** 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

Running the parameter from 0 to 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 ranges between the values and by

- .
- Using the chain rule,
the derivative of the curve
at a value
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 . By a simple modification, we can reparameterize the curve so that can range between any two values and . The resulting curve algorithms for can all be related to the algorithms for the originally defined .

2000-11-28