geodesic - geodesic spheres
geodesic [options] [freq]
Create higher frequency, plane-faced polyhedra or geodesic spheres.
- 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
A 4 frequency Class II icosahedral geodesic sphere
geodesic -o geo_sphere.off -c 2 4
A planar octahedron with a Class III 1,2 pattern repeated 3 times along
an edge
geodesic -p o -M p -o geo_octahedron.off -c 1,2 3
When an input file is specified the geodesic faces are coloured
the same as the base polyhedron face they corespond to.
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.
Next:
unipoly - uniform polyhedra (using Kaleido)
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