#Non Euclydina Geometry
9 messages in this thread
David,
This is a reply to a message that has probably since scrolled — I'm just now
getting to it.
Your problem was to create a set of cones which meet the following criteria:
1) The cones share a common "tip".
2) The "sides" of the cones are tangent.
3) The "bases" of the cones approximate a shpere.
Geometry is my forte, so without going into detail, here is one
possible model of thought. Consider a geometic die or approximated
sphere. If the base of your cone was inscribed in a polygon of the
geo die, the conditions of your problem would be satisfied. The rest
is just math. If you like I can elaborate.
Hi … I've been working on this problem with Dave and your answer
was on of the ones we cam up with, however … how do I create the
geometry in 3DS _accurately_ ? I'd do it in AutoCAD if that would
help …
Gimme a clue …
Thanks,
|-\\/
Well, I can see two alternatives:
The automated way:
1) Get out the pencil and paper and develop a formula for finding the various
angles, sizes, etc. Then, write a quick script with scriptable software like
ACAD that would generate your geometry.
The manual/aproximated way:
2) Achieve a close approximation by doing the following:
a) Create a geo sphere in 3DS using snap so you know where
the center is.
b) Create a circle in one of the polygons of the geo sphere. To
do this you will need to align your view with an n-gon face to
get the true size of the polygon. This is your base size.
c) Since you know where the center of the polygon and geo
sphere are, you can figure out the height of the cone.
d) With the above, you can align your view to each n-gon, and
key-in the cone. Then slide it around so the base of the cone
is inscribed in the n-gon. Repeat this step until all cones are
created.
Note: The manual way will take a long time with any geo sphere with more that
16 sides or so — and it is not as accurate. I haven't developed the
algorithm/formulas for the automated method, but
I can assist if you would rather go that route.
— James —
James – As the source of this little construction problem, I have to mention
that the problem with using a GSPHERE in 3DS as a guide (I've tried it) is that
the sizes of the faces varies with the distance from the poles/equator. My
model (mental) calls for the cones to be tangent in the object, but identical
in all dimensions otherwise.
David {still causing trouble} Taffet
James –
I'd be _very_ interested to see some of your calcs for this thing.
I was a math major, many moons ago, and since I've been outta the
more or less theoretical stuff for a while, I've been having trouble
going beyond the mathematics of a cone created as a surfrev, 45 deg.
Obviously, that makes a "cube" of cones … 4 of them.
If you have time, I'd enjoy revving up my old math skills.
|-\\/
Todd – Our cube of cones had 6 cones in it. Someone else has pointed out to me
that there are a limited number of spheroids that use regular polygons on the
surface. One can be formed with pentagons (a dodecahedron) for example. The few
(less than 10) solutions to forming these spheroids will be the same for
forming a sphere of cones. (*I* should have realized this!) This person created
a sphere of cones, in ACAD, using a dodecahedron as a model and EMAILed it to
me. I have not had a chance to look at it, yet, as I'm still at work. i"ve done
a little more mental work on this myself. If the diameter of one of the cones
is X, the distance the base of a cone needs to rotate up is .8660245X in order
to become tangent to two other cones. This yields a 3-cone clump that could be
used as a basis for completing a sphere. However, there are a limited number of
solutions for an integer number of facets/cones of the sphere. (Hm. Was that
sentence English? Did you follow it? Did anyone?) Any idea where to get that
list of possible solutions? If we got it, it would be easy to make each of the
possible conic constructions in ACAD.
DT
Further development. I remembered another one of these regular
polyhedrons. Icosahedron. It has 20 equilateral triangles as facets.
I *think* it has the most sides of any regular polyhedron. Can you
make one in ACAD and inscribe a cone into each facet? That would
certainly do it for me. I looked at the 12-sided one John XXXX made
for me and, while it's *very* cool, it isn't busy enough to satisfy
my muse. (Hey, you didn't have anything else to do with your "free"
time, did you? <BG>
David,
I'd been passing over this thread, (not even looking at it), because I
didn't consider myself into Non Euclydian Geometry, (I'm not even
sure I know what that means), so I didn't know it was about the
sphere/cones problem. Maybe I'll try to see what an Icosahedron looks
like. And, yea…I thought 12 sides wouldn't be enough, but just
wanted to show the principle.
John XXXX <g> "free" time? <g>
p.s. 12 sides was "easy"; I'm not sure about 20! <g>
John – Sorry I didn't record your last name. My software is only so
friendly and, if I don't plan ahead, loses a lot of info for me. I
hope I didn't sound ungrateful; I'm very grateful. I was tossing that
to Todd to see if he "wanted" to build this thing in *his* "free"
time. He's the first ACADer I snagged into helping on this.
I found an icosahedron in Webster's dictionary. My copy had a picture,
but it basically looks like a ball made out of triangles. If you want
to invest more time in construction for me I can offer you my eternal
gratitude (though not much else <g>).