[css-easing-2] Complex easing/timing functions

[nearestnabors]

In light of the Webkit team's implementation of spring(), it's apparent we need to attend to the issue of complex timing functions sooner rather than later.

The problem

Designers often need more advanced timing functions than can be described with cubic-beziers. They are not limited to spring functions, either. A common problem is there is no effective way to export a timing graph from Adobe After Effects to a timing function that could be used with CSS or with the Web Animations API. Currently designers have to hack together individual timing functions using CSS animation keyframes, which is impossible to do by hand in all but the most trifling instances.

spring() is just a bandaid.

The solution

We need a format to write functions like spring() in, one that we can export to from software like AfterEffects and prototyping tools that have yet to be built.

I am not in a position to propose the technical specifications of this solution. But there are people who have that knowledge. I have invited them (@visiblecode) to share their proposals below.

[birtles]

There is some previous discussion of this topic, thread starting here: https://lists.w3.org/Archives/Public/public-fx/2015JulSep/0034.html

[visiblecode]

Cubic beziers aren't a bad choice for timing functions. They're very easy to evaluate (just a few additions and multiplications). Also, most animation tools use some form of cubic spline to specify animation paths, which can be converted into cubic bezier easings with good fidelity.

There are two problems with cubic-bezier() as it is currently specced, however:

1. You can't specify a spline of multiple segments
2. cubic-bezier() is abusing 2d parametric curves to specify 1d functions

The first issue can be addressed by, essentially, just adding more points to the cubic-bezier() syntax. I believe this has already been proposed in various forms.

The second issue is pretty significant, though, and deserves some unpacking. cubic-bezier() is specified as a 2d bezier, a 2d parametric curve. This has a lot of implications, including:

1. Without additional constraints, the resulting curve isn't guaranteed to have a singular y value for a given x and can't be used to define a mathematical function. This has been dealt with by restricting x to the range [0, 1], but that doesn't actually address the root (no pun) problem -- because a cube root is involved there are still two extra solutions in the complex plane.
2. As mentioned, implementors must solve cube roots to work backwards from x to the implicit t parameter, before they get to the normal (and much simpler/exact!) bezier calculation to evaluate the easing function.
3. Tool authors who want to do a direct conversion of cubic spline segments to cubic-bezier()s have no use at all for the extra dimension, but must clutter the output with decimal approximations of 1/3 and 2/3.

The extra dimension isn't useful and makes life harder for implementors and tool authors alike. Ideally, instead of having to specify a two-dimensional curve like cubic-bezier(0.33333, 0.4, 0.66666, 0.6), we ought to have gotten one-dimensional curves, e.g. cubic-bezier(0.4, 0.6).

With one dimension, instead of having to work backwards from (normalized) time to an implicit t, the normalized time could be used directly as the input without having to solve cube roots first.

I guess if I were going to make a concrete proposal, I'd like to see syntax for specifying 1d bezier splines, in a format something like cubic-spline(c, [c,t,c,]* c) (where the cs and ts are numbers).

The first and last values are exactly like the control points in cubic-bezier(), except since we're only dealing with one dimension, the "control points" are specified by scalar values, rather than pairs of them.

Between those end "control values", you could have zero or more (control value, time, control value) triples representing intermediate knots with their adjacent control "points". The knot times (the middle value of the triples) would have to be monotonically increasing, in the range (0, 1).

As a concrete example, let's take the following complex easing:

 @keyframes example {
     from {
         some-property: 100px;
         animation-timing-function: cubic-bezier(0.33333, 0.25, 0.66666, 0.8);
     }
     50% {
         some-property: 75px;
         animation-timing-function: cubic-bezier(0.33333, 0.5, 0.66666, 0.6);
     }
     to {
         some-property: 50px;
     }
 }

Given the aforementioned cubic-spline() function, you could define the same animation more simply as:

 @keyframes example {
     from {
         some-property: 100px;
         animation-timing-function: cubic-spline(0.125, 0.4,0.5,0.75, 0.8);
     }
     to {
         some-property: 50px;
     }
 }

IMO this seems better for both export tooling and hand authoring.

One thing this proposal doesn't address are use cases where you want easings/animations with sudden jumps in them. (i.e. not C0 continuous) Having multiple intermediate CSS keyframes may still be a good answer for animations with discontinuities (rather than trying to define timing functions with discontinuities), though in order to do that there would need to be a mechanism to specify different incoming and outgoing property values for a CSS keyframe.

