Closest point on a cubic Bezier curve?

graphics geometry bezier spline curve
Adrian Lopez picture Adrian Lopez · Apr 30, 2010 · Viewed 29.2k times · Source

How can I find the point B(t) along a cubic Bezier curve that is closest to an arbitrary point P in the plane?

Answer

Adrian Lopez picture Adrian Lopez · May 1, 2010

After lots of searching I found a paper that discusses a method for finding the closest point on a Bezier curve to a given point:

Improved Algebraic Algorithm On Point Projection For Bezier Curves, by Xiao-Diao Chen, Yin Zhou, Zhenyu Shu, Hua Su, and Jean-Claude Paul.

Furthermore, I found Wikipedia and MathWorld's descriptions of Sturm sequences useful in understanding the first part of the algoritm, as the paper itself isn't very clear in its own description.

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