From 7b09710392d35fb55b52031d447a542d99fc6b4b Mon Sep 17 00:00:00 2001 From: Tavian Barnes Date: Tue, 19 Aug 2014 17:10:03 -0400 Subject: Modularize the libdimension codebase. --- libdimension/math/matrix.c | 388 +++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 388 insertions(+) create mode 100644 libdimension/math/matrix.c (limited to 'libdimension/math/matrix.c') diff --git a/libdimension/math/matrix.c b/libdimension/math/matrix.c new file mode 100644 index 0000000..25590d8 --- /dev/null +++ b/libdimension/math/matrix.c @@ -0,0 +1,388 @@ +/************************************************************************* + * Copyright (C) 2009-2014 Tavian Barnes * + * * + * This file is part of The Dimension Library. * + * * + * The Dimension Library is free software; you can redistribute it and/ * + * or modify it under the terms of the GNU Lesser General Public License * + * as published by the Free Software Foundation; either version 3 of the * + * License, or (at your option) any later version. * + * * + * The Dimension Library 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 * + * Lesser General Public License for more details. * + * * + * You should have received a copy of the GNU Lesser General Public * + * License along with this program. If not, see * + * . * + *************************************************************************/ + +/** + * @file + * Matrix function implementations. + */ + +#include "internal.h" +#include "dimension/math.h" +#include + +// Identity matrix +dmnsn_matrix +dmnsn_identity_matrix(void) +{ + return dmnsn_new_matrix(1.0, 0.0, 0.0, 0.0, + 0.0, 1.0, 0.0, 0.0, + 0.0, 0.0, 1.0, 0.0); +} + +// Scaling matrix +dmnsn_matrix +dmnsn_scale_matrix(dmnsn_vector s) +{ + return dmnsn_new_matrix(s.x, 0.0, 0.0, 0.0, + 0.0, s.y, 0.0, 0.0, + 0.0, 0.0, s.z, 0.0); +} + +// Translation matrix +dmnsn_matrix +dmnsn_translation_matrix(dmnsn_vector d) +{ + return dmnsn_new_matrix(1.0, 0.0, 0.0, d.x, + 0.0, 1.0, 0.0, d.y, + 0.0, 0.0, 1.0, d.z); +} + +// Left-handed rotation matrix; theta/|theta| = axis, |theta| = angle +dmnsn_matrix +dmnsn_rotation_matrix(dmnsn_vector theta) +{ + // Two trig calls, 25 multiplications, 13 additions + + double angle = dmnsn_vector_norm(theta); + if (fabs(angle) < dmnsn_epsilon) { + return dmnsn_identity_matrix(); + } + dmnsn_vector axis = dmnsn_vector_div(theta, angle); + + // Shorthand to make dmnsn_new_matrix() call legible + + double s = sin(angle); + double t = 1.0 - cos(angle); + + double x = axis.x; + double y = axis.y; + double z = axis.z; + + return dmnsn_new_matrix( + 1.0 + t*(x*x - 1.0), -z*s + t*x*y, y*s + t*x*z, 0.0, + z*s + t*x*y, 1.0 + t*(y*y - 1.0), -x*s + t*y*z, 0.0, + -y*s + t*x*z, x*s + t*y*z, 1.0 + t*(z*z - 1.0), 0.0 + ); +} + +// Find the angle between two vectors with respect to an axis +static double +dmnsn_axis_angle(dmnsn_vector from, dmnsn_vector to, dmnsn_vector axis) +{ + from = dmnsn_vector_sub(from, dmnsn_vector_proj(from, axis)); + to = dmnsn_vector_sub(to, dmnsn_vector_proj(to, axis)); + + double fromnorm = dmnsn_vector_norm(from); + double