Mathematical formula for Bezier curves

Question

What mathematical formula does Blender use to calculate Bezier paths?

score 15 · Answer 1 · edited Jan 12 '17 at 03:13

15

For each segment of the path between two control points, it approximates a cubic Bezier curve:

Bezier Curve formula

where P₀ and P₃ are the control points and P₁ and P₂ are the handles.

Visual control points

edited Jan 12 '17 at 03:13

greatwolf

answered Feb 02 '14 at 02:31

Garrett

How do you generalize this for a curve of n segments? – Iván Dec 03 '23 at 08:54

ideasman42 · Answer 2 · 2015-09-09T07:13:46.103

10

Even though this isn't a question about Blender's code exactly, linking to Blender's code for calculating cubic bezier curves may help understanding.

2D masks can also use a different (more straightforward but slower) method where you can get any point on along the spline using a factor:
See BKE_mask_point_segment_co (trivial to adapt for 3D curves too).
3D curves use an optimized (less straightforward), fixed step, subdivision method:
See BKE_curve_forward_diff_bezier

To see how this method works, see wikipedia's description of Higher order curves

edited Sep 09 '15 at 07:13

answered Feb 02 '14 at 04:32

ideasman42

I read that bezier curves commonly are interpolated with Casteljau's algorithm. – pink vertex Feb 02 '14 at 10:38

2 Answers2