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RATIONAL BEZIER CURVE
Affine parametrization: Rational algebraic curve of degree £n. |
Given a sequence of weighted points - the control
points - ,
the associated rational Bezier curve is the curve with the aforementioned
parametrization ; the curve passing through A0
(for t = 0) and An (for
t= 1) and has its portion joining these points traced in the convex
envelope of the control points ; the tangent at A0
is (A0A1)
and the one at An is (An-1An).
When the weights (ak)
are equal, we get polynomial Bezier curves.
This curve is the conical projection (transformation )
of the 3D polynomial Bezier curve whose control points are the points Ãk
defined by
.
Therefore, rational Bezier curves include all rational
curves.
The parabola being the only polynomial conic, simple Bezier curves cannot represent a circle exactly, but it is possible with rational Bezier curves.
For example, if you take a control polygon formed by two orthogonal line segments of equal length, then the simple Bezier curve is a parabola; if you double the weight of one of the extremities, you get a circle:
In bold, the rational Bezier curve: a circle.
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© Robert FERRÉOL
2017