Point-normal triangle

The curved point-normal triangle, in short PN triangle, is an interpoaltion algorithm to retrieve a cubic Bézier triangle from a regular flat triangles vertex coordinates and normal vectors. The PN triangle retains the vertices of the flat triangle as well as the corresponding normals. It was first introduced by A. Vlachos et al. in 2001 and is primarily used in the field of computer graphics. The usage of 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

With information of the given vertex positions ${\textstyle \mathbf {P} _{1},\mathbf {P} _{2},\mathbf {P} _{3}\in \mathbb {R} ^{3}}$ of a flat triangle and the according normal vectors ${\textstyle \mathbf {N} _{1},\mathbf {N} _{2},\mathbf {N} _{3}}$ at the vertices a cubic Bézier triangle is constructed. In contrast to the notation of the Bézier triangle page the nomenclatur follows G. Farin (2002), therefore we denote the 10 control points as ${\textstyle \mathbf {b} _{ijk}}$ with the positive indices holding the condition ${\textstyle i+j+k=3}$.

The first three control points are equal to the given vertices.$${\begin{aligned}\mathbf {b} _{300}&=\mathbf {P} _{1},&\mathbf {b} _{030}&=\mathbf {P} _{2},&\mathbf {b} _{003}&=\mathbf {P} _{3}\end{aligned}}$$Six control points related to the triangle edges, i.e. ${\textstyle i,j,k=\left\{0,1,2\right\}}$ are computed as$${\begin{aligned}\mathbf {b} _{012}&={\frac {1}{3}}\left(2\mathbf {P} _{3}+\mathbf {P} _{2}-\omega _{32}\mathbf {N} _{3}\right),&\mathbf {b} _{021}&={\frac {1}{3}}\left(2\mathbf {P} _{2}+\mathbf {P} _{3}-\omega _{23}\mathbf {N} _{2}\right),&&\\\mathbf {b} _{102}&={\frac {1}{3}}\left(2\mathbf {P} _{3}+\mathbf {P} _{1}-\omega _{31}\mathbf {N} _{3}\right),&\mathbf {b} _{201}&={\frac {1}{3}}\left(2\mathbf {P} _{1}+\mathbf {P} _{3}-\omega _{13}\mathbf {N} _{1}\right),&&\\\mathbf {b} _{120}&={\frac {1}{3}}\left(2\mathbf {P} _{2}+\mathbf {P} _{1}-\omega _{21}\mathbf {N} _{2}\right),&\mathbf {b} _{210}&={\frac {1}{3}}\left(2\mathbf {P} _{1}+\mathbf {P} _{2}-\omega _{12}\mathbf {N} _{1}\right)&\qquad {\text{with}}\quad \omega _{ij}&=\left(\mathbf {P} _{j}-\mathbf {P} _{i}\right)\cdot \mathbf {N} _{i}.\\\end{aligned}}$$This defintion ensures that the original vertex normals are reproduced in the interpolated triangle.

Finally the internal control point ${\textstyle (i=j=k=1)}$is derived from the previously calculated control points as$${\begin{aligned}\mathbf {b} _{111}&=\mathbf {E} +{\frac {1}{2}}\left(\mathbf {E} -\mathbf {V} \right)\\{\text{with}}&\quad \mathbf {E} ={\frac {1}{6}}\left(\mathbf {b} _{012}+\mathbf {b} _{021}+\mathbf {b} _{102}+\mathbf {b} _{201}+\mathbf {b} _{120}+\mathbf {b} _{210}\right)\\{\text{and}}&\quad \mathbf {V} ={\frac {1}{3}}\left(\mathbf {P} _{1}+\mathbf {P} _{2}+\mathbf {P} _{3}\right).\end{aligned}}$$