tonorm = dmnsn_vector_norm(to); + if (fromnorm < dmnsn_epsilon || tonorm < dmnsn_epsilon) { + return 0.0; + } + + from = dmnsn_vector_div(from, fromnorm); + to = dmnsn_vector_div(to, tonorm); + + double angle = acos(dmnsn_vector_dot(from, to)); + + if (dmnsn_vector_dot(dmnsn_vector_cross(from, to), axis) > 0.0) { + return angle; + } else { + return -angle; + } +} + +// Alignment matrix +dmnsn_matrix +dmnsn_alignment_matrix(dmnsn_vector from, dmnsn_vector to, + dmnsn_vector axis1, dmnsn_vector axis2) +{ + double theta1 = dmnsn_axis_angle(from, to, axis1); + dmnsn_matrix align1 = dmnsn_rotation_matrix(dmnsn_vector_mul(theta1, axis1)); + from = dmnsn_transform_direction(align1, from); + axis2 = dmnsn_transform_direction(align1, axis2); + + double theta2 = dmnsn_axis_angle(from, to, axis2); + dmnsn_matrix align2 = dmnsn_rotation_matrix(dmnsn_vector_mul(theta2, axis2)); + + return dmnsn_matrix_mul(align2, align1); +} + +// Matrix inversion helper functions + +/// A 2x2 matrix for inversion by partitioning. +typedef struct { double n[2][2]; } dmnsn_matrix2; + +/// Construct a 2x2 matrix. +static dmnsn_matrix2 dmnsn_new_matrix2(double a1, double a2, + double b1, double b2); +/// Invert a 2x2 matrix. +static dmnsn_matrix2 dmnsn_matrix2_inverse(dmnsn_matrix2 A); +/// Negate a 2x2 matrix. +static dmnsn_matrix2 dmnsn_matrix2_negate(dmnsn_matrix2 A); +/// Subtract two 2x2 matricies. +static dmnsn_matrix2 dmnsn_matrix2_sub(dmnsn_matrix2 lhs, dmnsn_matrix2 rhs); +/// Add two 2x2 matricies. +static dmnsn_matrix2 dmnsn_matrix2_mul(dmnsn_matrix2 lhs, dmnsn_matrix2 rhs); + +/// Invert a matrix with the slower cofactor algorithm, if partitioning failed. +static dmnsn_matrix dmnsn_matrix_inverse_generic(dmnsn_matrix A); +/// Get the [\p row, \p col] cofactor of A. +static double dmnsn_matrix_cofactor(dmnsn_matrix A, size_t row, size_t col); + +// Invert a matrix, by partitioning +dmnsn_matrix +dmnsn_matrix_inverse(dmnsn_matrix A) +{ + // Use partitioning to invert a matrix: + // + // [ P Q ] -1 + // [ R S ] + // + // = [ PP QQ ] + // [ RR SS ], + // + // with PP = inv(P) - inv(P)*Q*RR, + // QQ = -inv(P)*Q*SS, + // RR = -SS*R*inv(P), and + // SS = inv(S - R*inv(P)*Q). + + // The algorithm uses 2 inversions, 6 multiplications, and 2 subtractions, + // giving 52 multiplications, 34 additions, and 8 divisions. + + dmnsn_matrix2 P, Q, R, S, Pi, RPi, PiQ, RPiQ, PP, QQ, RR, SS; + double Pdet = A.n[0][0]*A.n[1][1] - A.n[0][1]*A.n[1][0]; + + if (dmnsn_unlikely(fabs(Pdet) < dmnsn_epsilon)) { + // If P is close to singular, try a more generic algorithm; this is very + // unlikely, but not impossible, eg. + // [ 1 1 0 0 ] + // [ 1 1 1 0 ] + // [ 0 1 1 0 ] + // [ 0 0 0 1 ] + return dmnsn_matrix_inverse_generic(A); + } + + // Partition the matrix + P = dmnsn_new_matrix2(A.n[0][0], A.n[0][1], + A.n[1][0], A.n[1][1]); + Q = dmnsn_new_matrix2(A.n[0][2], A.n[0][3], + A.n[1][2], A.n[1][3]); + R = dmnsn_new_matrix2(A.n[2][0], A.n[2][1], + 0.0, 0.0); + S = dmnsn_new_matrix2(A.n[2][2], A.n[2][3], + 0.0, 1.0); + + // Do this inversion ourselves, since