pol_recip  polar reciprocals (duals)
Usage: pol_recip [options] [input_file]
Read a file in OFF format and make a polar reciprocal from the face
planes. 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
c <cent> reciprocation centre (default: C)
X,Y,Z  centre with these coordinates
C  vertex centroid, M  midsphere (approximate)
e  edge balanced, E  edge balanced with inversion
v  vert/face balanced, V  vert/face balanced with inversion
a  v/f/e balanced A  v/f/e balanced with inversion
C <init> initial value for a centre calculation, in form 'X,Y,Z',
or C to use centroid (default), M to calculate approx midsphere
r <rad> reciprocation radius (default: calculated)
radius value  use this value as the radius
v  nearest vertex distance, V  furthest vertex distance
e  nearest edge distance, E  furthest edge distance
f  nearest face distance, F  furthest face distance
X  topological dual using face centroids for dual vertices
or a comma separated list of vertex indices (starting from 0)
and the distance is to the space containing those vertices
R <rad> initial value for a radius calculation (default: calculated)
i transform dual by reflecting in centre point
I <dist> maximum distance to project any normal or infinite dual vertex
(default: 1e+15), if 0 then use actual distances and delete
infinite points
x exclude extra elements added to duals with ideal vertices
n <itrs> maximum number of iterations (default: 10000)
l <lim> minimum distance change to terminate, as negative exponent
(default: 12 giving 1e12)
a append dual to original polyhedron
o <file> write output to file (default: write to standard output)
Make the dual of an octahedron, a cube.
pol_recip oct  antiview
Make the dual pair of an octahedron and a cube.
pol_recip a oct  antiview
Make the dual pair of a cuboctahedron and rhombic dodecahedron,
with the reciprocation radius touching the nearest faces (first
command) and farthest faces (second command) causing a dual vertex
to lie on faces at this distance
pol_recip a r f cubo  off_color r A0.4,f  antiview
pol_recip a r F cubo  off_color r A0.4,f  antiview
Specify a face plane of a polyhedron by three of its index numbers, make
a dual with a vertex that lies on this face plane, no matter where the
reciprocation centre is
pol_recip a c 0,0,0.2 r 2,3,5 pyr5  off_color r A0.4,f  antiview
pol_recip a c 0,0,0.4 r 2,3,5 pyr5  off_color r A0.4,f  antiview
A canonical antihemaphrodite is selfdual, but not in the centroid
(first command). The correct centre can be found as a midsphere centre
(second command)
polygon ant s antih 4  canonical  pol_recip a  antiview
polygon ant s antih 4  canonical  pol_recip a c M n 10000  antiview
Pyramids have a number interesting reciprocation centres, e.g. a
default "midcentre" (first command), an external "midcenter" (second
command), a selfdual "edgebalanced" centre (third command)
polygon l 1.5 pyr 7  off_color f blue  pol_recip c M n 50000 a  antiview
polygon l 1.5 pyr 7  off_color f blue  pol_recip C 0,0,1 c M n 50000 a  antiview
polygon l 1.5 pyr 7  off_color f blue  pol_recip c e n 50000 a  antiview
When a polyhedron has a face plane passing through the reciprocation
centre, this is dualised into a an ideal vertex at infinity. To maintain
the visual symmetry of the dual, and emphasise that the vertex can be
reached in two opposite directions, the vertex is placed far away (control
distance with I, and suppress this with x). The first two
commands show the dual as the reciprocation centre is above and below
the base face of the triangular pyramid. Then on the plane (command 3),
and again, but with the duplicated vertices surpressed (command 4)
pol_recip c 0,0,0.05 r 0,1 a pyr3  antiview R 90,0,0
pol_recip c 0,0,0.05 r 0,1 a pyr3  antiview R 90,0,0
pol_recip c 0,0,0.00 r 0,1 a pyr3  antiview R 90,0,0
pol_recip c 0,0,0.00 x r 0,1 a pyr3  antiview R 90,0,0
A dual of a convex polyhedron is normally made with the centre of
reciprocation at the polyhedron centre and the radius to just touch
the edges.
Some polyhedra have faces passing through their natural centre. This
causes a problem when making a dual because the vertex which is dual
to this face should be infinitely far away. pol_recip allows these
vertices to be included by placing them at a specified (probably very large)
distance normal to the face. Any programs dealing with these distant vertices
(e.g. povray) can interpret these distant vertices accordingly.
The default reciprocation centre and radius are found by the following
algorithm. It aims to find a reciprocation sphere that is balanced,
in the sense that the polyhedron and its dual have the same relationship
with the sphere.
centre = centroid of vertices of base polyhedron
radius = average distance from centre to edges
LOOP:
dual = polar reciprocal of base, using centre and radius
invert dual in centre point
edge_centroid = centroid of the nearest points to the centre
on the base's edges and duals edges
radius_sum_base = sum of distances from the centre to the nearest
point to the centre on the base's edges
radius_sum_dual = sum of distances from the centre to the nearest
point to the centre on the dual's edges
if loop count is even:
centre = 0.9*centre + 0.1*edge_centroid
if loop count is odd:
radius = radius * sqrt(rad_sum_g/rad_sum_d)
finish loop if change in centre and radius are small enough
The aim is that this will be a similar reciprocation method.
That is to say
 It will be reciprocal  it will always reciprocate a polyhedron
into the same polyhedron dual, and it will always reciprocate the dual
into the original polyhedron
 It will respect similarity  similar polyhedra will have similar
dualpairs and similar duals
The default method above tends to reciprocate in the midsphere, if
it exists.
The other available balanced reciprocation methods use the centroid
of combined face nearpoints and vertex offsets, or the centroid
of the nearpoints of all three elements combined. For all three
cases the dual may be inverted in the reciprocation centre before
the centroid calculation.
r e may select a sphere suitable for selfduality.
r v and r V may always give the same results.
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geodesic  geodesic spheres
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