Geometric form of Simplicial Complexes

In the geometric form each vertex has a cartesian coordinate instead of a symbol or an index.

In this form we have additional constraints on what makes a valid simplicial complex, the faces are not allowed to overlap.

In 1 Dimension

  Not valid simplicial complex Convert to simplicial complex
These lines overlap so they do not form a valid simplicial complex. We can detect this because line 'A' has an endpoint inside line 'B' and line 'B' has an endpoint inside line 'A' simplicial complex simplicial complex
It is not always the case that each line has an endpoint inside the other. In this example line 'A' does not have an endpoint inside line 'B' and line 'B' has two endpoints inside line 'A'. So we have to check every endpoint and every line to see if the endpoint is inside the line. simplicial complex simplicial complex

So when we are testing for an invalid overlap, we need to check each line segment that make up the triangles, if two line segments overlap then we have a non-valid case.

In 2 Dimensions

  Not valid simplicial complex Convert to simplicial complex
These triangles overlap so they do not form a valid simplicial complex. simplicial complex simplicial complex
It is not always the case that an overlap will have a vertex inside the triangle. This example is not valid but does not have a vertex inside either triangle. simplicial complex simplicial complex

So when we are testing for an invalid overlap, we need to check each line segment that make up the triangles, if two line segments overlap then we have a non-valid case.

We can test for overlap of line segments by:

  • Find intersection point of the two lines.
  • Check if the intersection is inside both line segments.
line segments

Finding the intersection point we need to solve two simultaneous equations.

In 3 Dimensions

  Not valid simplicial complex
This is a lot more difficult, we need to test that no triangles in tetrahedron 'A' pass through the triangles in tetrahedron 'B'. overlap simplical complex

There are well known algorithms for determining whether 2 triangles in 3 dimensions intersect. For instance:

more info on site here.

In 'n' Dimensions

Is there a generic test that will work in any number of dimensions?

Optimisation

These checks for overlapping faces could use a lot of CPU time and therefore ways to speed things up could be worthwhile, the following approaches are possible:

Bounding Boxes

It is very simple to generate an 'n' dimensional rectangle (bounding box) around each faces and it is very simple to check these rectangles for overlap. If these bounding boxes do not overlap then the faces inside them cannot overlap. So we can test all combinations of bounding boxes for overlap and then only those that do will require the more expensive (in CPU time) tests.

Moving Line

Rather than test every combination of faces we imagine a line moving across the space and check for overlaps as we go.

flag flag flag flag flag flag Graphs, Surfaces and Homology - Peter Giblin.
Builds up to homology groups via graphs and simplicial complexes.

 


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Correspondence about this page

Book Shop - Further reading.

Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them.

flag flag flag flag flag flag Mathematics++: Selected Topics Beyond the Basic Courses (Student Mathematical Library) Kantor, Ida.

Chapters:

  1. Measure
  2. High Dimensional Geometry
  3. Fourier Analysis
  4. Representations of Finite Groups
  5. Polynomials
  6. Topology

Chapter 6 - Topology. Contains a relatively gentle introduction to homology.

 

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