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geodesic - geodesic spheres

Usage

Synopsis

Description

Options

freq

For Class I and II patterns this is the number of divisions along an edge, for Class III patterns (and those specified by two numbers) it is the number of times the pattern is repeated along an edge. Default is one repetition.

-h

program help

-p <poly>

type of poly: i - icosahedron (default), o - octahedron, t - tetrahedron, T - triangle

M <method>

Method used when applying the 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.

-c <class>

face division pattern, 1 (Class I, default) or 2 (Class II), or two numbers separated by a comma to determine the pattern.

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 face edges in order it is posible, starting at a face vertex, to step a units in the direction of one edge, then b units in the direction of the following edge and, if lying 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 with the frequency value corresponding to divisions along an edge. The Class II pattern is equivalent to 1,1 with frequency value corresponding to half the divisions along an edge. 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.

Another way of understanding the pattern formed by the integer pair is that if the frequency is f then it is possible to move between face vertices by moving fa vertices along one line, then turning and moving fb along another line.

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.

-C <cent>

centre of points, in form x_val,y_val,z_val (default 0,0,0), used for geodesic spheres

-i <file>

oriented input file in OFF format containing the base triangle-faced polyhedron to be divided. If '-' then read file from stdin

-o <file>

write output to file, if this option is not used the program writes to standard output

Examples

   geodesic -o geo_sphere.off -c 2 4

   geodesic -p o -M p -o geo_octahedron.off -c 1,2 3

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.

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