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Point-normal triangle

From Wikipedia, the free encyclopedia

The curved point-normal triangle, in short PN triangle, is an interpolation algorithm to retrieve a cubic Bézier triangle from the vertex coordinates of a regular flat triangle and normal vectors. The PN triangle retains the vertices of the flat triangle as well as the corresponding normals. For computer graphics applications, additionally a linear or quadratic interpolant of the normals is created to represent an incorrect but plausible normal when rendering and so giving the impression of smooth transitions between adjacent PN triangles.[1] The usage of the PN triangle enables the visualization of triangle based surfaces in a smoother shape at low cost in terms of rendering complexity and time.

Mathematical formulation

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With information of the given vertex positions of a flat triangle and the according normal vectors at the vertices a cubic Bézier triangle is constructed. In contrast to the notation of the Bézier triangle page the nomenclature follows G. Farin (2002),[2] therefore we denote the 10 control points as with the positive indices holding the condition .

The first three control points are equal to the given vertices. Six control points related to the triangle edges, i.e. are computed asThis definition ensures that the original vertex normals are reproduced in the interpolated triangle.

Finally the internal control point is derived from the previously calculated control points as

An alternative interior control point was suggested in.[3]

References

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  1. ^ Vlachos, Alex; Peters, Jörg; Boyd, Chas; Mitchell, Jason L. (2001-03-01). Curved PN triangles. ACM. pp. 159–166. doi:10.1145/364338.364387. ISBN 978-1581132922. S2CID 5227025.
  2. ^ Farin, Gerald E. (2002). Curves and surfaces for CAGD : a practical guide (5th ed.). San Francisco, CA: Morgan Kaufmann. ISBN 9780080503547. OCLC 181100270.
  3. ^ USA 6,462,738, Kato, Saul S., "Curved Surface Reconstruction", published oct. 08, 2002