Title: CSS Easing Functions Level 1
Status: ED
Work Status: Refining
Shortname: css-easing
Level: 1
Group: csswg
ED: https://drafts.csswg.org/css-easing/
TR: https://www.w3.org/TR/css-easing-1/
Previous version: https://www.w3.org/TR/2018/WD-css-easing-1-20181009/
Previous version: https://www.w3.org/TR/2018/WD-css-timing-1-20180926/
Previous version: https://www.w3.org/TR/2017/WD-css-timing-1-20170221/
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
Former Editor: Shane Stephens, Google Inc, shans@google.com, w3cid 47691

Abstract: This CSS module describes a way for authors to define a transformation
    that controls the rate of change of some value.
    Applied to animations, such transformations 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.

!Participate: IRC: #css on W3C's IRC
Repository: w3c/csswg-drafts
!Tests: web-platform-tests css/css-easing

{
  "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 some value changes. 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 intuitive user 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. Similarly, controlling the rate of change of gradient interpolation can be used to produce different visual effects such as suggesting a concave or convex surface, or producing a striped effect. [=Easing functions=] provide a means to transform such values by taking an input progress value and producing a corresponding transformed output progress value.
Example of an easing function that produces an ease-in effect.
Example of an easing function that produces an ease-in effect.
Given an input progress of 0.7, the easing function scales the value to produce an output progress of 0.52.
Applying this easing function to an animation would cause it to progress more slowly at first but then gradually progress more quickly.

Value Definitions

This specification uses the value definition syntax from [[!CSS-VALUES-3]]. Value types not defined in this specification are defined in CSS Values & Units [[!CSS-VALUES-3]]. Combination with other CSS modules may expand the definitions of these value types.

Easing functions

An easing function takes an [=input progress value=] and produces an [=output progress value=]. An [=easing 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 [=easing functions=] are chained together. The output progress value is a real number in the range [-∞, ∞]. Some types of easing functions also take an additional boolean [=before flag=] input which is defined subsequently. This specification defines four types of easing functions whose definitions follow. The syntax for specifying an [=easing function=] is as follows:
<easing-function> = ''linear'' | <> | <>

The linear easing function: ''linear''

The linear easing 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 easing function=] is simply the linear keyword.

Cubic Bézier easing functions: ''ease'', ''ease-in'', ''ease-out'', ''ease-in-out'', ''cubic-bezier()''

A cubic Bézier easing function is a type of [=easing 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].
A cubic Bezier curve used as an easing function.
A cubic Bézier curve used as an easing 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.
A cubic Bézier easing function has the following syntax (using notation from [[!CSS-VALUES-3]]):
<cubic-bezier-easing-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 easing 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 easing functions produced by keyword values.
The easing functions produced by each of the cubic Bézier easing function keyword values.
### Output of a cubic bézier easing function ### {#cubic-bezier-algo} 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]]. 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, ∞].

Step easing functions: ''step-start'', ''step-end'', ''steps()''

A step easing function is a type of [=easing function=] that divides the input time into a specified number of intervals that are equal in length. It is defined by a number of steps, and a step position. It has following syntax:
<step-easing-function> = ''step-start'' | ''step-end'' | steps(<>[, <>]?) <step-position> = ''jump-start'' | ''jump-end'' | ''jump-none'' | ''jump-both'' | ''start'' | ''end''
The meaning of each value is as follows:
: step-start :: Computes to ''steps(1, start)'' : step-end :: Computes to ''steps(1, end)''
Example step easing keywords.
Example step easing function keyword values.
: steps(<integer>[, <step-position> ]?) :: The first parameter specifies the number of intervals in the function. It must be a positive integer greater than 0 unless the second parameter is jump-none in which case it must be a positive integer greater than 1. The second parameter, which is optional, specifies the [=step position=] using one of the following values:
: jump-start :: The first rise occurs at [=input progress value=] of 0. : jump-end :: The last rise occurs at [=input progress value=] of 1. : jump-none :: All rises occur within the range (0, 1). : jump-both :: The first rise occurs at [=input progress value=] of 0 and the last rise occurs at [=input progress value=] of 1. : start :: Behaves as jump-start. : end :: Behaves as jump-end.
If the second parameter is omitted, the value ''end'' is assumed. These values are illustrated below:
Example step easing functions.
Example step easing functions.

Output of a step easing function

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 easing function=], if true, will cause the result of the function to correspond to the bottom of the step at the step point.
As an example of how the [=before flag=] affects the behavior of this function, consider an animation with a [=step easing 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 easing 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 easing function will correspond to the top of the first step.
For the purposes of calculating the [=output progress value=], the [=step position=] start is considered equivalent to jump-start. Likewise end is considered equivalent to jump-end. As a result, the following algorithm does not make explicit reference to start or end. Note: User agents must still differentiate between jump-start and start for the purpose of serialization (see [[#serialization]]). 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 one of: * jump-start, * jump-both, 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. Calculate |jumps| based on the [=step position=] as follows: : jump-start or jump-end :: [=steps=] : jump-none :: [=steps=] - 1 : jump-both :: [=steps=] + 1 1. If [=input progress value=] ≤ 1 and current step > |jumps|, let current step be |jumps|.
Steps 4 and 6 in this procedure ensure that given an [=input progress value=] in the range [0, 1], a step easing function does not produce an [=output progress value=] outside that range. For example, although mathematically we might expect that a step easing function with a [=step position=] of jump-start would step up (i.e. beyond 1) when the [=input progress value=] is 1, intuitively, when we apply such an easing 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 easing function with a [=step position=] of jump-end when applied to an animation during its delay phase.
1. The [=output progress value=] is current step / |jumps|.

Serialization

Easing 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 easing 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 ''jump-end'' or ''end'', serialize as steps(<integer>). 2. Otherwise, serialize as steps(<integer>, <step-position>). Privacy and Security Considerations {#priv-sec} =================================== This specification does not directly introduce any new capabilities to the Web platform but rather provides common definitions that may be referenced by other specifications. As a result, it does not introduce any new privacy and security concerns. Specifications referencing the features defined in this specification should consider that while easing functions most commonly take an [=input progress value=] in the range [0,1] and produce an [=output progress value=] in the range [0, 1], this is not always the case. Applications of easing functions should define the behavior for inputs and outputs outside this range to ensure they do not introduce new security considerations. Changes since last publication {#changes} ============================== The following changes have been made since the 9 October 2018 Working Draft: * Reworded specification to refer to “easing functions” rather than “timing functions” and use more general language that does not assume these functions are only applied to animations. 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.