Fundamentals to Detect Tensor Product Bézier Patches in Euclidean 3-Space

The aim of this paper is to deliver the fundamentals to detect Bézier patches of scanned objects

based on their normal congruence. In five-dimensional real projective space (P5), we introduce

a new approach for tensor product (TP) Bézier patch representation. For this reason, we use

Plücker coordinates which are a way to assign six homogeneous coordinates to each line in

three-dimensional projective space (P3). Derivatives, normal vectors of Bézier patches and some

of geometric properties of these patches are discussed. Further, the special case, biquadratic Bézier

patch is introduced. The Plücker coordinates of the normal congruence of the patch are functions of

order 14 in general, because of that high degree, it seems not to be of practical use to calculate the

focal points of the normal vectors of the patch in general. We try these calculations for the biquadratic

patches (m=n=2). Finally, we present a computational example to compute the two focal points of a normal of this patch.

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