next curve | previous curve | 2D curves | 3D curves | surfaces | fractals | polyhedra |
3D BÉZIER CURVE
Link to a figure manipulable by mouse
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_{n-1}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.
next curve | previous curve | 2D curves | 3D curves | surfaces | fractals | polyhedra |
© Robert FERRÉOL 2018