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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 A_{0
}: . |

Given a broken line
(called control polygon, (*A _{k}*) being the control points),
the associated (polynomial) Bezier curve is the curve with the aforementioned
parametrization ; the curve passing through

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 *A*_{0}
= *A*_{5}), 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