Surface of Bézier

The advantageous method to define a curve by the data of check-points can extend on a surface.

Definition

Being given a matrix points of space $A_ {I, J}$ , the corresponding surface of Bézier is the whole of the points M generated by the values ranging between 0 and 1 of the variables U and v of the polynomial: $\ overrightarrow {O M} = \ sum_ {i=o} ^n \ sum_ {j=o} ^m v^i (1-v) ^ {(nor)} u^j (1-u) ^ {(m-j)} \ overrightarrow {O A_ {I J}}$

Properties

The points thus defined are obviously independent of the choice of the point O.
Le particular case N = 1 (or m=1) corresponds to regulated surfaces. If m = N = 1, one obtains a surface twice regulated, which is a plan if the four points are coplanar, that is to say a hyperbolic paraboloid. While considering on such a surface the intersections of two pairs of close rules, one realizes that the data of four points does not make that to define the nature of surface, but is also used to stop the borders of them. In the general case, the curves of Bézier corresponding to the subsets of points $A_ {0, J}$ , $A_ {N, J}$ , $A_ {I, 0}$ , and $A_ {I, m}$ define the borders of surface.

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