On-Line Geometric Modeling Notes


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 $ \left\{
{\bf P} _0,
{\bf P} _0,
{\bf P} _n
\right\}$ where

$\displaystyle {\bf P} (t) = \sum_{i=0}^{n} {\bf P} _i B_{i,n}(t)

for $ 0 \leq t \leq 1$.

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 $ {\bf P} (t)$, we can develop a new parameterization of the curve where $ t$ ranges between the values $ a$ and $ b$ by

$\displaystyle {\bf P} _{\left[ a,b \right]} (t) = {\bf P} ( \frac{t-a}{b-a} )

We note that $ {\bf P} _{\left[ a,b \right]}$ and $ {\bf P} (t)$ are exactly the same curve, but traversed through different ranges of $ t$. This change impacts only a few of the Bézier curve properties, namely


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 $ {\bf P} _{\left[ a,b \right]}(t)$ can all be related to the algorithms for the originally defined $ {\bf P} (t)$.

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Ken Joy