Adrian Rossiter writes:
"Here are some animations showing a symmetric closing down of
the edges of a truncated cube. First O symmetry, showing
strings across alternate vertices and then showing the hull,
and then Th symmetry (4-5Mb)
http://www.antiprism.com/misc/tr_cube_strs.gif
http://www.antiprism.com/misc/tr_cube_hull.gif
http://www.antiprism.com/misc/tr_cube_th_strs.gif
http://www.antiprism.com/misc/tr_cube_th_hull.gif
An example Python animation script is below.
They are essentially made from a single point, which is
symmetrically repeated by hand to make a triangle and link
edge, and then these elements are symmetrically repeated by
poly_kscope.
In the development version of poly_kscope I have changed Th
symmetry so the three-fold axes are aligned with those of
O symmetry. They were previously aligned one vertically and
another in the xy-plane, which wasn't very convenient.
I am writing a program that creates a similar transformation
to the octahedral one on all the regular polyhedra. (I will
probably include it in the extras section.)"
Adrian.
--
Adrian Rossiter
adrian@antiprism.com
Home: http://antiprism.com/adrian
On Thu, 2 Oct 2008, Carmen Laski wrote:
> 360 triangles each. I would like to build an oval dome but cannot find
> the geodesic formulas to build the triangles. I saw these formulas
> about 2 years ago on the net but cannt find them anywhere. Any help
> out there with this?
You could calculate the lengths and angles with the Antiprism
programs.
http://www.antiprism.com/
Here is an example showing how you would do it.
Generate an F3 icosahedral geodesic sphere and scale it by 2
along the x-axis to make an oval base
geodesic 3 | off_trans -S 2,1,1 > oval_dome.off
You need to see the vertex numbers so open it in Antiprism's
antiview program like this
antiview -n v oval_dome.off
Alternatively make a VRML model to show the vertex numbers
off2vrml -n oval_dome.off > oval_dome.wrl
Here is the model of the geodesic ellipsoid with numbered
vertices
http://www.antiprism.com/misc/oval_dome.wrl
Oval dome 3d image
A base triangle runs between vertices 81, 80, and 12. Get the
strut lengths like this
off_query -e 81,80,80,12,12,81 Evl oval_dome.off
The output is
E0,81 80,1.4272883590923595
E1,80 12,0.7384708168734635
E2,12 81,0.73847081687346305
To find the mitre angles you first need to add a point
at the centre of the model. The joint "axes" will pass
through this point.
echo "0,0,0" | off_util > centre_point.off
off_util -o oval_dome.off oval_dome.off centre_point.off
Find how many vertices are in the model
off_report oval_dome.off
One less than this is the index of the centre point, in this
case 92.
To work out the angles of strut 20,12 at the end 12. First
find the joint axis angle at 12 (along 20,12,92)
off_query -f 20,12,92 Fa oval_dome.off
This gives the angles of the triangle 20,12,92 and the
angle at 12 is the second value, around 83 degrees
E0,58.553885026443339 82.925065091715098 38.521049881841577
It should be possible to find the mitre angles to cut at each
side of the joint axis by querying for the two dihedral angles
each side of the plane through the strut and joint axis, like
this
off_query -f 80,12,92 -f 0,12,92 -e 12,92 Ea oval_dome.off
Unfortunately, this doesn't work! There is a bug relating
to the reporting of dihedral angles when extra elements are
added using -e or -f. I will fix this.
However, if you think you would like to calculate your dome
like this I could write a short program that would produce
a list of the joint axis angles and the mitre angles at
each vertex of the dome. I think will write some code to do
this anyway as I imagine it may be of more general interest.
If you wanted extra vertices for the base, for example
between vertices 12 and 13 to split the wide triangle
12,81,80, you would have to add them by hand to the
off file produced by 'geodesic'. Then project onto
a sphere with 'off_util -S', and then scale as before
e.g 'off_trans -S 2,1,1'
Adrian.
--
Adrian Rossiter
adrian@antiprism.com
http://antiprism.com/adrian