Title: CSS Timing Functions Level 1
Status: ED
Work Status: Refining
Shortname: css-timing
Level: 1
Group: csswg
ED: https://drafts.csswg.org/css-timing/
TR: https://www.w3.org/TR/css-timing-1/
Editor: Brian Birtles, Mozilla https://www.mozilla.org/, bbirtles@mozilla.com, w3cid 43194
Editor: Dean Jackson, Apple Inc https://www.apple.com/, dino@apple.com, w3cid 42080
Editor: Matt Rakow, Microsoft, w3cid 62267
Editor: Shane Stephens, Google Inc, shans@google.com, w3cid 47691

Abstract: This CSS module describes a way for authors to define a transformation
    to be applied to the time of an animation. This can be used to produce
    animations that mimic physical phenomena such as momentum or to
    cause the animation to move in discrete steps producing robot-like
    movement.

{
  "FUND-COMP-GRAPHICS": {
    "title": "Fundamentals of Computer Graphics",
    "authors": [
      "Peter Shirley",
      "Michael Ashikhmin",
      "Steve Marschner"
    ],
    "date": "2009",
    "publisher": "A K Peters Limited"
  }
}
Introduction {#introduction} ============================ This section is not normative. It is often desirable to control the rate at which an animation progresses. For example, gradually increasing the speed at which an element moves can give the element a sense of weight as it appears to gather momentum. This can be used to produce user intuitive interface elements or convincing cartoon props that behave like their physical counterparts. Alternatively, it is sometimes desirable for animation to move forwards in distinct steps such as a segmented wheel that rotates such that the segments always appear in the same position. [=Timing functions=] provide a means to transform animation time by taking an input progress value and producing a corresponding transformed output progress value.
Example of a timing function that produces an ease-in effect.
Example of a timing function that produces an ease-in effect. Given an input progress of 0.7, the timing function scales the value to produce an output progress of 0.52. By applying this timing function, the animation will progress more slowly at first but then gradually progress more quickly.
Timing functions {#timing-functions} ==================================== A timing function takes an [=input progress value=] and produces an [=output progress value=]. A [=timing function=] must be a pure function meaning that for a given set of inputs, it always produces the same [=output progress value=]. The input progress value is a real number in the range [-∞, ∞]. Typically, the [=input progress value=] is in the range [0, 1] but this may not be the case when [=timing functions=] are chained together. The output progress value is a real number in the range [-∞, ∞]. Some types of timing function also take an additional boolean [=before flag=] input which is defined subsequently. This specification defines four types of timing functions whose definitions follow. The syntax for specifying a [=timing function=] is as follows:
<timing-function> = ''linear'' | <> | <> | <>
The linear timing function: ''linear'' {#linear-timing-function-section} ------------------------------------------------------------------------ The linear timing function is an identity function meaning that its [=output progress value=] is equal to the [=input progress value=] for all inputs. The syntax for the [=linear timing function=] is simply the linear keyword. Cubic Bézier timing functions: ''ease'', ''ease-in'', ''ease-out'', ''ease-in-out'', ''cubic-bezier()'' {#cubic-bezier-timing-functions} --------------------------------------------------------------------- A cubic Bézier timing function is a type of [=timing function=] defined by four real numbers that specify the two control points, P1 and P2, of a cubic Bézier curve whose end points P0 and P3 are fixed at (0, 0) and (1, 1) respectively. The x coordinates of P1 and P2 are restricted to the range [0, 1]. The mapping from input progress to output progress is performed by determining the corresponding y value ([=output progress value=]) for a given x value ([=input progress value=]). The evaluation of this curve is covered in many sources such as [[FUND-COMP-GRAPHICS]].
A cubic Bezier curve used as a timing function.
A cubic Bézier curve used as a timing function.
The shape of the curve is determined by the location of the control points P1 and P2.
Input progress values serve as x values of the curve, whilst the y values are the output progress values.
For [=input progress values=] outside the range [0, 1], the curve is extended infinitely using tangent of the curve at the closest endpoint as follows: * For [=input progress values=] less than zero, 1. If the x value of P1 is greater than zero, use a straight line that passes through P1 and P0 as the tangent. 1. Otherwise, if the x value of P2 is greater than zero, use a straight line that passes through P2 and P0 as the tangent. 1. Otherwise, let the [=output progress value=] be zero for all [=input progress values=] in the range [-∞, 0). * For [=input progress values=] greater than one, 1. If the x value of P2 is less than one, use a straight line that passes through P2 and P3 as the tangent. 1. Otherwise, if the x value of P1 is less than one, use a straight line that passes through P1 and P3 as the tangent. 1. Otherwise, let the [=output progress value=] be one for all [=input progress values=] in the range (1, ∞]. A cubic Bézier timing function may be specified as a string using the following syntax (using notation from [[!CSS3VAL]]):
<cubic-bezier-timing-function> = ''ease'' | ''ease-in'' | ''ease-out'' | ''ease-in-out'' | cubic-bezier(<>, <>, <>, <>)
The meaning of each value is as follows:
: ease :: Equivalent to cubic-bezier(0.25, 0.1, 0.25, 1). : ease-in :: Equivalent to cubic-bezier(0.42, 0, 1, 1). : ease-out :: Equivalent to cubic-bezier(0, 0, 0.58, 1). : ease-in-out :: Equivalent to cubic-bezier(0.42, 0, 0.58, 1). :
cubic-bezier(<>, <>, <>, <>)
:: Specifies a cubic Bézier timing function. The four numbers specify points P1 and P2 of the curve as (x1, y1, x2, y2). Both x values must be in the range [0, 1] or the definition is invalid.
