geodesic - geodesic spheres

Usage

Usage: geodesic [options] [input_file]

Read a file in OFF format and make a higher frequency, plane-faced
polyhedron or geodesic sphere. If input_file is not given the program
reads from standard input

Options
  -h,--help this help message (run 'off_util -H help' for general help)
  --version version information
  -f <freq> pattern frequency, a positive integer (default: 1) giving the
            number of repeats of the specified pattern along an edge
  -F <freq> final step frequency, minimum number of edges to move between
            base vertices in the geodesic model. For a pattern m,n the
            step frequency is pattern_frequency/(m+n)
  -c <clss> face division pattern,  1 (Class I, default), 2 (Class II), or
            two numbers separated by a comma to determine the pattern
            (Class III, but n,0 or 0,n is Class I, and n,n is Class II)
  -M <mthd> Method of applying the frequency pattern:
            s - geodesic sphere (default). The pattern grid is formed
                from divisions along each edge that make an equal angle
                at the centre. The geodesic vertices are centred at the
                origin and projected on to a unit sphere.
            p - planar. The pattern grid is formed from equal length
                divisions along each edge, the new vertices lie on the
                surface of the original polyhedron.
  -C <cent> centre of points, in form \"x_val,y_val,z_val\" (default: 0,0,0)
            used for geodesic spheres
  -o <file> write output to file (default: write to standard output)

Examples

A Class II icosahedral geodesic sphere with a pattern frequency of 4

geodesic -c 2 -f 4 ico | antiview

geodesic -c 2 -F 4 ico | antiview

geodesic -M p -c 1,2 -f 3 oct | antiview

Notes

Geodesic faces may bridge across an edge of the base polyhedron. If the edge belongs to only one face, or is shared by faces with opposite orientations, the geodesic faces that bridge the edge will not be included in the output.

All the patterns may be specified by a pair of integers. If the integers are a and b, a triangular grid is laid out on the polyhedron face, having (a² + ab + b²)/highest common factor(a, b) divisions. Taking the faces in order it is posible, starting at a face vertex, to step a units in a direction between the edges, then turn left and step another b units and, if the point lies on the face, this point will be a geodesic vertex. The process can be repeated three times from this geodesic vertex, finding the original face vertex and up to two new geodesic vertices. The process is continued until all the geodesic vertices covering the face have been found.

0,6	1,5	2,4	3,3	4,2	5,1	6,0
F6 Class I	1x 1,5 Class III	2x 1,2 Class III	F6 Class II	2x 2,1 Class III	1x 5,1 Class III	F6 Class I

In terms of the general pattern, the Class I pattern is equivalent to 0,1 and the Class II pattern is equivalent to 1,1. Any pattern a,b with a, b > 0 and a ≠ b is a Class III pattern. Class III patterns are chieral, with a,b and b,a being mirror images of each other.

A pattern given by a general integer pair has the property that if the pattern repeat frequency is f then it is possible to move between base vertices by moving fa vertices along one line of edges, then turning and moving fb along another line of edges. This establishes the relationship F = f(a+b) between the -f and -F options.

For some patterns there will be geodesic vertices lying on the polyhedron edges between the face vertices. There will be f x Highest Common Factor(a, b) steps between these geodesic vertices along each polyhedron edge.

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