,
we can define a Bézier curve
of degree n by either of the following definitions:
On-Line Geometric Modeling Notes
Overview
The Bézier curve representation is one that is utilized most frequently in computer graphics and geometric modeling. The curve is defined geometrically, which means that the parameters have geometric meaning - they are just points in three-dimensional space. It was developed by two competing European engineers in the late 1960s to attempt to draw automotive components.
In these notes, we develop the mathematical description for the Bézier curve of arbitrary degree by generalizing the development for the quadratic and cubic Bézier curves, creating a parameterized version of the curve.
To get a postscript version of these notes look here.
Given the set of control points,
,
we can define a Bézier curve
of degree n by either of the following definitions:
where
are the
Bernstein polynomials
of degree n, and t ranges between zero and one - 
.
where
where t ranges between zero and one - 
.
Properties of the Bézier Curve
The Bézier curve has properties similar to that of the quadratic and cubic curve. These can be verified directly from the equations above.
and 
are on the curve.
is the line
. The tangent to the curve at the point
is the line 
.
is a convex combination of the
points 
and 
.
, 
, ..., 
are all on the curve only if the curve is linear.
Summary
Given a sequence of n+1 control points, one can specify a Bézier curve of degree n defined by these points. Two definitions of the curve can be given: an analytic definition specifying the blending of the control points with Bernstein polynomials, and a geometric definition specifying a recursive generation procedure that calculates successive points on line segments developed from the control point sequence.
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This document maintained by
Ken Joy
All contents copyright (c) 1996, 1997 |
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