Isomorphic Triangulation of Simple Polygons Diana Garroway

Isomorphically triangulating two simple polygons.
The algorithm outlined here was presented in [4]. It will produce isomorphic triangulations of any two simple polygons.  We will show the algorithm using the following two polygons as an example.

Step 1: Adjust the cardinality of the polygons.
If the two polygons do not have the same number of vertices, then add vertices to the smaller polygon, along its edges, until they both have the same number.  The green vertices have been added in the following figure.

Step 2: Pick two corresponding vertices.
Before creating the triangulations, we decide upon the correspondence between the vertices of the 2 polygons.  By picking 1 vertex from each to be a corresponding pair, we then map the rest of the vertices in a clockwise order from these two.  In the figure below, the red vertices were chosen to be the corresponding pair.

Step 3: Triangulate and Find the dual tree
The next step is to triangulate each polygon, using any triangulation algorithm, and then to find the dual tree of each polygon.  This step is shown in the following figure.

Step 4: Find the centre vertex of the dual trees.
For each of the dual trees, find the centre vertex of the tree.  In the figure below, the centre vertex is shown as the large black dot.

Step 5: Assign weights to each of the polygon vertices.
For each of the vertices, assign it a weight based on the following method:
1.  For each vertex in the dual tree of the polygon, calculate the length of the path from it to the center vertex.
2.  Each of these vertices is at the centre of a triangle of the original triangulation.  Assign each triangle the path length +1 as its weight.
3.  Each of the vertices of the original polyon is a vertex of at least 1 triangle.  For each vertex, look at the triangle(s) that it is a part of and find                  the triangle with the minimum weight.  Assign the vertex to have this weight.

This concept is shown in the following figure.  Notice that each polygon vertex is a part of one or more triangles and the each vertex of the dual tree is at the centre of one triangle.  The weight given to each polygon vertex is the minimum path length of the dual tree vertices of the associated triangles, to the centre vertex (+1).  The weights are shown at each vertex in the following figure and the 3 vertices which form the center triangle are highlighted in black.

Step 6: Adjust the vertex weights to be the maximum of the corresponding pairs.
For each of corresponding vertex pairs, assign the one with lower weight the weight of the other vertex.  In the polygon on the left, since the vertex we chose as the corresponding vertex is also a center vertex, it is colored black instead of red, but it is still our corresponding vertex.

Step 7: Find the spiderweb connectivity for the vertex weights and embed it into each polygon.
For each polygon, take the vertex weights as the sequence for the spiderweb connectivity and embed the spiderweb into the polygons.  The figure below shows the spiderweb connectivity for the vertex weights and the following figure shows the connectivity embedded into the polygons.  Notice that for each vertex of the polygon, the mimimum path from that vertex to the center vertex the vertex weight.