The discussion focuses on the mathematical description of a super-helix, specifically how a helix can trace out a helical trajectory. The base helix is defined with parametric equations: x = R cos(t), y = R sin(t), z = ct, where "c" controls the pitch. To derive the equations for the super-helix, one must calculate the normal and bi-normal vectors of the base helix, which serve as axes for the new helix. The final parametric equations for the super-helix combine the components of the axial helix with those of the desired helix, incorporating radius and pitch parameters.
PREREQUISITESMathematicians, physics students, and engineers interested in advanced geometry, particularly those working with curves and helices in three-dimensional space.
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.