[visiblecode]

One of the things that bothers me about spring() specifically is that it requires finding an approximate solution to a differential equation, just like cubic-bezier() requires approximate solving of cube roots (only moreso). Browsers' approximations will vary, and there are performance/quality tradeoffs.

Using (1d) bezier splines for defining easing functions is far simpler than something like spring() computationally, and is easy to pull off with good precision. It's also relatively easy for tools to "bake" a physically-based animation (including ones much more complex than spring simulations) into a spline, which animation tools do anyway for better reproduceability and performance.

[visiblecode]

Thinking about it overnight, cubic-polybezier() might be a more accurate/less misleading name than cubic-spline(). I'll let other people bikeshed about naming though.

[visiblecode]

I'm also an idiot; those triples need to be four-tuples, there's an extra position parameter which I left out. So that example/proposal is wrong.

[visiblecode]

Adding the missing position parameter to the two control values, the syntax would be something like cubic-spline(c, [c, t, p, c,]* c). And that example becomes:

@keyframes {
    from {
        some-property: 100px;
        animation-timing-function: cubic-spline(0.125, 0.4, 0.5,0.5, 0.75, 0.8);
    }
    to {
        some-property: 50px;
    }
}

[visiblecode]

Having thought about this for a while and chatted with others, I think the minimum required features for representing complex easing functions boil down to:

• piecewise
• cubic functions
• with jump discontinuities
• of chosen directional continuity

Piecewise cubics are pretty much the common denominator for animation software, which either uses some form of piecewise cubic spline directly, or else curves which are reasonably easy to convert to piecewise cubics.

The main exception are animation tools which are doing physical simulations. Even then, some simple physical systems can be directly represented. For example, piecewise cubics are more than enough to precisely represent physically-accurate "bouncing ball" easings, since ballistic trajectories are simple parabolic arcs.

Oscillating spring (or pendulum) easings are sinusoidal and not exactly representable using simple polynomials, but you can still do a good job of approximating them by gluing multiple cubic segments together. If spring() became part of the standard, it might make sense to define it in terms of an appropriate piecewise cubic approximation.

I don't know how often jump continuities are really needed in easing functions, but they're certainly required if you want to be able to define a single easing function to recreate existing complex @keyframes animations which use timing functions like step-start() or step-end(). So it seems worth including them.

[visiblecode]

It's probably worth observing that motion curves (as opposed to timing/easing curves) in animation software are a little bit of a different story. There you're more likely to find NURBS or other more sophisticated types of curves which require a bit more work to approximate with cubic segments.

On the 1d versus 2d issue -- From what I can see so far, After Effects uses one-dimensional piecewise functions for timing curves, although I don't have a copy to play with directly. However, poking at the implementation of Blender, it appears Blender f-curve segments do work similarly to cubic-bezier(), using (suitably constrained) 2d bezier curves to define the 1d function.

[visiblecode]

Additional observation: while you can muddle through with multiple keyframes and cubic-bezier() easings for most purposes, if you need smooth curves joined by a jump discontinuity, @keyframes as currently specified can't be made to work. Best you can currently do to simulate such a jump discontinuity is to define two keyframes very close together and use a step easing between them:

@keyframes example {
    from {
        blah: 10px;
        animation-timing-function: cubic-bezier( ... stuff ... );
    }
    49.99999% {
        blah: 40px;
        animation-timing-function: step-end;
    }
    50% {
        blah: 100px;
        animation-timing-function: cubic-bezier( ... stuff ...);
    }
    to {
        blah: 150px;
    }
}

[birtles]

Best you can currently do to simulate such a jump discontinuity is to define two keyframes very close together and use a step easing between them

Yes, this is actually quite common to see. The Web Animations API allows you to overlap keyframe offsets so that you can add discontinuities and I believe there has been discussion in the past of adding syntax to allow you to do this in CSS keyframes (e.g. 50%+).

[visiblecode]

One suggestion made in the Slack discussion, which I had promised to document here, was to use an SVG path to specify an easing function.

In that case, any SVG path could be allowed, at least provided it ranged between 0 and 1 in the x dimension, and the curve's x component was continuous and monotonic over that range as well. In that case, the M operator could used to indicate jump discontinuities.

So, for two cubic segments joined by a jump discontinuity 50% of the way through, you might have something like: animation-timing-function: path("M 0,0 C 0.17,0.1, 0.33,0.2 0.5,0.4 M 0.5,0.8 C 0.67,0.8 0.83,0.9, 1,1")

I have mixed feelings about this, but it would work, aside from the issue of needing to indicate whether you wanted left- or right- continuity at any jumps.