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Desperado

Is there an implicit formula for an elliptic torus?

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Hello folks, I have an elliptic torus of the parametric form x = a1(a4 + cos^E1(u))(cos^E2(v)) y = a2(a4 + cos^E1(u))(sin^E2(v)) z = a3 (sin^E1(u) + cos^E1 (u)) Does this shape have an implicit formula (specifically: Inside/Outside function) like regular tori do? Thanks for your attention [Edited by - Desperado on June 18, 2008 5:09:48 AM]

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The rest of the world
http://mathworld.wolfram.com/EllipticTorus.html
thinks that an elliptic torus is obtained by rotating an ellipse around an axis.

x = (c + a * cos u)cos v
y = (c + a* cos u)sin v
z = b * sin u

The plain torus has b=a.

Your formula, instead, allows for all sorts of self intersections and is nothing like a torus; where does it come from?

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Well, raising the sines and cosines to arbitrary exponents E1 and E2, not to mention deforming the shape with factors a1, a2 and a2, turns a well-behaved formula like the one you linked to into a completely different mess.

Secondarily, what Paul Bourke calls an elliptic torus is one instance (45° angle) of a slight generalization (slanted ellipse axes) of the standard elliptic torus (horizontal and verical axes).

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