The discussion revolves around the mathematical description of a curve formed by a helix tracing out a helical trajectory, specifically focusing on the concept of a "super-helix" where the axis itself is a helix. Participants explore the mathematical formulation and underlying concepts related to this topic.
Participants appear to agree on the general approach to describing the helical trajectory, but there is no consensus on the specific mathematical details or the terminology used, such as "binormal." The discussion remains exploratory and unresolved regarding the best methods and resources for learning.
Participants express uncertainty about specific mathematical concepts and terminology, indicating a need for further learning and clarification. The discussion does not resolve the complexities involved in describing super helices.
HallsofIvy said: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.