Consider the analogy of complex numbers. The equation z = x + i*y represents a point in this context. An equation like b = cos(a)*l + i*sin(a)*l serves as a parameter, while
c = cos(ANGLE)*DECAY + i*sin(ANGLE)*DECAY
remains constant.
Initially, you start with z0 = 0 and b0 = c*LENGTH/DECAY. With each iteration, the following calculations take place:
b(k+1) = b(k)*c
z(k+1) = z(k) - b
This leads to:
b1 = b0*c = c^2*LENGTH/DECAY
z1 = z0-b1 = -b1 = -c^2*LENGTH/DECAY
b2 = b1*c = c^3*LENGTH/DECAY
z2 = z1-b2 = -(c^2+c^3)*LENGTH/DECAY
⋮
zn = -(c^2+c^3+⋯+c^(n+1))*LENGTH/DECAY
Consulting Wolfram Alpha reveals that:
c^2+c^3+⋯+c^(n+1) = c^2*(c^n - 1)/(c - 1)
By manipulating the denominator, making it real and converting the formula back to real numbers, the equation can be expressed as:
c = cr + i*ci cr = cos(ANGLE)*DECAY ci = sin(ANGLE)*DECAY
d = c^n = dr + i*di dr = cos(n*ANGLE)*pow(DECAY, n) di = …
Subsequently, we arrive at the following expressions:
c^2*(d - 1)*(cr - i*ci - 1)/((cr + i*ci - 1)*(cr - i*ci - 1))
= ((cr + i*ci)*(cr + i*ci)*(dr + i*di - 1)*(cr - i*ci - 1)) /
((cr - 1)*(cr - 1)*ci*ci)
= ((cr^3*dr + cr*ci^2*dr - cr^2*ci*di - ci^3*di - cr^3 - cr*ci^2
- cr^2*dr + ci^2*dr + 2*cr*ci*di + cr^2 - ci^2) +
(cr^2*ci*dr + ci^3*dr + cr^3*di + cr*ci^2*di - cr^2*ci - ci^3
- 2*cr*ci*dr - cr^2*di + ci^2*di + 2*cr*ci))/((cr - 1)*(cr - 1)*ci*ci)
xn = -(cr^3*dr + cr*ci^2*dr - cr^2*ci*di - ci^3*di - cr^3 - cr*ci^2
- cr^2*dr + ci^2*dr + 2*cr*ci*di + cr^2 - ci^2) /
((cr - 1)*(cr - 1)*ci*ci) * LENGTH / DECAY
yn = -(cr^2*ci*dr + ci^3*dr + cr^3*di + cr*ci^2*di - cr^2*ci - ci^3
- 2*cr*ci*dr - cr^2*di + ci^2*di + 2*cr*ci) /
((cr - 1)*(cr - 1)*ci*ci) * LENGTH / DECAY
The above calculations were derived through computational algebraic analysis, and while the expressions can potentially be simplified, the presented form captures the essence of the mathematical relationships. Here is a practical demonstration showcasing the described concepts:
var ctxt = document.getElementById("MvG1").getContext("2d");
var sin = Math.sin, cos = Math.cos, pow = Math.pow;
var DECAY = 0.75;
var LENGTH = 150;
var ANGLE = 0.52;
var cr = cos(ANGLE)*DECAY, ci = sin(ANGLE)*DECAY;
var cr2 = cr*cr, ci2 = ci*ci, cr3 = cr2*cr, ci3 = ci2*ci;
var f = - LENGTH / DECAY / ((cr - 1)*(cr - 1)*ci*ci)
ctxt.beginPath();
ctxt.moveTo(100,450);
for (var n = 0; n < 20; ++n) {
var da = pow(DECAY, n), dr = cos(n*ANGLE)*da, di = sin(n*ANGLE)*da;
var xn, yn;
xn = (cr3*dr + cr*ci2*dr - cr2*ci*di - ci3*di - cr3 - cr*ci2
- cr2*dr + ci2*dr + 2*cr*ci*di + cr2 - ci2)*f;
yn = (cr2*ci*dr + ci3*dr + cr3*di + cr*ci2*di - cr2*ci - ci3
- 2*cr*ci*dr - cr2*di + ci2*di + 2*cr*ci)*f;
console.log([xn,yn]);
ctxt.lineTo(0.1*xn + 100, 0.1*yn + 450);
}
ctxt.stroke();
<canvas id="MvG1" width="300" height="500"></canvas>