algorithmic modeling for Rhino
Hi there!
I need to model a spiraling ramp with a constant pitch circumscribed in a hyperboloid. I manage to model the spiral but the pitch is not continuous...
I've been looking through many discussions with similar requirements (logarithmic spirals, helix, loxodromes, conic spirals...) but I couldn't find one to apply into a hyperbolic shape.
If someone sees the light, please let me know.
Thank you!
Toni
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Just to make sure we're on the same page... you're looking for a curve shape which would result in a flat graph rather than the bumpy one shown here right?
ps. I took the liberty of simplifying the file somewhat and also remove one of the sample points which resulted in an awkward lump at the top of the curve.
Exactly David!
I need the graph to be flat resulting a equiangular spiral.
Any suggestion?
Thanks a lot!
ps: Thanks for cleaning up the mess as well :-)
I can't seem to figure it out. I'm trying to manually tweak a sampling curve, hoping that by controlling how fast the UV points most along the surface I could adjust the tangent distribution, but it's like pushing air-bubbles underneath tape...
The exact mathematics are well beyond me so I definitely can't furnish you with an analytic solution.
Thanks for your help David!
Hi Toni,
This was a very interesting question that proved much harder than I thought. I assumed a logarithmic spiral projected from the xy plane onto the hyperboloid might work and it looked pretty good, except at the hyperboloid's "waist", where it deviated from constant slope. Using calculus, I came up with differential equation for the radius of the spiral versus the angle and it turned out that the radius is the hyperbolic cosine (cosh)of the angle. Coincidence???? You be the judge... Hyperbolic cosine looks like an exponential at large +/- values but it deviates near zero.
I generated a cosh spiral in the xy plane that fit between the large and small radii of the hyperbola and wound a specified number of times. The points of this spiral were essentially projected onto the hyperbola to generate a spiral about it. Using David's method you can see the deviation from constant slope is quite small.
Yesss!!!
Is brilliant to generate the hyperbolic spiral in the xy plane!!
That tiny variation on the slope will do the job perfectly!
Thousands thanks Ethan!
Best,
Toni
Gorgeous!
Toni and David, congratulations were premature. I went over my calculations again, and discovered that, in my eagerness to find an neat analytic expression, I left out a term which would have made my integral intractable instead of one where everything cancelled out so obligingly. Thus, my solution is really a first-order approximation -a pretty good one, but an approximation nonetheless. If you play around with some extreme values like a very narrow waist and a steep spiral you'll see it goes off somewhat. So, it's not gorgeous but it is good enough for government work.
Hi Toni,
Welcome to
Grasshopper
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