Robust and Controllable Quadrangulation of Triangular and Rectangular Regions

Abstract

Figure 1: Basic topological patterns used for quadrangulating rectangular and triangular regions. Vertices with a red circle represent corners. Vertices with orange and blue fill represent irregular vertices with valence three and five, respectively. pX and pY represent a regular grid of quadrilaterals padded to the corresponding side of the rectangle (only shown with pattern 1 for brevity). pB , pR, and pL are similarly defined for the triangle. α and β represent the number of times an edge flow is inserted at these locations. Compared to the patterns proposed by Nasri et al. [2009], all of our patterns have at least one side that has only one edge, which is important for guaranteeing valid quadrangulation of the region even when the specified number of edge subdivisions is very small and extreme.