle> type with deg unit.
skewX	<skewX()>	Number value interpreted as CSS <angle> type with deg unit.
skewY	<skewY()>	Number value interpreted as CSS <angle> type with deg unit.
matrix	<matrix()>

The SVG transform function rotate with 3 values can not be mapped to a corresponding CSS <transform-function>. The 2 optional number values represent a horizontal translation value cx followed by a vertical translation value cy. Both number values get interpreted as CSS <length> types with px units and define the origin for rotation. The behavior is equivalent to an initial translation by cx, cy, a rotation defined by the first number value interpreted as <angle> type with deg unit followed by a translation by -cx, -cy.

A transform attribute can be the start or end value of a CSS Transition. If the value of a transform attribute is the start or end value of a CSS Transition and the SVG transform list contains at least one rotate transform function with 3 values, the individual SVG transform functions must get post-multiplied and the resulting matrix must get mapped to a <matrix()> CSS <transform-function> and used as start/end value of the CSS Transition.

7.4. User coordinate space

For the pattern element, the patternTransform attribtue and transform property define an additional transformation in the pattern coordinate system. See patternUnits attribute for details [SVG2].

For the linearGradient and radialGradient elements, the gradientTransform attribtue and transform property define an additional transformation in the gradient coordinate system. See gradientUnits attribute for details [SVG2].

For the clipPath element, the transform attribtue and transform property define an additional transformation in the clipping path coordinate space. See clipPathUnits attribute for details [CSS-MASKING].

For all other transformable elements the transform attribute and transform property define a transformation in the current user coordinate system of the parent. All percentage values of the transform attribute are relative to the element’s reference 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 doesn’t have a bounding box. The reference box of the referencing rect 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 elements content.

7.5. SVG DOM interface for the transform attribute

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.

baseVal gives the author the possibility to access and modify the values of the SVG transform, patternTransform, gradientTransform attributes. 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.

8. SVG Animation

8.1. The `animate` and `set` element

With this specification, the animate element and the set element can animate the data type <transform-list>.

The animation effect is post-multiplied to the underlying value for additive animate 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 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 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 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 introduced presentation attributes transform, patternTransform, gradientTransform and transform-origin are animatable.

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 and the SVG set element. SVG animations must run the same animation steps as described in section Transitions and Animations between Transform Values.

Animatable data types
Data type	Additive?	`animate`	`set`	`animateColor`	`animateTransform`	Notes
<transform-list>	yes	yes	yes	no	yes	Additive for `animateTransform` means that a transformation is post-multiplied to the base set of transformations.

8.2. Neutral element for addition

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), scale(0), rotate(0), skewX(0), skewY(0).

Note: This paragraph focuses on the requirements of [SMIL] and the extension defined by [SVG11]. This specification does not provide definitions of neutral elements for the other transform functions than the functions listed above.

A by animation with a by value v_b 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 v_b, 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)).

8.3. The SVG 1.1 'attributeName' attribute

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.

9. The Transform Functions

The value of the transform property is a list of <transform-function>. The set of allowed transform functions is given below. In the following functions, a <zero> behaves the same as 0deg ("unitless 0" angles are preserved for legacy compat). A percentage for horizontal translations is relative to the width of the reference box. A percentage for vertical translations is relative to the height of the reference box.

9.1. 2D Transform Functions

matrix() = matrix( <number> [, <number> ]{5,5} )

specifies a 2D transformation in the form of a transformation matrix of the six values a, b, c, d, e, f.

translate() = translate( <length-percentage> [, <length-percentage> ]? )

specifies a 2D translation by the vector [tx, ty], where tx is the first translation-value parameter and ty is the optional second translation-value parameter. If <ty> is not provided, ty has zero as a value.

translateX() = translateX( <length-percentage> )

specifies a translation by the given amount in the X direction.

translateY() = translateY( <length-percentage> )

specifies a translation by the given amount in the Y direction.

scale() = scale( <number> [, <number> ]? )

specifies a 2D scale operation by the [sx,sy] scaling vector described by the 2 parameters. If the second parameter is not provided, it takes a value equal to the first. For example, scale(1, 1) would leave an element unchanged, while scale(2, 2) would cause it to appear twice as long in both the X and Y axes, or four times its typical geometric size.