The keyword values listed above are illustrated below.
The timing functions produced by keyword values.
The timing functions produced by each of cubic Bézier timing function keyword values.
Step timing functions: ''step-start'', ''step-end'', ''steps()'' {#step-timing-functions} ---------------------------------------------------------------- A step timing function is a type of timing function that divides the input time into a specified number of intervals that are equal in length. Some example step timing functions are illustrated below.
Example step timing functions.
Example step timing functions. In each case the domain is the input progress whilst the range represents the output progress produced by the step function.
The first row shows the function for each transition point when only one step is specified whilst the second row shows the same for three steps.
A [=step timing function=] is defined by a non-zero positive number of steps, and a step position property that may be either start or end. At the exact point where a step occurs the result of the function is conceptually the top of the step. However, an additional before flag passed as input to the [=step timing function=], if true, will cause the result of the function to correspond to the bottom of the step at the step point. The [=output progress value=] is calculated from the [=input progress value=] and [=before flag=] as follows: 1. Calculate the current step as floor([=input progress value=] × [=steps=]). 1. If the [=step position=] property is start, increment current step by one. 1. If both of the following conditions are true: * the [=before flag=] is set, and * [=input progress value=] × [=steps=] mod 1 equals zero (that is, if [=input progress value=] × [=steps=] is integral), then decrement current step by one. 1. If [=input progress value=] ≥ 0 and current step < 0, let current step be zero. 1. If [=input progress value=] ≤ 1 and current step > [=steps=], let current step be [=steps=].
Steps 4 and 5 in this procedure ensure that given an [=input progress value=] in the range [0, 1], a step timing function does not produce an [=output progress value=] outside that range. For example, although mathematically we might expect that a step timing function with a [=step position=] of start would step up when the [=input progress value=] is 1, intuitively, when we apply such a timing function to a forwards-filling animation, we expect it to produce an [=output progress value=] of 1 as the animation fills forwards. A similar situation arises for a step timing function with a [=step position=] of end when applied to an animation during its delay phase.
1. The [=output progress value=] is current step / [=steps=].
As an example of how the [=before flag=] affects the behavior of this function, consider an animation with a [=step timing function=] whose [=step position=] is start and which has a positive delay and backwards fill. For example, using CSS animation: 
animation: moveRight 5s 1s steps(5, start);
During the delay phase, the [=input progress value=] will be zero but if the [=before flag=] is set to indicate that the animation has yet to reach its animation interval, the timing function will produce zero as its [=output progress value=], i.e. the bottom of the first step. At the exact moment when the animation interval begins, the [=input progress value=] will still be zero, but the [=before flag=] will not be set and hence the result of the timing function will correspond to the top of the first step.
The syntax for specifying a step timing function is as follows:
<step-timing-function> = ''step-start'' | ''step-end'' | steps(<>[, [ ''start'' | ''end'' ] ]?)
The meaning of each value is as follows:
: step-start :: Equivalent to steps(1, start); : step-end :: Equivalent to steps(1, end); : steps(<integer>[, [ start | end ] ]?) :: Specifies a step timing function. The first parameter specifies the number of intervals in the function. It must be a positive integer greater than 0. The second parameter, which is optional, is either the value start or end, and specifies the [=step position=]. If the second parameter is omitted, it is given the value ''end''.
Frames timing functions: ''frames()'' {#frames-timing-functions} ---------------------------------------------------------------- A frames timing function is a type of timing function that divides the input time into a specified number of intervals of equal length, each of which is associated with an output progress value of increasing value. The difference between a [=frames timing function=] and a [=step timing function=] is that a frames timing function returns the [=output progress values=] 0 and 1 for an equal portion of the [=input progress values=] in the range [0, 1]. This makes it suitable, for example, for using in animation loops where the animation should display the first and last frame of the animation for an equal amount of time as each other frame during each loop. Some example frames timing functions are illustrated below.
Example frames timing functions.
Example frames timing functions. In each case the domain is the input progress whilst the range represents the output progress produced by the function.
A [=frames timing function=] is defined by an integral number of frames greater than one. As with [=step timing functions=], at the exact point where a step occurs the result of the function is conceptually the top of the step. The [=output progress value=] is calculated from the [=input progress value=] as follows: 1. Calculate the current frame as floor([=input progress value=] × [=frames=]). 1. Let the initial [=output progress value=] be current frame / ([=frames=] - 1). 1. If [=input progress value=] ≤ 1 and [=output progress value=] > 1, let [=output progress value=] be 1. The syntax for specifying a frames timing function is as follows:
<frames-timing-function> = frames(<>)
The parameter to the function specifies the number of [=frames=]. It must be a positive integer greater than 1. Serialization {#serializing-a-timing-function} ------------- Timing functions are serialized using the common serialization patterns defined in [[CSSOM]] with the following additional requirements: * The keyword values ''ease'', ''linear'', ''ease-in'', ''ease-out'', and ''ease-in-out'' are serialized as-is, that is, they are not converted to the equivalent ''cubic-bezier()'' function before serializing. * Step timing functions, whether they are specified using the ''steps()'' function or either of the ''step-start'' or ''step-end'' keywords, are serialized as follows: 1. If the [=step position=] is ''end'', serialize as steps(<integer>). 2. Otherwise, serialize as steps(<integer>, start). Acknowledgements {#acknowledgements} ================ This specification is based on the CSS Transitions specification edited by L. David Baron, Dean Jackson, David Hyatt, and Chris Marrin. The editors would also like to thank Douglas Stockwell, Steve Block, Tab Atkins, Rachel Nabors, Martin Pitt, and the Animation at Work slack community for their feedback and contributions.