On-Line Geometric Modeling Notes

Reparameterizing Bezier Curves

Overview

The Bézier curve is the representation that is most utilized in computer graphics and geometric modeling. This curve is usually defined by a set of control points where

for .

Running the parameter t from 0 to 1 gives a simple analytic and geometric definition of the curve. However, when we wish to examine general B-spline curves, which are piecewise Bézier curves, we will need to vary this parameter over an arbitrary interval. This is actually quite simple, and is discussed in the sections below.

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Defining the Reparameterized Curve

Given a Bézier curve , we can develop a new parameterization of the curve where t ranges between the values a and b by

We note that and are exactly the same curve, but traversed through different ranges of t. This change impacts only a few of the Bézier curve properties, namely

• .
• Using the chain rule, the derivative of the curve at a value t is equal to
  
  
• Subdividing the curve at the point , is equivalent to subdividing the curve at the point .

Summary

The Bézier curve is normally developed by using a parameter that ranges between 0 and 1. By a simple modification, we can reparameterize the curve so that t can range between any two values a and b. The resulting curve algorithms for can all be related to the algorithms for the originally defined .

This document maintained by Ken Joy Comments to the Author All contents copyright (c) 1996, 1997 Computer Science Department, University of California, Davis All rights reserved.