scaleX() = scaleX( <number> )

specifies a 2D scale operation using the [sx,1] scaling vector, where sx is given as the parameter.

scaleY() = scaleY( <number> )

specifies a 2D scale operation using the [1,sy] scaling vector, where sy is given as the parameter.

rotate() = rotate( [ <angle> | <zero> ] )

specifies a 2D rotation by the angle specified in the parameter about the origin of the element, as defined by the transform-origin property. For example, rotate(90deg) would cause elements to appear rotated one-quarter of a turn in the clockwise direction.

skew() = skew( [ <angle> | <zero> ] [, [ <angle> | <zero> ] ]? )

specifies a 2D skew by [ax,ay] for X and Y. If the second parameter is not provided, it has a zero value.

skew() exists for compatibility reasons, and should not be used in new content. Use skewX() or skewY() instead, noting that the behavior of skew() is different from multiplying skewX() with skewY().

skewX() = skewX( [ <angle> | <zero> ] )

specifies a 2D skew transformation along the X axis by the given angle.

skewY() = skewY( [ <angle> | <zero> ] )

specifies a 2D skew transformation along the Y axis by the given angle.

9.2. Transform function primitives and derivatives

Some transform functions can be represented by more generic transform functions. These transform functions are called derived transform functions, and the generic transform functions are called primitive transform functions. Two-dimensional primitives and their derived transform functions are:

translate(): for <translateX()>, <translateY()> and <translate()>.
scale(): for <scaleX()>, <scaleY()> and <scale()>.

10. The Transform Function Lists

If a list of <transform-function>s is provided, then the net effect is as if each transform function had been specified separately in the order provided.

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 coordinate system of the ancestor to the local coordinate system of a given element. The resulting transform is the matrix multiplication of the list of transforms.

For example,

<div style="transform: translate(-10px, -20px) scale(2) rotate(45deg)"/>

is functionally equivalent to:

<div style="transform: translate(-10px, -20px)" id="root">
  <div style="transform: scale(2)">
    <div style="transform: rotate(45deg)">
    </div>
  </div>
</div>

If a transform function causes the current transformation matrix 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.

11. Interpolation of Transforms

Interpolation of transform function lists is performed as follows:

If both V_a and V_b are none:
- V_result is none.
Treating none as a list of zero length, if V_a or V_b differ in length:
- extend the shorter list to the length of the longer list, setting the function at each additional position to the identity transform function matching the function at the corresponding position in the longer list. Both transform function lists are then interpolated following the next rule.
For example, if V_a is scale(2) and V_b is none then the value scale(1) will be used for V_b and interpolation will proceed using the next rule. Similarly, if V_a is scale(1) and V_b is scale(2) rotate(50deg) then the interpolation will be performed as if V_a were scale(1) rotate(0).
Let V_result be an empty list. Beginning at the start of V_a and V_b, compare the corresponding functions at each position:
- While the functions have either the same name, or are derivatives of the same primitive transform function, interpolate the corresponding pair of functions as described in §12 Interpolation of primitives and derived transform functions and append the result to V_result.
- If the pair do not have a common name or primitive transform function, post-multiply the remaining transform functions in each of V_a and V_b respectively to produce two 4x4 matrices. Interpolate these two matrices as described in §13 Interpolation of Matrices, append the result to V_result, and cease iterating over V_a and V_b.
For example, if V_a is rotate(0deg) scale(1) translate(20px) and V_b is rotate(270deg) translate(10px) scale(2), the rotate(0deg) and rotate(360deg) functions will be interpolated according to §12 Interpolation of primitives and derived transform functions while the remainder of each list—scale(1) translate(20px) and translate(10px) scale(2)—will first be converted to equivalent 4x4 matrices and then interpolated as described in §13 Interpolation of Matrices.
A previous version of this specification did not attempt to interpolate matching pairs of transform functions unless all functions in the list matched. As a result, the two lists in this example would be interpolated using matrix interpolation only and the rotate(360deg) component of the second list would be lost.

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.

12. 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 <matrix()>.

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() 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.

13. Interpolation of Matrices

When interpolating between two matrices, each matrix is decomposed into the corresponding translation, rotation, scale, skew. Each corresponding component of the decomposed matrices gets 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.

