Parameters of helix approximated by parts of torus

[lewhoo]

Hello,

I am approximating a helix by parts of torus, to build an optical fiber wrapped around a cylinder simulation. Due to the software limitations, there is no easier way.

So I take a part of the torus, rotate it so the one end points slightly up, connect similar part of the torus to the rotated end and so on. The helix pitch, radius and its angle in respect to the z-axis are determined by the radius of the original torus, the length of the part of the torus (how much of the torus I cut out to make the part from which I make the helix) and the angle of rotation of the one of the torus part end. The problem is, I don't know how to calculate the helix parameters from these torus parts parameters.

I can calculate the coordinates of the end of each torus part in the general coordinates system. From this I could calculate coordinates in the system connected to helix (rotated so the z' axis goes along the helix axis) and compare with the helix analytical equations, to get the helix parameters and the coordinate system rotation parameters. Maple is trying to calculate it now, but I don't know if it will succeed - seems complicated.

Maybe one of you can see an easier solution to my problem?

 

[Stephen Tashi]

As I visualze it, the approximation you are making isn't a real helix, so the meaning of "what are the helix parameters?" isn't precisely defined.

If we are interesting in describing how far along the axis of the helix you go to complete one rotation then my (hasty) answer would be 4 R sin(theta) where R is the "major radius" of the torus (in the terminology of the current Wikiipedia article on "Torus"). and theta is the angle at which the plane of the torus is tilted to a plane normal to the axis of the pseudo-helix.

The "radius" of the pseudo helix is a more complicated question since the tilted half torus doesn't have a constant distance from the center of the tubel to the axis of the pseudo helix. As I visualize it, the ends of the tilted half torus are closer to the axis than it's middle is. If you need a single number to represent the radius of a helix, you have to consider how this number will be used in the calculations you care about.