NopSCADlib/utils/maths.scad

//
// NopSCADlib Copyright Chris Palmer 2018
// nop.head@gmail.com
// hydraraptor.blogspot.com
//
// This file is part of NopSCADlib.
//
// NopSCADlib is free software: you can redistribute it and/or modify it under the terms of the
// GNU General Public License as published by the Free Software Foundation, either version 3 of
// the License, or (at your option) any later version.
//
// NopSCADlib is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY;
// without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.
// See the GNU General Public License for more details.
//
// You should have received a copy of the GNU General Public License along with NopSCADlib.
// If not, see <https://www.gnu.org/licenses/>.
//

//
//! Maths utilities for manipulating vectors and matrices.
//
function sqr(x) = x * x;                        //! Square x
function radians(degrees) = degrees * PI / 180; //! Convert radians to degrees
function degrees(radians) = radians * 180 / PI; //! Convert degrees to radians

function sinh(x) = (exp(x) - exp(-x)) / 2;      //! hyperbolic sine
function cosh(x) = (exp(x) + exp(-x)) / 2;      //! hyperbolic cosine
function tanh(x) = sinh(x) / cosh(x);           //! hyperbolic tangent
function coth(x) = cosh(x) / sinh(x);           //! hyperbolic cotangent
function argsinh(x) = ln(x + sqrt(sqr(x) + 1)); //! inverse hyperbolic sine
function argcosh(x) = ln(x + sqrt(sqr(x) - 1)); //! inverse hyperbolic cosine
function argtanh(x) = ln((1 + x) / (1 - x)) / 2;//! inverse hyperbolic tangent
function argcoth(x) = ln((x + 1) / (x - 1)) / 2;//! inverse hyperbolic cotangent

function translate(v) = let(u = is_list(v) ? len(v) == 2 ? [v.x, v.y, 0] //! Generate a 4x4 translation matrix, ```v``` can be ```[x, y]```, ```[x, y, z]``` or ```z```
                                                         : v
                                           : [0, 0, v])
                          [ [1, 0, 0, u.x],
                            [0, 1, 0, u.y],
                            [0, 0, 1, u.z],
                            [0, 0, 0,   1] ];

function rotate(a, v) = //! Generate a 4x4 rotation matrix, ```a``` can be a vector of three angles or a single angle around ```z```, or around axis ```v```
    is_undef(v) ? let(av = is_list(a) ? a : [0, 0, a],
                      cx = cos(av[0]),
                      cy = cos(av[1]),
                      cz = cos(av[2]),
                      sx = sin(av[0]),
                      sy = sin(av[1]),
                      sz = sin(av[2]))
                          [
                              [ cy * cz, sx * sy * cz - cx * sz, cx * sy * cz + sx * sz, 0],
                              [ cy * sz, sx * sy * sz + cx * cz, cx * sy * sz - sx * cz, 0],
                              [-sy,      sx * cy,                cx * cy,                0],
                              [ 0,       0,                      0,                      1]
                          ]
                : let(s = sin(a),
                      c = cos(a),
                      C = 1 - c,
                      m = sqr(v.x) + sqr(v.y) + sqr(v.z), // m used instead of norm to avoid irrational roots as much as possible
                      u = v / sqrt(m))
                          [
                              [ C * v.x * v.x / m + c,       C * v.x * v.y / m - u.z * s, C * v.x * v.z / m + u.y * s, 0],
                              [ C * v.y * v.x / m + u.z * s, C * v.y * v.y / m + c,       C * v.y * v.z / m - u.x * s, 0],
                              [ C * v.z * v.x / m - u.y * s, C * v.z * v.y / m + u.x * s, C * v.z * v.z / m + c,       0],
                              [ 0,                           0,                           0,                           1]
                         ];

function rot3_z(a) = //! Generate a 3x3 matrix to rotate around z
    let(c = cos(a),
        s = sin(a))
        [ [ c, -s,  0],
          [ s,  c,  0],
          [ 0,  0,  1] ];

function rot2_z(a) = //! Generate a 2x2 matrix to rotate around z
    let(c = cos(a),
        s = sin(a))
        [ [ c, -s],
          [ s,  c] ];

