2D-catenary helper class for audio cable drawing in C++.

Catenary.h

/*
  ==============================================================================

    Catenary.h
    Created: 30 Jan 2021 7:26:28pm
    Author:  Adrian Pflugshaupt

    Calculates points on a catenary curve between two 2d points
    with addLength "rope" added to the distance.

    (c) 2021 Adrian Pflugshaupt

    Permission is hereby granted, free of charge, to any person obtaining a copy
    of this software and associated documentation files (the "Software"), to deal
    in the Software without restriction, including without limitation the rights
    to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
    copies of the Software, and to permit persons to whom the Software is
    furnished to do so, subject to the following conditions:

    The above copyright notice and this permission notice shall be included in all
    copies or substantial portions of the Software.

    THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
    IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
    FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
    AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
    LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
    OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
    SOFTWARE.
  ==============================================================================
*/

#pragma once
#include <algorithm> // for swap
#include <cmath>
#include <cassert>

namespace plap {

class Catenary {
public:
    Catenary(float x0, float y0, float x1, float y1, float addLength, int numIterations = 1) {
        assert(x0 != x1);
        assert(addLength > 0.f);
        // make sure x1 is right
        if (x0 > x1) {
            std::swap(x0, x1);
            std::swap(y0, y1);
        }

        auto x_d = x1 - x0;
        auto y_d = y1 - y0;
 
        auto d = std::hypot(x_d, y_d);
        auto L = d + addLength;

        auto x_f = std::sqrt(L*L - y_d*y_d)/x_d; // always > 1

        // need to solve sinh(xi)/xi - x_f = 0
        
  //  auto xi_approx = 1.15f*std::log(x_f - 0.8f) + 1.9f; // manual fit for xi>2, guaranteed too low.
        auto xi_approx = 1.16f*std::log(x_f - 0.75f) + 1.9f; // manual fit for large 2<xi<10..
        if (xi_approx < 2.f) {
            // taylor-approx: 1 + xi^2/6 + xi^4/120 + xi^6/5040 - x_f = 0
            // use inverse taylor series for smallish values. Always slightly too high.
            auto tmp = std::cbrt(std::sqrt(15680.f)*std::sqrt(x_f*(405.f*x_f + 198.f) + 62.f) + 2520.f*x_f + 616.f);
            xi_approx = std::sqrt(tmp - 84.f/tmp - 14.f);
        }
        
        auto xi = xi_approx;
        for (auto n = 0; n < numIterations; ++n) { // newton iterations to improve precision
            auto x = xi;
            auto exp_half = 0.5f*std::exp(x); // sinh and cosh at the same time from exp
            auto iexp_half = 0.25f/exp_half;
            auto sinhx = exp_half - iexp_half;
            auto coshx = exp_half + iexp_half;
            auto val = sinhx/x - x_f;
            auto der = (coshx*x - sinhx)/(x*x);
            xi -= val/der;
        }
                
        a = 0.5f*x_d/xi;
        inva = 1.f/a;
        b = 0.5f*(x0 + x1) - a*std::asinh(0.5f*y_d/(a*std::sinh(xi)));
        c = y0 - (a*std::cosh((x0 - b)*inva));
    }
    
    /// determine the y pos for a given x.
    float calcY(float x) const {
        auto y = a*std::cosh((x - b)*inva) + c;
        return y;
    }
    
    /// get curve vertex = lowest point
    std::pair<float, float> getVertex() const {
        return {b, a + c};
    }

private:
    float a; // curvature = radius of the circle fitting inside the curve at the vertex
    float inva; // inverse of a to avoid a division per point
    float b; // x offset. xpos of the vertex
    float c; // y offset. note that cosh(0) = 1
};

};