Created 2004-09-06 Modified 2007-01-01
Chelton Evans
Polygon Meshes
Intro
Indexing Points
Comparison of Triangle Table, Edge Table and the Double Connected Edge List
Generalized Triangle/Tetrahedron Mesh
Polygon List with Edges or Edge Table
Doubly Connected Edge List
Polygons as List of Boundary Polygons
Dual Mesh
"... the more explicitly the relations among polygons, vertexes, and edges are represented, the faster the operations are and the more space the representation requires." [1]
I am of the view that primitive shapes are the best tools for dealing with polygons. Operators defined on primitive shapes and meshes are the building blocks of other methods and views.
Polygon meshes are used to represent surfaces. They generalize to higher dimensions. The focus here is on triangle representations, because ultimately the surface has triangles for geometry which are processed. I will look at two traditional data structures for polygon representations and use them with triangles. Further I will come up with a new data structure called the triangle table to do the same thing and more.
The triangle table uses convexity and the order of the points/edges to define the structure.
All useful polygon representations use pointers, so a particular data structure may have certain pointers to edges, vertexes, and polygons. However not all pointers are created equal and just because a structure has a pointer is completely different to a structure which implements that pointer on an array. Integer pointers or integer indexing into stacks is probably one of the oldest techniques. Arrays are the most common data structure and can implement all other data structures.
So while it may appear that I have gone on a tangent, polygons are best implemented with integer indexes in arrays. Hybrids are extremely good too where the C++ pointers can be combined with arrays for some interesting data structures, but integer calculations are superior in the sense that we may be able to define inverse operations on integer indexes more easily than on a C/C++ style pointer arithmetic.
The most common polygon representation. If a program can output a list of points it can output a list of indexes to those points instead. This makes it ideal for communication between algorithms as their output can be compared. For example examining different tessellation output on the same data.
| Polygon | Points |
| 1 | 4,5,1 |
| 2 | 4,1,2 |
| 3 | 4,2,3 |
| Polygon | Points |
| 1 | 1,2,7,6,5 |
| 2 | 1,2,4,3,8,9,6,7 |
| Index | Points | Neighb. |
| 1 | 4,5,1 | 0,0,2 |
| 2 | 4,1,2 | 1,0,3 |
| 3 | 2,3,4 | 0,0,2 |
The neighbors correspond to the right of the directed edge. eg 4,5,1 produces edges 45, 51, 14 with neighbors 0,0,2 where 0 is no neighbor.
An extension to the polygon list is to include the edges.
| Poly | Edges |
| 1 | 6,5,7 |
| 2 | 5,4,3 |
| 3 | 3,1,2 |
The neighbors are left and right respectively of the directed edge.
| Edge | Points | Neigh |
| 1 | 2,3 | 3,0 |
| 2 | 3,4 | 3,0 |
| 3 | 4,2 | 3,2 |
| 4 | 1,2 | 2,0 |
| 5 | 4,1 | 2,1 |
| 6 | 5,1 | 1,0 |
| 7 | 4,5 | 1,0 |
| Edge | Points | Neigh | Next E0 | Next E1 |
| 1 | 2,3 | 3,0 | 3 | 2 |
| 2 | 3,4 | 3,0 | 1 | 7 |
| 3 | 4,2 | 3,2 | 2 | 4 |
| 4 | 1,2 | 2,0 | 5 | 1 |
| 5 | 4,1 | 2,1 | 7 | 6 |
| 6 | 5,1 | 1,0 | 7 | 4 |
| 7 | 4,5 | 1,0 | 5 | 6 |
The Next Ei is counter clockwise from the point.
| Poly | Edge Entree |
| 1 | 7 |
| 2 | 4 |
| 3 | 2 |
All share point table.
| Index | x y z |
| 1 | |
| 2 | |
| . | ... |
The second and third data structures have an edge and vertex perspective. They are the standard data structures.
The first data structure is homegrown and uses half the memory. More importantly its perspective is that of the shape, not the edges or vertexes.
I am amazed that this doesn't appear much more in the literature - especially since it uses half the memory!
A discussion of data structures always involves generality. Clearly the standard structures are "the most general" because every shape has vertexes and edges.
But what are they used for? Algorithms! So would it not be better to have a data structure that suits algorithms better? This sounds like it could shock a few people - common sense. To be direct because I am sure the subtle argument was missed, this IS GENERAL because its the perspective of the algorithms! Indeed this perspective shows how the other ones are NOT general by its existence. The end result is general from who's perspective, and we can always use more perspectives when algorithms are concerned.
Thats why I thought up the triangle structure - because I am designing algorithms and I wanted a simple way to think of transversing a surface.
After speaking with some computer scientists I get the feeling that they have no idea of generality in mathematics. Generality is not a standard, just because the Jones do it this way ... does not mean that is the only way of doing it. Instead generality is taking some property and seeing the flow.
As I said previously looking at vertexes and edges is not the only generalization of meshes, even if every single book I know does this to a large extent they are wrong not to look at problems differently. You do not know what you are talking about if you stick to these structures. They are at times too difficult to wield and are not general enough.
