1. 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.
1.1. Value Definitions
This specification uses the value definition syntax from [CSSVALUES3]. Value types not defined in this specification are defined in CSS Values & Units [CSSVALUES3]. Combination with other CSS modules may expand the definitions of these value types.
2. 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.
An example of when easing functions are chained together occurs in Web Animations [WEBANIMATIONS] where the output of the easing function specified on an animation effect may become the input to an easing function specified on one of the keyframes of a keyframe effect. In this scenario, the input to the easing function on the keyframe effect may be outside the range [0, 1].
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:
2.1. The linear easing function: linear()
A linear easing function is an easing function that interpolates linearly between its points.
A linear easing function has points, a list of linear easing points. Initially a new empty list.
A linear easing point is a struct that has:
 input

A number or null
Note: This is only null during create a linear easing function.
 output

A number
2.1.1. Syntax
A linear easing function has the following syntax:
<lineareasingfunction> = linear(<linearstoplist>) <linearstoplist> = [ <linearstop> ]# <linearstop> = <number> && <linearstoplength>? <linearstoplength> = <percentage>{1,2}
linear() is parsed into a linear easing function by calling create a linear easing function, passing in its <linearstoplist> as a list of <linearstop>s.
2.1.2. Parsing
To create a linear easing function given a list of <linearstop>s stopList, perform the following. It returns a linear easing function or failure.

Let function be a new linear easing function.

Let largestInput be negative infinity.

If there are less than two items in stopList, then return failure.

For each stop in stopList:

Let point be a new linear easing point with its output set to stop’s <number> as a number.

If stop has a <linearstoplength>, then:

Set point’s input to whichever is greater: stop’s <linearstoplength>'s first <percentage> as a number, or largestInput.

Set largestInput to point’s input.

If stop’s <linearstoplength> has a second <percentage>, then:

Let extraPoint be a new linear easing point with its output set to stop’s <number> as a number.

Set extraPoint’s input to whichever is greater: stop’s <linearstoplength>'s second <percentage> as a number, or largestInput.

Set largestInput to extraPoint’s input.



Otherwise, if stop is the first item in stopList, then:

Set point’s input to 0.

Set largestInput to 0.


Otherwise, if stop is the last item in stopList, then set point’s input to whichever is greater: 1 or largestInput.


For runs of items in function’s points that have a null input, assign a number to the input by linearly interpolating between the closest previous and next points that have a nonnull input.

Return function.
2.1.3. Serializing
For example:

linear(0, 0.25, 1) serializes as linear(0 0%, 0.25 50%, 1 100%)

linear(0 20%, 0.5 10%, 1) serializes as linear(0 20%, 0.5 20%, 1 100%)

linear(0, 0.25 25% 75%, 1) serializes as linear(0 0%, 0.25 25%, 0.25 75%, 1 100%)
To get a linear easing function's (linearEasingFunction) serialized computed value, perform the following. It returns a string.

Let output be "
linear(
". 
Append "
)
" to output. 
Return output.
2.1.4. Output of a linear easing function
To calculate linear easing output progress for a given linear easing function linearEasingFunction, and an input progress value inputProgress, perform the following. It returns an output progress value.

Let points be linearEasingFunction’s points.

Let pointAIndex be index of the last item in points with an input less than or equal to inputProgress, or 0 if there is no match.

If pointAIndex is equal to points size minus 1, decrement pointAIndex by 1.
Note: This ensures we have a "next" point to compare to.

Let pointA be points[pointAIndex].

Let pointB be points[pointAIndex + 1].

If pointA’s input is equal to pointB’s input, return pointB’s output.

Let progressFromPointA be inputProgress minus pointA’s input.

Let pointInputRange be pointB’s input minus pointA’s input.

Let progressBetweenPoints be progressFromPointA divided by pointInputRange.

Let pointOutputRange be pointB’s output minus pointA’s output.

Let outputFromLastPoint be progressBetweenPoints multiplied by pointOutputRange.

