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BEZIER SURFACE
Pierre Bezier (1910  1999): engineer at Régie Renault. 
Affine parametrization: .
Polynomial surface of degree . 
Given points (called control points), the associated Bezier surface, or "tile", is the surface with the above parametrization; the portion of the surface for u and v 0 is included in the convex hull of the control points.
Example with n = 1 and m = 3 (6 control points)



If we write the point with parameter t of the Bezier curve with control points , and the point with parameter (u,v) of the Bezier surface with control points , then we have the relation: , which proves that the Bezier surface is the reunion of Bezier curves in two ways.
In particular, it contains the 4 Bezier curves with control points , , and .
For n = m = 1 (4 control points), the Bezier surface is none other than the hyperbolic paraboloid the generatrices of which are the 4 lines .
The Bezier surfaces are therefore special cases of spline surfaces.
There exists another
kind of Bezier surface, defined by a triangulation instead of a "tiling".


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© Robert FERRÉOL
2017