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What mathematical formula does Blender use to calculate Bezier paths?

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2 Answers 2

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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

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  • $\begingroup$ How do you generalize this for a curve of n segments? $\endgroup$
    – Iván
    Dec 3, 2023 at 8:54
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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

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