In the following we differ between the interpolation of two 2D matrices and the interpolation of two matrices where at least one matrix is not a 2D matrix.

If one of the matrices for interpolation is non-invertible, the used animation function must fall-back to a discrete animation according to the rules of the respective animation specification.

13.1. Supporting functions

The pseudo code in the next subsections make use of the following supporting functions:

Supporting functions (point is a 3 component vector, matrix is a 4x4 matrix, vector is a 4 component vector):
  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
  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
  double  abs(double)                  returns the absolute value of the passed value
  double  rad2deg(double)              transforms a value in radian to degree and returns it
  double  deg2rad(double)              transforms a value in degree to radian and returns it

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

13.2. Interpolation of 2D matrices

13.2.1. Decomposing a 2D matrix

The pseudo code below is based upon the "unmatrix" method in "Graphics Gems II, edited by Jim Arvo".

Matrices in the pseudo code use the column-major order. The first index on a matrix entry represents the column and the second index represents the row.

Input:  matrix      ; a 4x4 matrix
Output: translation ; a 2 component vector
        scale       ; a 2 component vector
        angle       ; rotation
        m11         ; 1,1 coordinate of 2x2 matrix
        m12         ; 1,2 coordinate of 2x2 matrix
        m21         ; 2,1 coordinate of 2x2 matrix
        m22         ; 2,2 coordinate of 2x2 matrix
Returns false if the matrix cannot be decomposed, true if it can


double row0x = matrix[0][0]
double row0y = matrix[0][1]
double row1x = matrix[1][0]
double row1y = matrix[1][1]

translate[0] = matrix[3][0]
translate[1] = matrix[3][1]

scale[0] = sqrt(row0x * row0x + row0y * row0y)
scale[1] = sqrt(row1x * row1x + row1y * row1y)

// If determinant is negative, one axis was flipped.
double determinant = row0x * row1y - row0y * row1x
if (determinant < 0)
    // Flip axis with minimum unit vector dot product.
    if (row0x < row1y)
        scale[0] = -scale[0]
    else
        scale[1] = -scale[1]

// Renormalize matrix to remove scale.
if (scale[0])
    row0x *= 1 / scale[0]
    row0y *= 1 / scale[0]
if (scale[1])
    row1x *= 1 / scale[1]
    row1y *= 1 / scale[1]

// Compute rotation and renormalize matrix.
angle = atan2(row0y, row0x);

if (angle)
    // Rotate(-angle) = [cos(angle), sin(angle), -sin(angle), cos(angle)]
    //                = [row0x, -row0y, row0y, row0x]
    // Thanks to the normalization above.
    double sn = -row0y
    double cs = row0x
    double m11 = row0x
    double m12 = row0y
    double m21 = row1x
    double m22 = row1y
    row0x = cs * m11 + sn * m21
    row0y = cs * m12 + sn * m22
    row1x = -sn * m11 + cs * m21
    row1y = -sn * m12 + cs * m22

m11 = row0x
m12 = row0y
m21 = row1x
m22 = row1y

// Convert into degrees because our rotation functions expect it.
angle = rad2deg(angle)

return true

13.2.2. Interpolation of decomposed 2D matrix values

Before two decomposed 2D matrix values can be interpolated, the following

Input: translationA ; a 2 component vector
       scaleA       ; a 2 component vector
       angleA       ; rotation
       m11A         ; 1,1 coordinate of 2x2 matrix
       m12A         ; 1,2 coordinate of 2x2 matrix
       m21A         ; 2,1 coordinate of 2x2 matrix
       m22A         ; 2,2 coordinate of 2x2 matrix
       translationB ; a 2 component vector
       scaleB       ; a 2 component vector
       angleB       ; rotation
       m11B         ; 1,1 coordinate of 2x2 matrix
       m12B         ; 1,2 coordinate of 2x2 matrix
       m21B         ; 2,1 coordinate of 2x2 matrix
       m22B         ; 2,2 coordinate of 2x2 matrix


// If x-axis of one is flipped, and y-axis of the other,
// convert to an unflipped rotation.
if ((scaleA[0] < 0 && scaleB[1] < 0) || (scaleA[1] < 0 && scaleB[0] < 0))
    scaleA[0] = -scaleA[0]
    scaleA[1] = -scaleA[1]
    angleA += angleA < 0 ? 180 : -180

// Don’t rotate the long way around.
if (!angleA)
    angleA = 360
if (!angleB)
    angleB = 360

if (abs(angleA - angleB) > 180)
    if (angleA > angleB)
        angleA -= 360
    else
        angleB -= 360

Afterwards, each component of the decomposed values translation, scale, angle, m11 to m22 of the source matrix get linearly interpolated with each corresponding component of the destination matrix.

