# Gregory triangle patch, calculating $b_{i,j,k}$

I am trying to implement the Triangular Gregory patch from this paper. The goal is to create a $$G^1$$ continuity across the patch boundaries. So that the normal vectors of neighboring triangles are equal.

Right now I'm trying to understand how to calculate the Bézier ordinates: $$b_{i,j,k}$$ where $$i,j,k \geq 0$$ and $$i+j+k = n$$

The second thing I'm not sure about is the degree $$n$$. How to choose a good $$n$$?

If you want to implement triangular Gregory patches then you do not have to choose $$n$$ since a triangular Gregory patch is special kind of quartic B'ezier triangle, so $$n = 4$$. The patch has special inner control points which are constructed from cubic B'ezier curves on the edges of the triangles.
I take your question about how to compute the control points $$b_{ijk}$$ (ordinates) to be how to go from a standard triangular mesh to a mesh of Gregory patches. The easiest way to construct B'ezier control points (ordinates) from a triangular mesh is to first construct vertex normals (by averaging incident face normals) and then to use the procedure from PN triangles (Section 3.1) to construct the edge control points of each triangle. Each one of these edges is then a cubic B'ezier curve which can be used as input to construct, through the Chiyokura-Kimura basis patch technique, the inner control points of the Gregory patch.