Created 2007-01-04 Modified 2007-02-16
Chelton Evans
Dual Convex Mesh Algorithm
Intro
Definitions
Partitioning a Plane from a Convex Mesh
Calculating Centroid from Surface
Algorithm
Generalized Duality
Dual Graph
References
A dual of a mesh is some point x inside a mesh element of mesh M1 taken to be a point in mesh M2. If two points in M2 share a surface they are connected.
Dual meshes are also used in partitioning. We need to consider polygons with infinite area when partitioning the plane. For a convex 2D mesh the dual is a partition of the plane.
The notion of an inverse with regard to the dual exists. Applying some function f to the mesh, and then applying the dual operator to the mesh to restore the original mesh.
is said to be the dual of mesh
Iterate over the shapes in M0
Choose an arbitary point inside the shape
adding it to M2
Iterate over the shapes in M0
If this shape has a neighbor
(ie the shapes share an edge) connect
their interior points by a straight line which is added
to M2
Here is an example of a convex triangular mesh and its dual.
Here is a partition of the plane formed by extending lines from the surface triangles at right angles to the surface.
The dual can be represented by linked shapes. But when a shape has infinite area it becomes necessary to represent it. Here is one possibility where all 0 linkes are treated as the polynomial going to infinity in that direction.
Note that the dual of the infintite partition is the original convex mesh. From a logical point of view the partition is the meshes dual - assuming all outside infinite regions are convex.
From a computational point of view it would be nice to construtct betoh
For a convex mesh its partition has infinite polyogons at its surface. Since these can be identified as such, they can be used to get back the hull of the convex mesh.
For example we can define a function that for edges on the hull creates a dual point outside the hull. The start of the line in the triangle is the centroid and it intersects the edge at right angles. The point a is choosen so that a ratio of the distances to the edge is constant and hence give point a and centroid c the edge can be reconstructed. Doing this for all the duals edges and finding the intersection points generates the convex hull.
A similar function can be constructed for 3D.
Solve the outer points on the hull of the convex mesh. Then the problem becomes given the hull and dual construct the mesh.
For each dual point construct a triangle.
Link the triangles together.
Place the triangles in a list.
Iterate over the triangles list
case: Zero or One known point:
Push the current triangle to the end of the list.
Continue
case : Two known points:
Solve for the third point.
Continue
case : Three known points:
Continue
The dual mesh is talked about in regard to 2D, why not generalize this to arbitary dimensions. The idea of shapes sharing a common surface being linked either with a line or a surface. Where I define a surface as 1 dimension lesss that the space.
I am not clear how this is generalized.
In 3D lines extending out from the surface shapes would not intersect for a convex mesh in 3D. The surface could be mapped to a sphere and retail the surface partition information. Sending a line through the centroid of each of the surfaces 2D shapes and connecting adjacent lines by a line that moves over time to sweep a surface.
For example given two finite lines in space, create a surface by linearly interpolating a point on both lines. Then join these points with a straight line. Now for the above problem one end of the lines is infinite so the surface is infinite.
Now the 3D space is partitioned. This surface is not a plane. Although a partition is enough for many practical applications so there is no need for an inverse to the partition.
A dual graph is similar without the restriction that a point from G to a neighboring point does not have to pass throught the face of G* where G* is the dual.
Put simply the mesh does not have to be tiled with convex shapes. This may also generalizes into any dimension, though I would like a geometric reasoning program to help.
Ore[1] defines the dual by constructing an edge though each edge in G. However his dual construction is not unique. For example a dual G* can be constructed with one more point. This is inherent in the problem because the outside points of G* can be interpreted as having gone to infinity.
I have quoted Ore's graph and showed another pattern - that being the red points are in one to one correspondence with the original graph the edges of G. This can be important if we wish to have operators such that dual(dual(G*))==G*. With the dual that I am suggesting this property is preserved. With Ore's dual I can not see how this property can be implemented.