13.2.3. Recomposing to a 2D matrix

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 pseudo code below.

Matrices in the pseudo code use the column-major order. The first index on a matrix entry represents the column and the second index represents the row.

Input:  translation ; a 2 component vector
        scale       ; a 2 component vector
        angle       ; rotation
        m11         ; 1,1 coordinate of 2x2 matrix
        m12         ; 1,2 coordinate of 2x2 matrix
        m21         ; 2,1 coordinate of 2x2 matrix
        m22         ; 2,2 coordinate of 2x2 matrix
Output: matrix      ; a 4x4 matrix initialized to identity matrix


matrix[0][0] = m11
matrix[0][1] = m12
matrix[1][0] = m21
matrix[1][1] = m22

// Translate matrix.
matrix[3][0] = translate[0] * m11 + translate[1] * m21
matrix[3][1] = translate[0] * m12 + translate[1] * m22

// Rotate matrix.
angle = deg2rad(angle);
double cosAngle = cos(angle);
double sinAngle = sin(angle);

// New temporary, identity initialized, 4x4 matrix rotateMatrix
rotateMatrix[0][0] = cosAngle
rotateMatrix[0][1] = sinAngle
rotateMatrix[1][0] = -sinAngle
rotateMatrix[1][1] = cosAngle

matrix = post-multiply(rotateMatrix, matrix)

// Scale matrix.
matrix[0][0] *= scale[0]
matrix[0][1] *= scale[0]
matrix[1][0] *= scale[1]
matrix[1][1] *= scale[1]

14. Mathematical Description of Transform Functions

Mathematically, all transform functions can be represented as 4x4 transformation matrices of the following form:

$\begin{bmatrix} m11 & m21 & m31 & m41 \\ m12 & m22 & m32 & m42 \\ m13 & m23 & m33 & m43 \\ m14 & m24 & m34 & m44 \end{bmatrix}$

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:

$\begin{bmatrix} a & c & 0 & e \\ b & d & 0 & f \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}$
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:

$\begin{bmatrix} 1 & \tan(\alpha) & 0 & 0 \\ \tan(\beta) & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}$
A 2D skew transformation along the X axis with the parameter alpha is equivalent to the matrix:

$\begin{bmatrix} 1 & \tan(\alpha) & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}$
A 2D skew transformation along the Y axis with the parameter beta is equivalent to the matrix:

$\begin{bmatrix} 1 & 0 & 0 & 0 \\ \tan(\beta) & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}$

15. Privacy and Security Considerations

UAs must implement transform operations in a way attackers can not infer information and mount a timing attack.

A timing attack is a method of obtaining information about content that is otherwise protected, based on studying the amount of time it takes for an operation to occur.

At this point there are no information about potential privacy or security concerns specific to this specification.

Changes

Since the 30 November 2018 Working Draft

No substantive changes
Boilerplate, styling updates for CR

Since the 30 November 2017 Working Draft

Remove specification text that makes patternTransform, gradientTransform presentation attributes representing the transform property. That is going to get specified by SVG 2 [SVG2].
Added privacy and security section.
Use [SVG2] definitions for transformable elements.
Added special syntax for transform, gradientTransform and patternTransform attributes.
Clarify multiplication order by using terms post-multiply and pre-multiply.
Clarify index order of matrix entries in pseudo-code.
Clarify multiplication order in recomposition pseudo-code.
Clarify behavior of transform on overflow area.
Remove translateX(0), translateY(0), scaleX(0), scaleY(0) from the list of neutral elements.
Remove any reference of 3D transformations of transform function definitions.
Specify interpolation between <transform-list>s to match lengths and avoid matrix interpolation for the common prefix of the two lists.
No transform on non-replaced inline boxes, table-column boxes, and table-column-group boxes.
Define target coordinate space for transformations on pattern, linearGradient, radialGradient and clipPath elements.
Remove 3-value <rotate()> from transform function primitives.
Be more specific about computation of transformation matrix and current transformation matrix.
Define reference box for paint servers and clipPath element.
Specify behavior of transform presentation attribute with 3-value-rotate as start or end value of a transition.
Add stroke-box and content-box to transform-box. Align box mapping behavior across all specifications.
Editorial changes.