For those of you with an open mind the following is really simple, perhaps already done by many other people but I did not know where to find it in the literature.
The work of tessellation in 2D and 3D with the most primitive regions in their dimension is working with the triangle and tetrahedron respectively. They are the building blocks. How can these be related and represented?
When I used a triangle mesh with triangle representation that
took into account triangles windings the triangle data structure was
born. eg 1 4 5 1 0 0 2 for index, 3 points and 3 neighbors.
This had a counter clockwise winding of points 4 5 1 with the order
of neighbors from the position of the next edge in the winding.
While this structure was successful it did not generalize to 3D. What does adjacent mean to a point in 3D. Further it was not till I started working in 3D that I realized this, else I would have implemented the data structure slightly differently. Instead of having the next or neighboring edge I would have the opposite edge. The counter clockwise winding remains but this time the data structure does generalize from 2D to 3D with the same meaning. The generality has been captured and makes sense.
The mesh itself acted also as an iterator with a current state or current pointer, similar to a pointer in a doubly linked list. With triangles from the base there are two sides - say left and right. Transversal algorithms could choose one of the two when moving through the mesh, the third side could be the previous move.
This generalized for tetrahedrons which have three possible moves. If you like you could further define the orientation but this way was good enough for my purposes.
In 2D an edge is opposite a point.
In 3D a surface is opposite a point.
The previous triangle table is easily rewritten.
| Index | Points | Neighb. |
| 1 | 4,5,1 | 0,2,0 |
| 2 | 4,1,2 | 0,3,1 |
| 3 | 2,3,4 | 0,2,0 |
Between the point and the opposite line/face is within the shapes region. For a tetrahedron the first 3 points describe the base, the next point is on the side of the base with the arms cross product from the first point.
are points of tetrahedron,
The boundary is always one dimension less and have the boundary in an anticlockwise winding then the figure can be described in the following way.
Polygon Points Neighbor 1: 2,1,6,7 2 2,7,5 0 1,2,5 0 5,6,7 0 1,6,5 0 2: 1,6,7,2 1 2,4,9,7 0 1,2,4,3 0 8,9,7,6 0 3,8,6,1 0 3,4,9,8 0
Although this representation seems unusual, it has the property that each face is mapped to another polygon.
For simplexes this has the special property that the face need only be represented by one number - the number opposite the associated faces vertex. Hence simplex data structures have the links with less memory.
Since the links are unique, we can use it as a key or function argument to operations on the neighbor. eg given 1,6,7,2 what is the local index of the neighboring face of polygon 2 which has an answerer of 1 if we have +ve indexes only (count from 1) and use zero as there being no such face hence if 0 were returned the two polygons would have no faces in common.
What is unusual is that this above would probably process quite quickly! Its a strange world.
A dual mesh is made from a simplex mesh by making each simplex a node and connecting it with the neighboring nodes.
In its simplest form this uses no geometry. For example are the nodes simplexes or do we construct a point inside the simplex and make this the node? Here the node know is a point in space and not a simplex in space.
With this definition linked simplexes is a dual mesh because it forms such connections.
The dual of a triangular mesh produces a mesh of polygons. It would be interesting to consider this as a linked structure where each polygon is uniquely associated with a point in the triangle mesh. The points on the outside are not associated with any of the polygons. Let each face of the polygon point to a point. Have a table of points pointing to the polygon that they are uniquely associated with.
While this structure is clean it does not answers questions for points not on the boundary. For example for non-leaf points - those points not on the boundary, by iterating about their polygon all their neighboring points can be visited. It would be nice to have this property for all points so that you could iterate about any point in the simplex mesh.
If we extend the definition so that all the points in the triangle mesh have a unique polygon then the polygons are divided into two types - those on the inside and then those on the boundary. The polygons on the boundary are infinite. Let the inner points be generated from the average of the three points. Let the outer polygon be defined by generating edges perpendicular to the mesh and through the newely generated dual point within the mesh. This lets the space be uniquely partitioned, something which a Voronoi Diagram dotes.
For partition algorithms this is very useful to divide the spaces into unique regions.
What I with for is a practical algorithm/function to reverse the dual operation to return the original simplex mesh back. Is this possible for convex meshes? Actually I believe that this is the property of the Voronoi Diagram. ie the Voronoi diagram and Delaunay triangulation are eachothers duals. It would be good to see software that can convert between the two meshes.
Another look at dual meshes is as a new maths. For example a two adjacent triangles mesh has a two points connected by a line dual. A three fan triangle has a three point line as a dual. Is there a maths theory of any use here? Is this already known? Is there any maths here?
Narrow this for a convex hull. In 2D apply to inner simplexes. The outer polygons are created by having rays extend from each outer edge point off to infinity.
The centroid is always inside the triangle. For an arbitary triangle on the convex hulls boundary, construct a line at right angles to the boundary starting from the centroid and extending out to infinity.
While this partitions the space, is it a reversible operation?