Narrowing-Cascade splines for control nets that shed mesh lines

Quad-dominant meshes are popular with animation designers and can efficiently be generated from point clouds. To join primary regions, quad-dominant meshes include non-4-valent vertices and non-quad regions. To transition between regions of rich detail and simple shape, quad-dominant meshes commonly use a cascade of $n-1$ triangles that reduce the number of parallel quad strips from $n+1$ to 2. For these cascades, the Narrowing-Cascade spline, short NC${}^n$, provides a new shape-optimized $G^1$ spline surface. NC${}^n$ can treat cascade meshes as B-spline-like control nets. For $n>3$, as opposed to $n=2,3$, cascades have interior points that both guide and complicate the construction of the output tensor-product NC${}^{}$spline. The NC${}^n$ spline follows the input mesh, including interior points, and delivers a high-quality curved surface of low degree.