EDIT 2014:This article ends up in an updated library available on NPM (`bezier-easing`

) and available on Github. It has been used by Apple for the mac-pro page and by Velocity.js. You can also find its usage in the glsl-transition examples.

Many animation libraries are today using **easing functions** – functions of time returning a progression percentage value. This is required to perform such cool effects:

But most of these libraries implement a huge collection of functions. We will see how we can generalize them with bezier curves.

For instance, we use to do this:

```
EasingFunctions = {
linear: function (t) { return t },
easeInQuad: function (t) { return t*t },
easeOutQuad: function (t) { return t*(2-t) },
easeInOutQuad: function (t) { return t<.5 ? 2*t*t : -1+(4-2*t)*t },
easeInCubic: function (t) { return t*t*t },
easeOutCubic: function (t) { return (--t)*t*t+1 },
easeInOutCubic: function (t) { return t<.5 ? 4*t*t*t : (t-1)*(2*t-2)*(2*t-2)+1 },
easeInQuart: function (t) { return t*t*t*t },
easeOutQuart: function (t) { return 1-(--t)*t*t*t },
easeInOutQuart: function (t) { return t<.5 ? 8*t*t*t*t : 1-8*(--t)*t*t*t },
easeInQuint: function (t) { return t*t*t*t*t },
easeOutQuint: function (t) { return 1+(--t)*t*t*t*t },
easeInOutQuint: function (t) { return t<.5 ? 16*t*t*t*t*t : 1+16*(--t)*t*t*t*t }
}
```

Defining such functions is lot of math fun but it is very **specific** and not really customizable. Hopefully, we can generalize these easing functions. With **Bezier curves**.

In fact, this work has already been done in CSS Transitions and CSS Animations specifications! You can use `transition-timing-function`

CSS property and give a `cubic-bezier(x1, y1, x2, y2)`

value (all **ease, linear, ease-in, ease-out, ease-in-out** values are just fallbacking on this cubic-bezier usage).

In a bezier curve based easing function, the X axis is the **time axis** whereas the Y axis represents the **percentage of progress** of the animation.

The two points P1 and P2 are called **handles** and you can (exclusively) control their X and Y positions to generate every possible cubic timing function.

### Live demo

Try to interact with the handles:

## Implementation

Ok, so, this bezier curve concept is great but how can I implement it?

I’ve read here how simple is it to **compute many points of a Bezier curve** and potentially draw them:

```
function B1(t) { return t*t*t }
function B2(t) { return 3*t*t*(1-t) }
function B3(t) { return 3*t*(1-t)*(1-t) }
function B4(t) { return (1-t)*(1-t)*(1-t) }
function getBezier(percent,C1,C2,C3,C4) {
var pos = new coord();
pos.x = C1.x*B1(percent) + C2.x*B2(percent) + C3.x*B3(percent) + C4.x*B4(percent);
pos.y = C1.y*B1(percent) + C2.y*B2(percent) + C3.y*B3(percent) + C4.y*B4(percent);
return pos;
}
```

But it’s not enough. We need to project a point to the Bezier curve, in other words, we need to get the Y of a given X in the bezier curve, and we can’t just get it with the `percent`

parameter of the Bezier computation.

**We need an interpolation.**

### Deep into Firefox implementation

In Mozilla Firefox, The bezier curve interpolation is implemented in nsSMILKeySpline.cpp : .

What we can learn from it is:

- A first optimization store
**sample values of the bezier curve**in a small table used to roughly find a initial X guess. - Then, it use two different implementation strategies: One use the Newton’s method and the other is just a dichotomic search (binary subdivision).
- A
**criteria**based on the**slope**give the best strategy to take.

These sub-optimizations probably make the difference for the C++ version but are not really relevant for the JavaScript implementation. Moreover, I have only used the Newton’s method algorithm.

And this is the code:

Now we can just alias some classic easing function – like CSS does.

I’m working on the next version of Slider.JS which relies on 3 different technologies for image transitions: **CSS Transitions**, **Canvas** and **GLSL shaders (from WebGL)**.

I have now found **a common way to describe easing functions for both CSS-based and Javascript-based animations**!

This example has shown that sometimes, finding a larger solution for a problem is more interesting than having specific solutions.

**This is called the Inventor’s paradox.**