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BEZIER CURVE

Curve studied by Pierre Bezier in 1956 and, independently, by Paul de Casteljau in 1958.
Pierre Bezier (1910 - 1999): engineer for Régie Renault.
See Centrale PC 99 exam paper.

 
Affine parametrization:  (i.e. ) where  are the Bernstein polynomials: .
Polynomial algebraic curve of degree £ n.
Curvature at A0 .

Given a broken line  (called control polygon, (Ak) being the control points), the associated (polynomial) 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).

Every line intersects the curve a number of times smaller than or equal to the number of vertices in the control polygon.

Recursive construction (algorithm invented by Casteljau, engineer for Citroën):
The point  is the barycentre of  and  where  are the current points respectively on the Bezier curves whose control polygons are  and  ; moreover, the line  is tangent in  to the Bezier curve.

Conversely, every polynomial algebraic curve is a Bezier curve with a one-to-one correspondence with the control polygon, once its extremities are chosen arbitrarily on the curve.

Here is an example of a closed Bezier curve and its control curve in red:

Is it a circle?

No! Since there are 6 control points (with A0 = A5), it is a quintic ; here is an (almost) complete flattened representation:

Bezier curves are used in drawing software such as Illustrator, with the tool "pen tool"; on the left, the cubic Bezier curve with ABCD as control polygon was traced using this tool.

Bezier curves are examples of spline curves and can be generalized into rational Bezier curves.
See also Lagrange curves, 3D Bezier curves, Bezier surfaces.

Link: A superb java applet that traces Bezier curves (make sure to tick "construction" to see the construction with Casteljau's algorithm).

Also look at:
 www.people.nnov.ru/fractal/Splines/Bezier.htm
demonstrations.wolfram.com/SimpleSplineCurves
 
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© Robert FERRÉOL  2017