Helix Tracing Helical Trajectory

[stevenphy2]

hi all,
I am wondering how to mathematically describe a curve formed by having a helix tracing out a helical trajectory? Any idea?

 

[stevenphy2]

So, I mean the mathematical description of a super-helix.

 

[HallsofIvy]

Any helix "traces out a helical trajectory"! Do you mean a helix whose axis is a helix?

I would do it this way: first assuming the "base helix"- that is the one forming the axis of the helix we want- has the z-axis as axis, we can write it as x= R cos(t), y= R sin(t), z= ct[/itex] where "c" controls the "pitch" of the helix. Now, the hard part: Find the normal and bi-normal to that curve. Those you can use as axes to give the same parametric equations for the "real" helix you want, with, say, radius r and pitch d. The parametric equations for that helix will be the sum of the two sets of parametric equations- you get the point on the axial helix and then add the components out to the "real" helix.

 

[stevenphy2]

Any helix "traces out a helical trajectory"! Do you mean a helix whose axis is a helix?

I would do it this way: first assuming the "base helix"- that is the one forming the axis of the helix we want- has the z-axis as axis, we can write it as x= R cos(t), y= R sin(t), z= ct[/itex] where "c" controls the "pitch" of the helix. Now, the hard part: Find the normal and bi-normal to that curve. Those you can use as axes to give the same parametric equations for the "real" helix you want, with, say, radius r and pitch d. The parametric equations for that helix will be the sum of the two sets of parametric equations- you get the point on the axial helix and then add the components out to the "real" helix.

HallsofIvy, thanks. That's what I meant.
I think it is the way to go, but I do not know things like "binormal" etc. I should learn it now. Do you have textbooks to recommend for learning these things? Or standard text which teach you how to do these super helices things? Thanks.