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3D BÉZIER CURVE
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Curve studied by Bézier in 1954 and by de Casteljau.
Pierre Bézier (1910  1999): engineer for Régie Renault. 
Affine parametrization:
(i.e. )
where the
are the Bernstein polynomials: .
Polynomial algebraic curve of degree n. Curvature at A_{0 }: , torsion at A_{0 }: . 
Given a broken line (called control polygon, the A_{k} being the control points), the associated Bézier curve is the curve with the above parametrization; the curve goes through A_{0} (for t = 0) and A_{n }(for t = 1), and the portion that links these points is traced inside the convex hull of the control points; the tangent at A_{0 }is (A_{0}A_{1}) and the tangent at A_{n} is (A_{n1}A_{n}). 

Animation of the evolution of a cubic Bézier curve with 4 control points, where A_{1} and A_{4} are fixed, A_{2} and A_{3} move on lines. 
Recursive construction (de Casteljau algorithm):
The point
is the barycenter of
and where
are the respective current points of the Bézier curves with control
points and ;
moreover, the line
is the tangent at
to the Bézier curve.
Conversely, any polynomial 3D algebraic curve is a Bézier curve associated to a unique polygon, once the vertices of the polygon are chosen arbitrarily on the curve.
Their planar conical projections are the plane
rational Bézier curves.
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© Robert FERRÉOL 2018