we already have the determinant + Pi = dmnsn_new_matrix2( P.n[1][1]/Pdet, -P.n[0][1]/Pdet, + -P.n[1][0]/Pdet, P.n[0][0]/Pdet); + + // Calculate R*inv(P), inv(P)*Q, and R*inv(P)*Q + RPi = dmnsn_matrix2_mul(R, Pi); + PiQ = dmnsn_matrix2_mul(Pi, Q); + RPiQ = dmnsn_matrix2_mul(R, PiQ); + + // Calculate the partitioned inverse + SS = dmnsn_matrix2_inverse(dmnsn_matrix2_sub(S, RPiQ)); + RR = dmnsn_matrix2_negate(dmnsn_matrix2_mul(SS, RPi)); + QQ = dmnsn_matrix2_negate(dmnsn_matrix2_mul(PiQ, SS)); + PP = dmnsn_matrix2_sub(Pi, dmnsn_matrix2_mul(PiQ, RR)); + + // Reconstruct the matrix + return dmnsn_new_matrix(PP.n[0][0], PP.n[0][1], QQ.n[0][0], QQ.n[0][1], + PP.n[1][0], PP.n[1][1], QQ.n[1][0], QQ.n[1][1], + RR.n[0][0], RR.n[0][1], SS.n[0][0], SS.n[0][1]); +} + +// For nice shorthand +static dmnsn_matrix2 +dmnsn_new_matrix2(double a1, double a2, double b1, double b2) +{ + dmnsn_matrix2 m = { { { a1, a2 }, + { b1, b2 } } }; + return m; +} + +// Invert a 2x2 matrix +static dmnsn_matrix2 +dmnsn_matrix2_inverse(dmnsn_matrix2 A) +{ + // 4 divisions, 2 multiplications, 1 addition + double det = A.n[0][0]*A.n[1][1] - A.n[0][1]*A.n[1][0]; + return dmnsn_new_matrix2( A.n[1][1]/det, -A.n[0][1]/det, + -A.n[1][0]/det, A.n[0][0]/det); +} + +// Also basically a shorthand +static dmnsn_matrix2 +dmnsn_matrix2_negate(dmnsn_matrix2 A) +{ + return dmnsn_new_matrix2(-A.n[0][0], -A.n[0][1], + -A.n[1][0], -A.n[1][1]); +} + +// 2x2 matrix subtraction +static dmnsn_matrix2 +dmnsn_matrix2_sub(dmnsn_matrix2 lhs, dmnsn_matrix2 rhs) +{ + // 4 additions + return dmnsn_new_matrix2( + lhs.n[0][0] - rhs.n[0][0], lhs.n[0][1] - rhs.n[0][1], + lhs.n[1][0] - rhs.n[1][0], lhs.n[1][1] - rhs.n[1][1] + ); +} + +// 2x2 matrix multiplication +static dmnsn_matrix2 +dmnsn_matrix2_mul(dmnsn_matrix2 lhs, dmnsn_matrix2 rhs) +{ + // 8 multiplications, 4 additions + return dmnsn_new_matrix2( + lhs.n[0][0]*rhs.n[0][0] + lhs.n[0][1]*rhs.n[1][0], + lhs.n[0][0]*rhs.n[0][1] + lhs.n[0][1]*rhs.n[1][1], + lhs.n[1][0]*rhs.n[0][0] + lhs.n[1][1]*rhs.n[1][0], + lhs.n[1][0]*rhs.n[0][1] + lhs.n[1][1]*rhs.n[1][1] + ); +} + +// Invert a matrix, if partitioning failed (|P| == 0) +static dmnsn_matrix +dmnsn_matrix_inverse_generic(dmnsn_matrix A) +{ + // For A = [ A' b ] A^-1 = [ A'^-1 -(A'^-1)*b ] + // [ 0 ... 0 1 ], [ 0 ... 0 1 ]. + // + // Invert A' by calculating its adjucate. + dmnsn_matrix inv; + double det = 0.0, C; + + // Perform a Laplace expansion along the first row to give us the adjugate's + // first column and the determinant + for (size_t j = 0; j < 3; ++j) { + C = dmnsn_matrix_cofactor(A, 0, j); + det += A.n[0][j]*C; + inv.n[j][0] = C; + } + + // Divide the first column by the determinant + for (size_t j = 0; j < 3; ++j) { + inv.n[j][0] /= det; + } + + // Find the rest of A' + for (size_t j = 0; j < 3; ++j) { + for (size_t i = 1; i < 3; ++i) { + inv.n[j][i] = dmnsn_matrix_cofactor(A, i, j)/det; + } + inv.n[j][3] = 0.0; + } + + // Find the translational component of the inverse + for (size_t i = 0; i < 3; ++i) { + for (size_t j = 0; j < 3; ++j) { + inv.n[i][3] -= inv.n[i][j]*A.n[j][3]; + } + } + + return