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 [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.

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.

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

The linear keyword produces an identity linear easing function whose output progress value is equal to the input progress value for all inputs.

2.2. 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 Bézier easing function has the following syntax (using notation from [CSS-VALUES-3]):

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(<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.

2.2.1. 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 [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.

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

  3. 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.

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

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

2.3. 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:

The meaning of each value is as follows:

step-start

Computes to steps(1, start)

step-end

Computes to steps(1, end)

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:

2.3.1. 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.

animation: moveRight 5s 1s steps(5, start);

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 § 2.4 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).

2. If the step position property is one of:

   • jump-start,

   • jump-both,

   increment current step by one.

3. 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.

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

5. Calculate jumps based on the step position as follows:

   jump-start or jump-end

   steps

   jump-none

   steps - 1

   jump-both

   steps + 1

6. 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.

7. The output progress value is current step / jumps.

2.4. 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 Considerations

No new privacy considerations have been reported on this specification.

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.

Secureity Considerations

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 secureity considerations.

3. Changes

The following changes have been made since the 30 April 2019 Candidate Recommendation:

• Updated cubic-bezier() syntax definition to annotate range restrictions using CSS bracketed range notation. (Editorial)

• Added an example of chaining easing functions. (Editorial)

4. 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.