Geometry: Approximating 3D with 2D Figures

[OSHMUNNIES]

Before I ask my question, allow me to explain how I thought of it, so that we're on the same page. I would like to reproduce a physics experiment in which a marble rolls down the concave surface of a spherical bowl and back up the opposite side. Hypothetically, the more friction the surface has, the greater the height the marble will reach on the opposite side, because its energy is devoted to translational, and not rotational, motion (I digress, sorry).

I would like to test this using several materials with different friction coefficients to line the inside of the bowl--but my real question here is directed more toward my curiosity of the following geometric concept, rather than its actual application to this experiment.

Here it is: Is there a preferred two-dimensional geometry to be used for approximating the shape of a sphere? In other words, for the purposes of my experiment; if I were to use sandpaper to coat the inside of my spherical bowl, what, if any, would be the optimum shape(s) to cut out?? Why??

Thanks in advance!

[tomgwyther]

If the purpose of the experiment is to see how far up the other side the ball rolls. Could you not use a U shape? a bit like a skateboarding half-pipe.

Or does the ball have to roll back and forth utilising the 3D shape of the bowl?

[OSHMUNNIES]

Yes, I suppose I could use a half-pipe for the actual experiment, but my question is regarding the geometry of spheres, not the spatial limitations of my experiment.

[psychlone]

So, when you say the “optimum shape” you’re actually referring to the geometrical shape that will generate the greatest velocity up and down the slope, assuming that the greater the height achieved up the other side of the sphere (“sphere like shape”) is dependent upon maximising velocity?

 

But from memory the equal frictional forces acting on both the bowl’s and marble’s surfaces are dependant not so much on the frictional coefficient of the surface selected, but dependant more on the mass of the object exerting its self on the stationary surface. The small changes in the magnitude of the frictional coefficients has only a marginal impact on the actual friction force applied.

Edited March 29, 2010 by psychlone

[OSHMUNNIES]

My question has NOTHING to do with the physics of the experiment. Forget the experiment. I'll ask again another way...If I had to approximate the shape of a hollow sphere using ONLY two-dimensional polygons to fabricate its surface, what polygon(s) (if any) would most precisely replicate the three-dimensional curvature of the sphere?

 

(p.s., psychlone, the only variable for that experiment would be the friction coefficient; the same marble would be used, so the same normal force would be applied each run....but this isn't the physics forum:doh:)

[psychlone]

I didn't make an error, I stand by what I wrote. I'm not talking down to you in any way but! a very important word of advice; if you want answers to questions you seek, then you must make the effort to write questions that make sense! (my apologies to the administrators, and to the Science Forums community)

Edited March 30, 2010 by psychlone
reply edit

[OSHMUNNIES]

Sorry if I came off as abrasive, pyschlone, I'm really not lookin' to butt heads with anyone...Maybe these images will help knock some sense into my question (maybe they answer it?):

 

http://www.imsc.res.in/~sitabhra/teaching/cmp03/dome.jpg

 

http://www.cameronius.com/games/antipod/map-final-2c.jpg

 

http://people.sc.fsu.edu/~burkardt/latex/vt_2009/sphere.png

 

http://tugendbol.de/wings3d/UV_challange/sphere_sph_mapping_05_result.png

[psychlone]

The answer to your question; how many polygon(s) to theoretically construct a 2D sphere/ sphere cross section it’s limitless.

 

But check out this link, it might be of some interest to you.

Cheers.

 

http://en.wikipedia.org/wiki/Method_of_exhaustion

Edited April 2, 2010 by psychlone
NO LINK

[Amr Morsi]

It's well known in differential geometry that a sphere cannot be flattened to a 2-dimensional shape. But, a cylinder can be flattened.

 

But, I think you can cut it down to many 6-side polygons. There will be spaces between them and compactness in some places.