I think the simplest visualization of what I am trying to parameterize is a slinky lifted straight 'up' right to the point at which the lowest full circle is still on the 'ground'. What I'm after are the 'formulae' for describing that "spiral on a cyinder" in which the angle with the horizontal (or vertical if you want) is not constant.
In a "normal" (spiral on a right regular cylinder), helix like the core of a roll of paper towels, if one 'unwraps' that cardboard core, one ends up with a trapezoidal piece of cardboard. If one then circumscribes that trapezoid with the smallest rectangle possible, the sides of the trapezoid will be at a constant angle to the sides of that rectangle. That angle is constant wherever you choose to measure it, and wrt whatever reference (vertical or horizontal) you care to use.
Now imagine that the "line" made by the edge of the unrolled paper towel core is NOT a straight line. (The angle is NOT constant.) Instead it is described by some non-linear function, say logarithmic. Now wrap THAT core back into a cylinder. In order for the resulting cylinder to be a "right regular" cylinder, the spiral made by the edge of the cardboard will have to get "tighter" and "tighter" as it goes to one end. (The "pitch" get higher and higher.) Just like the coils in the slinky will be "tighter" and "tighter" as one progresses from the end above the 'ground' to the end still on the ground.
So... Let me state my problem differently: Let's say I know that the rate of rotation (pitch) on my helix is 1 revolution per cm at one end of my cylinder, and I know that the rate of rotation is 1 revolution in 10 cm at the other end. What is the 'angle of attack' at any point on the cylinder? Or better yet, what is the generalized formulae for specifying that non-linear helix given:
1) The diameter (radius) of the helix ('cylinder'), 2) The starting rotations-per-unit-height, (starting pitch), 3) The ending rotations-per-unit-height, (ending pitch), and 4) The height of the cylinder.
x = a cos(t), and y = a sin(t), and z = bt, where
a = the radius of the helix, and b = the helical angle,
I can imagine that a "non-linear" helix is created by making "b" be some non-linear function of "t".
Is that "all there is to it"?
What I'm after is a way to parameterize the equations not knowing the non-linear function that "b" would be, but rather knowing the four specific items I listed above, especially 2 and 3.