Parametric Equations for a 2-D, Hyperbolic Golden-ratio Helix

Jinne

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Jan 30, 2020
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I'm trying to formulate parametric equations for a 2-d helix, or sine-wave (you'll see why I'm using this term). 1. Where at t= every whole number multiple of the golden angle (i.e., φ [1.618...] to the minus 1 times 2 π), the pitch or distance to the lower loop of the helix is a power of φ; 2. Where the helix has points of tangency to the hyperbola(s) x*y=±1.; 3. Where the helix starts at (0, 1). As a result of "2.", the equations should take the form, x, y=sin(t)*(1/f(t)), f(t). (Please see the attached images to get an idea of what I'm looking for; also note that I want this indexed such that 2 times the golden angle yields a distance φ to the minus 1.) (The powers of φ go like this, ...φ^-2, φ^-1, φ^0, φ^1, φ^2.....)

Ok, so I have an f(t) that conforms to "1." as a cylindrical helix (i.e., when used in the form x, y=sin(t), f(t)), however, when used in the form x, y=sin(t)*(1/f(t)), f(t), it no longer conforms to "1." Here is a graph of these equations: https://www.desmos.com/calculator/i21lzu0hfx

And, of course, there are infinitely many f(t) functions that conform to "2." and "3."; so, what I need help with is finding an f(t) such that x, y=sin(t)*(1/f(t)), f(t) retains "1.," "2.," and "3."

Thank you all!
 

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