Bezier Curves

Bezier curves were developed by Pierre Bezier for designing Renault automobile bodies. Bezier curves use 4 input points with the tangent vectors at the end points being determined by line segments, e.g., the line segment from P1 to P2

Therefore the relationship between the Hermite Geometry matrix, GH and Bezier geometry matrix GB is:

    |P1|   | 1 0  0 0| |P1| 
GH =|P4| = | 0 0  0 1| |P4| 
    |R1|   |-3 3  0 0| |P3| 
    |R4|   | 0 0 -3 3| |P4| 
= MHB*GB

Putting this back into our eqn. for Hermite coordinates we get

x(t) = TMHGHx = TMHMHBGBx

Let MB = MH * MHB then we have Bezier form x(t) = TMBGBX with:

     |-1  3 -3 1 |
MB = | 3 -6  3 1 |
     |-3  3  0 0 |
     | 1  0  0 0 |

So we have an equation for x(t) with similar equation sfor y(t) and z(t)). As with the Hermite case we multiply T with Mb and get:

TMb = [(-t^3 + 3t^2 - 3t + 1) (3t^3 - 6t^2 + 3t) (-3t^3 + 3t^2)(t^3)]

Post Multiply by GB and get (the t terms are the Bezier blending functions):

x(t) = P1x(-t^3 + 3t^2 - 3t + 1) + P2x(3t^3 - 6t^2 + 3t) + P3x(-3t^3 + 3t^2) + P4x(t^3)

We can derive similar equations for y and z. Note that Bezier curves, unlike Hermite curves, have a bounding convex hull, defined by the control points. Also, for curve segments, we have C(1) continuity if and only if P3, P4, P5 (etc.) are colinear.

A program illustrating Bezier Splines.


Splines main page
HyperGraph Table of Contents.
HyperGraph Home page.

Last changed June 22, 1999, G. Scott Owen, owen@siggraph.org