# Torus geometry using villarceau circles

a torus geometry generated using villarceau circles would be useful for many advanced proyects.

anyone its up for the chalenge?

A few examples of such projects?

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a torus generated in that way has a more solid structure, so in architecture gives way more solid meshes that a standard torus, also itâ€™s more easy to map into a surface so getting templates itâ€™s easier

then we go to more esoteric spaces like physics, topology, electronics.

have hear about of Rodin coils? are incredibly efficient because the structure close resemblance to how the magnetic field projects from a magnet. so its like going on rails to the current.

but on the side of 3d art, the resulting uv map of such a torus gets really interesting, there are imges on google, have fun browsing

Yes, I agree that in the physical world a torus with some internal reinforcement is much better. However, in computer graphics this would be just a waste of resources. The mesh will not be stronger, only slower to render (or to compute).

Here is a quick test for circles and coils (not geometries in the Three.js sense):

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iâ€™m not convinced

if we go that road, on the video you got the slices intercepting each other. so this means more apparent surface. take the standard torus mesh and apply this same level of detail. itâ€™s formed by squares so 2 triangles per face. each diamond here has the same two triangles. so in the end the same amount of triangles but better distributed

but on the side of computing it, i agree, but only in part, a villarceau torus only needs one parameter: detail. a standard torus mech needs 2. but the villarcieau torus use a square root on one of the formulas, so its a little worse. then again, once you got the mesh, itâ€™s all the same

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If only the torus surface is needed, circles and coils may produce faces arranged like fish scales. Compared to the â€śordinaryâ€ť torus faces, it is similar to how the icosahedron sphere is compared to â€śordinaryâ€ť uv-sphere.

It appears you have thought a lot about this. Have you tried to implement it? Does it show some interesting properties? Generally, if for some surface there is a parametric formula, it should be easy to implement it in Three.js. So, I do not see graphical or programming difficulty in the challenge, it is 90% math.

BTW some of the feature of the traditionally arranged torus will be lost (e.g. drawing part of the torus within a given angular range, a simple example is a half torus). Another disadvantage could be issues while implementing torus knots with circles (coils might work without issues).

iâ€™m currently trying to implement it, itâ€™s the math that got me stuck. i have use the parametric formulas of wolfram alph and those of the wikipedia, but those donâ€™t translate well, but probably itâ€™s just me not been good enough on math

Iâ€™m not even sure that a parametric solution itâ€™s the right approach, i think its better to get the plane that generates the circles, rotate a copy, get the line that intercept the planes, store those lines, an see where they intercept to get the vertexes, and from there build the torus

messier, i know. but on a parametric solution you get a vertex on a villarceau circle, but that not guarantees that the vertex itâ€™s on an interception, only that is evenly distributed