1. 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.
2. 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:
2.1. The linear timing function: linear
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.
2.2. Cubic Bézier timing functions: ease, easein, easeout, easeinout, cubicbezier()
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 [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, ∞].

A cubic Bézier timing function may be specified as a string using the following syntax (using notation from [CSS3VAL]):
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>, <number>, <number>, <number>)

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.
2.3. Step timing functions: stepstart, stepend, steps()
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.
A step timing function is defined by a nonzero 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:

Calculate the current step as
floor(input progress value × steps)
. 
If the step position property is start, 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.

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 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 timing function with a step position of end when applied to an animation during its delay phase.

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:
The meaning of each value is as follows:
 stepstart

Equivalent to steps(1, start);
 stepend

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.
2.4. Frames timing functions: frames()
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.
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:

Calculate the current frame as
floor(input progress value × frames)
. 
Let the initial output progress value be
current frame / (frames  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:
The parameter to the function specifies the number of frames. It must be a positive integer greater than 1.
2.5. Serialization
Timing 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() function before serializing.

Step timing 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 end, serialize as steps(<integer>).

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

3. 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.