Written by Vincent

Rapidly calculating Bezier points

The standard cubic Bezier curve is given by
B(t) = (1-t)³p0 + 3(1-t)²tp1 + 3(1-t)t²p2 + t³p3
where p0, p1, p2, p3 are the control points and t is in the interval [0, 1].

Very elegant, but not very practical. Too many multiplications and additions to be used in a fast-paced environment such as game development. If it’s a 3D Bezier curve, 3 separate calculations are needed for the x-, y- and z-component for a given t value. What we need is a simplication of the equation.

Expanding the polynomial equation, we have
B(t)
= (1 – 3t + 3t² – t³)p0
+ (3t -6t² + 3t³)p1
+ (3t² – 3t³)p2
+ t³p3

Rearranging the coefficients of tⁿ, we have
B(t)
= t³(-p0 + 3p1 – 3p2 + p3)
+ t²(3p0 – 6p1 + 3p2)
+ t(-3p0 + 3p1)
+ p0

The coefficients can now be reduced to constants by precalculating them, and calculation of a Bezier point takes less computation.

In matrix form, the equation looks like
Coefficient matrix form of Bezier curve

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Tags: bezier-curve, game-development, polynomial