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Created 2003-01-04   Modified 2007-02-13
Chelton Evans

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Intro
Source
Convexity and Polygon Representation in Algebra
Results
TODO
Triangle and Tetrahedron Generalization

Intro

Is a point within a triangle? While a straight forward question, I was surprised with the issues it raised.

img01.png

Source

Makefile

Partitioning a plane is a generalization. A line is one instance of this generalization. A triangle is another. This concept also appears as a fundamental cornerstone in the mathematics of shapes.

Convexity and Shape Representation in Algebra

Shapes can be constructed as an AND and OR expression. General containers d2and and d2or do the job.

Results

belowline insidetriangle shape04.png
shape07.png

TODO

  • Program the maths shape algorithm construction by finding algebra expression.
  • Extend 2D Partition to return the line or segment where the point breaks down. For dt simulation this would be the wall that the particle just went through.

Partitioning the half-space

In 2D a line cuts the plane in half. In 3D the plane cuts space in half.

The testing for which side of the partition is on using vector algebra is exactly the same for 2D and 3D. Shift the cutter and point to the origin, dot product this with the cutters tangent.

Triangle and Tetrahedron Generalization

The tetrahedron is exactly the same as the triangle in the next higher dimension. They could both be viewed as generalized triangles.

For example the operation(partition) of inside a triangle corresponds with inside a tetrahedron.

The representations can be made similar. Let the edge have the same index as the point opposite it. In 3D this becomes let the face have the same index as the point opposite it.

In 1D the greater than operator partitions a line using a 0D point. In 2D the greater than operator generalizes to above a line and partitions a plane using a 1D line. In 3D the greater than operator generalizes above a plane and partitions the the volume using a 2D plane. The cutter is one dimension less than the space it cuts in two.

greater than operator generalizes to above a line and partitions a plane using a 1D line. In 3D the greater than operator generalizes above a plane and partitions the the volume using a 2D plane. The cutter is one dimension less than the space it cuts in two.