What math Adobe After Effects uses for animation?

Question

I need to recreate animation curves from adobe after effects in my program.

For instance, the red curve on this chart (with corresponding speed chart in white):

I tried to represent it with hermite spline using the value and speed extracted from both ends of the curve. But since AE curve has the additional "influence" parameters my hermite spline failed to reproduce the result.

What kind of curves do they use? It doesn't look like a cubic spline.

Answer 1

After Effects' graph editor uses Bezier curves, which use influence points (attached to handles for better visual connection) to influence the curve's path between keyframes.

Control points

A Bezier curve can be described using a mathematical formula: Given the coordinates of control points P_i: the first control point has coordinates P₁ = (x₁, y₁), the second: P₂ = (x₂, y₂), and so on, the curve coordinates are described by the equation that depends on the parameter t from the segment [0,1].

For 4 control points:

P = (1−t)³P₁ + 3(1−t)²tP₂ +3(1−t)t²P₃ + t³P₄

Lots more detail about Bezier curves can be found here: https://javascript.info/bezier-curve