inv; +} + +// Gives the cofactor at row, col; the determinant of the matrix formed from the +// upper-left 3x3 corner of A by ignoring row `row' and column `col', +// times (-1)^(row + col) +static double +dmnsn_matrix_cofactor(dmnsn_matrix A, size_t row, size_t col) +{ + // 2 multiplications, 1 addition + double n[4]; + size_t k = 0; + for (size_t i = 0; i < 3; ++i) { + for (size_t j = 0; j < 3; ++j) { + if (i != row && j != col) { + n[k] = A.n[i][j]; + ++k; + } + } + } + + double C = n[0]*n[3] - n[1]*n[2]; + if ((row + col)%2 == 0) { + return C; + } else { + return -C; + } +} + +// 4x4 matrix multiplication +dmnsn_matrix +dmnsn_matrix_mul(dmnsn_matrix lhs, dmnsn_matrix rhs) +{ + // 36 multiplications, 27 additions + dmnsn_matrix r; + + r.n[0][0] = lhs.n[0][0]*rhs.n[0][0] + lhs.n[0][1]*rhs.n[1][0] + lhs.n[0][2]*rhs.n[2][0]; + r.n[0][1] = lhs.n[0][0]*rhs.n[0][1] + lhs.n[0][1]*rhs.n[1][1] + lhs.n[0][2]*rhs.n[2][1]; + r.n[0][2] = lhs.n[0][0]*rhs.n[0][2] + lhs.n[0][1]*rhs.n[1][2] + lhs.n[0][2]*rhs.n[2][2]; + r.n[0][3] = lhs.n[0][0]*rhs.n[0][3] + lhs.n[0][1]*rhs.n[1][3] + lhs.n[0][2]*rhs.n[2][3] + lhs.n[0][3]; + + r.n[1][0] = lhs.n[1][0]*rhs.n[0][0] + lhs.n[1][1]*rhs.n[1][0] + lhs.n[1][2]*rhs.n[2][0]; + r.n[1][1] = lhs.n[1][0]*rhs.n[0][1] + lhs.n[1][1]*rhs.n[1][1] + lhs.n[1][2]*rhs.n[2][1]; + r.n[1][2] = lhs.n[1][0]*rhs.n[0][2] + lhs.n[1][1]*rhs.n[1][2] + lhs.n[1][2]*rhs.n[2][2]; + r.n[1][3] = lhs.n[1][0]*rhs.n[0][3] + lhs.n[1][1]*rhs.n[1][3] + lhs.n[1][2]*rhs.n[2][3] + lhs.n[1][3]; + + r.n[2][0] = lhs.n[2][0]*rhs.n[0][0] + lhs.n[2][1]*rhs.n[1][0] + lhs.n[2][2]*rhs.n[2][0]; + r.n[2][1] = lhs.n[2][0]*rhs.n[0][1] + lhs.n[2][1]*rhs.n[1][1] + lhs.n[2][2]*rhs.n[2][1]; + r.n[2][2] = lhs.n[2][0]*rhs.n[0][2] + lhs.n[2][1]*rhs.n[1][2] + lhs.n[2][2]*rhs.n[2][2]; + r.n[2][3] = lhs.n[2][0]*rhs.n[0][3] + lhs.n[2][1]*rhs.n[1][3] + lhs.n[2][2]*rhs.n[2][3] + lhs.n[2][3]; + + return r; +} + +// Give an axis-aligned box that contains the given box transformed by `lhs' +dmnsn_aabb +dmnsn_transform_aabb(dmnsn_matrix trans, dmnsn_aabb box) +{ + // Infinite/zero bounding box support + if (isinf(box.min.x)) { + return box; + } + + // Taking the "absolute value" of the matrix saves some min/max calculations + for (int i = 0; i < 3; ++i) { + for (int j = 0; j < 3; ++j) { + trans.n[i][j] = fabs(trans.n[i][j]); + } + } + + dmnsn_vector Mt = dmnsn_matrix_column(trans, 3); + dmnsn_aabb ret = { Mt, Mt }; + + dmnsn_vector Mz = dmnsn_matrix_column(trans, 2); + ret.min = dmnsn_vector_add(ret.min, dmnsn_vector_mul(box.min.z, Mz)); + ret.max = dmnsn_vector_add(ret.max, dmnsn_vector_mul(box.max.z, Mz)); + + dmnsn_vector My = dmnsn_matrix_column(trans, 1); + ret.min = dmnsn_vector_add(ret.min, dmnsn_vector_mul(box.min.y, My)); + ret.max = dmnsn_vector_add(ret.max, dmnsn_vector_mul(box.max.y, My)); + + dmnsn_vector Mx = dmnsn_matrix_column(trans, 0); + ret.min = dmnsn_vector_add(ret.min, dmnsn_vector_mul(box.min.x, Mx)); + ret.max = dmnsn_vector_add(ret.max, dmnsn_vector_mul(box.max.x, Mx)); + + return ret; +} -- cgit v1.2.3