Acknowledgments

The editors would like to thank Robert O’Callahan, Cameron McCormack, Tab Atkins, Gérard Talbot, L. David Baron, Rik Cabanier, Brian Birtles, Benoit Jacob, Ken Shoemake, Alan Gresley, Maciej Stochowiak, Sylvain Galineau, Rafal Pietrak, Shane Stephens, Matt Rakow, XiangHongAi, Fabio M. Costa, Nivesh Rajbhandari, Rebecca Hauck, Gregg Tavares, Graham Clift, Erik Dahlström, Alexander Zolotov, Amelia Bellamy-Royds and Boris Zbarsky for their careful reviews, comments, and corrections.

Conformance

Document conventions

Conformance requirements are expressed with a combination of descriptive assertions and RFC 2119 terminology. The key words “MUST”, “MUST NOT”, “REQUIRED”, “SHALL”, “SHALL NOT”, “SHOULD”, “SHOULD NOT”, “RECOMMENDED”, “MAY”, and “OPTIONAL” in the normative parts of this document are to be interpreted as described in RFC 2119. However, for readability, these words do not appear in all uppercase letters in this specification.

All of the text of this specification is normative except sections explicitly marked as non-normative, examples, and notes. [RFC2119]

Examples in this specification are introduced with the words “for example” or are set apart from the normative text with class="example", like this:

Informative notes begin with the word “Note” and are set apart from the normative text with class="note", like this:

Advisements are normative sections styled to evoke special attention and are set apart from other normative text with <strong class="advisement">, like this: UAs MUST provide an accessible alternative.

Conformance classes

A style sheet is conformant to this specification if all of its statements that use syntax defined in this module are valid according to the generic CSS grammar and the individual grammars of each feature defined in this module.

A renderer is conformant to this specification if, in addition to interpreting the style sheet as defined by the appropriate specifications, it supports all the features defined by this specification by parsing them correctly and rendering the document accordingly. However, the inability of a UA to correctly render a document due to limitations of the device does not make the UA non-conformant. (For example, a UA is not required to render color on a monochrome monitor.)

An authoring tool is conformant to this specification if it writes style sheets that are syntactically correct according to the generic CSS grammar and the individual grammars of each feature in this module, and meet all other conformance requirements of style sheets as described in this module.

Requirements for Responsible Implementation of CSS

The following sections define several conformance requirements for implementing CSS responsibly, in a way that promotes interoperability in the present and future.

Partial Implementations

So that authors can exploit the forward-compatible parsing rules to assign fallback values, CSS renderers must treat as invalid (and ignore as appropriate) any at-rules, properties, property values, keywords, and other syntactic constructs for which they have no usable level of support. In particular, user agents must not selectively ignore unsupported property values and honor supported values in a single multi-value property declaration: if any value is considered invalid (as unsupported values must be), CSS requires that the entire declaration be ignored.

Implementations of Unstable and Proprietary Features

Implementations of CR-level Features

Once a specification reaches the Candidate Recommendation stage, implementers should release an unprefixed implementation of any CR-level feature they can demonstrate to be correctly implemented according to spec, and should avoid exposing a prefixed variant of that feature.

To establish and maintain the interoperability of CSS across implementations, the CSS Working Group requests that non-experimental CSS renderers submit an implementation report (and, if necessary, the testcases used for that implementation report) to the W3C before releasing an unprefixed implementation of any CSS features. Testcases submitted to W3C are subject to review and correction by the CSS Working Group.

CR exit criteria

For this specification to be advanced to Proposed Recommendation, there must be at least two independent, interoperable implementations of each feature. Each feature may be implemented by a different set of products, there is no requirement that all features be implemented by a single product. For the purposes of this criterion, we define the following terms: