BEZIER SURFACE
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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 (8 control points)
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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