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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 triangle (under the usual polygon definition of a triangle) and normal vectors. The PN triangle retains the vertices of the original 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.

With information of the given vertex positions ${\textstyle \mathbf {P} _{1},\mathbf {P} _{2},\mathbf {P} _{3}\in \mathbb {R} ^{3}}$ of a 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. Following the notation and nomenclature of G. Farin (2002),^[2] one denotes 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.$${\displaystyle {\begin{aligned}\mathbf {b} _{300}&=\mathbf {P} _{1},&\mathbf {b} _{030}&=\mathbf {P} _{2},&\mathbf {b} _{003}&=\mathbf {P} _{3}\end{aligned}}}$$ Six control points related to the triangle edges, i.e. ${\textstyle i,j,k=\left\{0,1,2\right\}}$ are computed as$${\displaystyle {\begin{aligned}\mathbf {b} _{012}&={\frac {1}{3}}\left(2\mathbf {P} _{3}+\mathbf {P} _{2}-\omega _{32}\mathbf {N} _{3}\right),&\mathbf {b} _{021}&={\frac {1}{3}}\left(2\mathbf {P} _{2}+\mathbf {P} _{3}-\omega _{23}\mathbf {N} _{2}\right),&&\\\mathbf {b} _{102}&={\frac {1}{3}}\left(2\mathbf {P} _{3}+\mathbf {P} _{1}-\omega _{31}\mathbf {N} _{3}\right),&\mathbf {b} _{201}&={\frac {1}{3}}\left(2\mathbf {P} _{1}+\mathbf {P} _{3}-\omega _{13}\mathbf {N} _{1}\right),&&\\\mathbf {b} _{120}&={\frac {1}{3}}\left(2\mathbf {P} _{2}+\mathbf {P} _{1}-\omega _{21}\mathbf {N} _{2}\right),&\mathbf {b} _{210}&={\frac {1}{3}}\left(2\mathbf {P} _{1}+\mathbf {P} _{2}-\omega _{12}\mathbf {N} _{1}\right)&\qquad {\text{with}}\quad \omega _{ij}&=\left(\mathbf {P} _{j}-\mathbf {P} _{i}\right)\cdot \mathbf {N} _{i}.\\\end{aligned}}}$$This definition 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 $${\displaystyle {\begin{aligned}\mathbf {b} _{111}&=\mathbf {E} +{\frac {1}{2}}\left(\mathbf {E} -\mathbf {V} \right)\\{\text{with}}&\quad \mathbf {E} ={\frac {1}{6}}\left(\mathbf {b} _{012}+\mathbf {b} _{021}+\mathbf {b} _{102}+\mathbf {b} _{201}+\mathbf {b} _{120}+\mathbf {b} _{210}\right)\\{\text{and}}&\quad \mathbf {V} ={\frac {1}{3}}\left(\mathbf {P} _{1}+\mathbf {P} _{2}+\mathbf {P} _{3}\right).\end{aligned}}}$$

An alternative interior control point $${\displaystyle {\begin{aligned}\mathbf {b} _{111}&=\mathbf {E} +5\left(\mathbf {E} -\mathbf {V} \right)\end{aligned}}}$$ was suggested in Kato (2002).^[3]