Return pointA’s output plus outputFromLastPoint.
2.1.5. Examples
For example, linear(0, 0.25, 1) produces an easing function that moves linearly from 0, to 0.25, then to 1:
For example, linear(0, 0.25 75%, 1) produces the following easing function, which spends 75% of the time transitioning from 0 to .25, then the last 25% transitioning from .25 to 1:
For example, linear(0, 0.25 25% 75%, 1) is equivalent to linear(0, 0.25 25%, 0.25 75%, 1), producing the following easing function:
For example, here are the implicit values from the previous function:
For example, here’s how linear() could be used to create a reusable "bounce" easing function:
:root{ bounce : linear ( /* Start to 1st bounce */ 0 , 0.063 , 0.25 , 0.563 , 1 36.4 % , /* 1st to 2nd bounce */ 0.812 , 0.75 , 0.813 , 1 72.7 % , /* 2nd to 3rd bounce */ 0.953 , 0.938 , 0.953 , 1 90.9 % , /* 3rd bounce to end */ 0.984 , 1 100 % 100 % ); } .example{ animationtimingfunction : var ( bounce); }
The definition ends 1 100% 100%
to create two final points,
so inputs greater than 1 always output 1.
More points could be used to create a smoother result, which may be needed for slower animations.
2.2. The linear easing keyword: linear
The linear keyword produces a linear easing function with two points:
Note: This results in an identity function, meaning that its output progress value is equal to the input progress value for all inputs.
Note: Although this produces a linear easing function, uses of the keyword linear always serialize asis, to linear. Whereas the function equivalent linear(0, 1) will serialize to linear(0 0%, 1 100%). These rules are in Serialization.
2.3. Cubic Bézier easing functions: ease, easein, easeout, easeinout, cubicbezier()
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 Bézier easing function has the following syntax (using notation from [CSSVALUES3]):
The meaning of each value is as follows:
 ease

Equivalent to cubicbezier(0.25, 0.1, 0.25, 1).
 easein

Equivalent to cubicbezier(0.42, 0, 1, 1).
 easeout

Equivalent to cubicbezier(0, 0, 0.58, 1).
 easeinout

Equivalent to cubicbezier(0.42, 0, 0.58, 1).
 cubicbezier(<number [0,1]>, <number>, <number [0,1]>, <number>)

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.
2.3.1. Output of a cubic bézier easing function
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 [FUNDCOMPGRAPHICS].
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,

If the x value of P1 is greater than zero, use a straight line that passes through P1 and P0 as the tangent.

Otherwise, if the x value of P2 is greater than zero, use a straight line that passes through P2 and P0 as the tangent.

Otherwise, let the output progress value be zero for all input progress values in the range [∞, 0).


For input progress values greater than one,

If the x value of P2 is less than one, use a straight line that passes through P2 and P3 as the tangent.

Otherwise, if the x value of P1 is less than one, use a straight line that passes through P1 and P3 as the tangent.

Otherwise, let the output progress value be one for all input progress values in the range (1, ∞].

2.4. Step easing functions: stepstart, stepend, 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:
<stepposition> = jumpstart  jumpend  jumpnone  jumpboth  start  end
The meaning of each value is as follows:
 stepstart

Computes to steps(1, start)
 stepend

Computes to steps(1, end)
 steps(<integer>, <stepposition>?)

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 jumpnone 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:
 jumpstart

The first rise occurs at input progress value of 0.
 jumpend

The last rise occurs at input progress value of 1.
 jumpnone

All rises occur within the range (0, 1).
 jumpboth

The first rise occurs at input progress value of 0 and the last rise occurs at input progress value of 1.
 start

Behaves as jumpstart.
 end

Behaves as jumpend.
If the second parameter is omitted, the value end is assumed.
These values are illustrated below:
2.4.1. 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 : moveRight5 s 1 s 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 jumpstart. Likewise end is considered equivalent to jumpend. As a result, the following algorithm does not make explicit reference to start or end.
Note: User agents must still differentiate between jumpstart and start for the purpose of serialization (see § 2.5 Serialization).
The output progress value is calculated from the input progress value and before flag as follows:

Calculate the current step as
floor(input progress value × steps)
. 
If the step position property is one of:
increment current step by one.

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.


If input progress value ≥ 0 and current step < 0, let current step be zero.

Calculate jumps based on the step position as follows:

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 jumpstart would step up (i.e. beyond 1) when the input progress value is 1, intuitively, when we apply such an easing function to a forwardsfilling 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 jumpend when applied to an animation during its delay phase.

The output progress value is
current step / jumps
.
2.5. Serialization
Easing functions are serialized using the common serialization patterns defined in [CSSOM] with the following additional requirements:

The keyword values ease, linear, easein, easeout, and easeinout are serialized asis, that is, they are not converted to the equivalent cubicbezier() or linear() function before serializing.

Step easing functions, whether they are specified using the steps() function or either of the stepstart or stepend keywords, are serialized as follows:

If the step position is jumpend or end, serialize as steps(<integer>).

Otherwise, serialize as steps(<integer>, <stepposition>).


A linear easing function created via linear() is serialized by getting its serialized computed value.
3. Privacy and Security Considerations
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.
4. Changes
4.1. Additions Since Level 1

Added linear() function.
5. 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.