function scale(v) = let(s = is_list(v) ? v : [v, v, v]) //!  Generate a 4x4 matrix that scales by ```v```, which can be a vector of xyz factors or a scalar to scale all axes equally
                        [
                          [s.x, 0,   0,   0],
                          [0,   s.y, 0,   0],
                          [0,   0,   s.z, 0],
                          [0,   0,   0,   1]
                        ];

function vec3(v) = [v.x, v.y, v.z]; //! Return a 3 vector with the first three elements of ```v```
function vec4(v) = [v.x, v.y, v.z, 1]; //! Return a 4 vector with the first three elements of ```v```
function transform(v, m) = vec3(m * [v.x, v.y, v.z, 1]); //! Apply 4x4 transform to a 3 vector by extending it and cropping it again
function transform_points(path, m) = [for(p = path) transform(p, m)]; //! Apply transform to a path
function unit(v) = let(n = norm(v)) n ? v / n : v; //! Convert ```v``` to a unit vector

function transpose(m) = [ for(j = [0 : len(m[0]) - 1]) [ for(i = [0 : len(m) - 1]) m[i][j] ] ]; //! Transpose an arbitrary size matrix

function identity(n, x = 1) = [for(i = [0 : n - 1]) [for(j = [0 : n - 1]) i == j ? x : 0] ]; //! Construct an arbitrary size identity matrix

function reverse(v) = let(n = len(v) - 1) n < 0 ? [] : [for(i = [0 : n]) v[n - i]]; //! Reverse a vector

function angle_between(v1, v2) = acos(v1 * v2 / (norm(v1) * norm(v2))); //! Return the angle between two vectors

// https://www.gregslabaugh.net/publications/euler.pdf
function euler(R) = let(ay = asin(-R[2][0]), cy = cos(ay)) //! Convert a rotation matrix to a Euler rotation vector.
    cy ? [ atan2(R[2][1] / cy, R[2][2] / cy), ay, atan2(R[1][0] / cy, R[0][0] / cy) ]
       : R[2][0] < 0 ? [atan2( R[0][1],  R[0][2]),  180, 0]
                     : [atan2(-R[0][1], -R[0][2]), -180, 0];

module position_children(list, t) //! Position children if they are on the Z = 0 plane when transformed by t
    for(p = list)
        let(q = t * p)
            if(abs(transform([0, 0, 0], q).z) < 0.01)
                multmatrix(q)
                    children();

// Matrix inversion: https://www.mathsisfun.com/algebra/matrix-inverse-row-operations-gauss-jordan.html

function augment(m) = let(l = len(m), n = identity(l)) [ //! Augment a matrix by adding an identity matrix to the right
    for(i = [0 : l - 1])
        concat(m[i], n[i])
];

function rowswap(m, i, j) = [ //! Swap two rows of a matrix
    for(k = [0 : len(m) - 1])
        k == i ? m[j] : k == j ? m[i] : m[k]
];

function solve_row(m, i) = let(diag = m[i][i]) [ //! Make diagonal one by dividing the row by it and subtract from other rows to make column zero
    for(j = [0 : len(m) - 1])
        i == j ? m[j] / diag : m[j] - m[i] * m[j][i] / diag
];

function nearly_zero(x) = abs(x) < 1e-5; //! True if x is close to zero

function solve(m, i = 0, j = 0) = //! Solve each row ensuring diagonal is not zero
    i < len(m) ?
        assert(i + j < len(m), "matrix is singular")
         solve(!nearly_zero(m[i + j][i]) ? solve_row(j ? rowswap(m, i, i + j) : m, i) : solve(m, i, j + 1), i + 1)
        : m;

function invert(m) = let(n =len(m), m = solve(augment(m))) [ //! Invert a matrix
    for(i = [0 : n - 1]) [
        for(j = [n : 2 * n - 1])
             each m[i][j]
    ]
];

function circle_intersect(c1, r1, c2, r2) =     //! Calculate one point where two circles in the X-Z plane intersect, clockwise around c1
    let(
        v = c1 - c2,                            // Line between centres
        d = norm(v),                            // Distance between centres
        a = atan2(v.z, v.x) - acos((sqr(d) + sqr(r2) - sqr(r1)) / (2 * d * r2)) // Cosine rule to find angle from c2
     ) c2 + r2 * [cos(a), 0, sin(a)];           // Point on second circle