/************************************************************************* * 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 * Core geometric types like vectors, matricies, and rays. */ #include #include /** A vector in 3 dimensions. */ typedef struct dmnsn_vector { double x; /**< The x component. */ double y; /**< The y component. */ double z; /**< The z component. */ } dmnsn_vector; /** A standard format string for vectors. */ #define DMNSN_VECTOR_FORMAT "<%g, %g, %g>" /** The appropriate arguements to printf() a vector. */ #define DMNSN_VECTOR_PRINTF(v) (v).x, (v).y, (v).z /** A 4x4 affine transformation matrix, with implied [0 0 0 1] bottom row. */ typedef struct dmnsn_matrix { double n[3][4]; /**< The matrix elements in row-major order. */ } dmnsn_matrix; /** A standard format string for matricies. */ #define DMNSN_MATRIX_FORMAT \ "[%g\t%g\t%g\t%g]\n" \ "[%g\t%g\t%g\t%g]\n" \ "[%g\t%g\t%g\t%g]\n" \ "[%g\t%g\t%g\t%g]" /** The appropriate arguements to printf() a matrix. */ #define DMNSN_MATRIX_PRINTF(m) \ (m).n[0][0], (m).n[0][1], (m).n[0][2], (m).n[0][3], \ (m).n[1][0], (m).n[1][1], (m).n[1][2], (m).n[1][3], \ (m).n[2][0], (m).n[2][1], (m).n[2][2], (m).n[2][3], \ 0.0, 0.0, 0.0, 1.0 /** A line, or ray. */ typedef struct dmnsn_line { dmnsn_vector x0; /**< A point on the line. */ dmnsn_vector n; /**< A normal vector; the direction of the line. */ } dmnsn_line; /** A standard format string for lines. */ #define DMNSN_LINE_FORMAT "(<%g, %g, %g> + t*<%g, %g, %g>)" /** The appropriate arguements to printf() a line. */ #define DMNSN_LINE_PRINTF(l) \ DMNSN_VECTOR_PRINTF((l).x0), DMNSN_VECTOR_PRINTF((l).n) /** An axis-aligned bounding box (AABB). */ typedef struct dmnsn_bounding_box { dmnsn_vector min; /**< The coordinate-wise minimum extent of the box. */ dmnsn_vector max; /**< The coordinate-wise maximum extent of the box. */ } dmnsn_bounding_box; /** A standard format string for bounding boxes. */ #define DMNSN_BOUNDING_BOX_FORMAT "(<%g, %g, %g> ==> <%g, %g, %g>)" /** The appropriate arguements to printf() a bounding box. */ #define DMNSN_BOUNDING_BOX_PRINTF(box) \ DMNSN_VECTOR_PRINTF((box).min), DMNSN_VECTOR_PRINTF((box).max) /* Constants */ /** The smallest value considered non-zero by some numerical algorithms. */ #define dmnsn_epsilon 1.0e-10 /** The zero vector. */ static const dmnsn_vector dmnsn_zero = { 0.0, 0.0, 0.0 }; /** The x vector. */ static const dmnsn_vector dmnsn_x = { 1.0, 0.0, 0.0 }; /** The y vector. */ static const dmnsn_vector dmnsn_y = { 0.0, 1.0, 0.0 }; /** The z vector. */ static const dmnsn_vector dmnsn_z = { 0.0, 0.0, 1.0 }; /* Scalar functions */ /** Find the minimum of two scalars. */ DMNSN_INLINE double dmnsn_min(double a, double b) { return a < b ? a : b; } /** Find the maximum of two scalars. */ DMNSN_INLINE double dmnsn_max(double a, double b) { return a > b ? a : b; } /** Convert degrees to radians. */ DMNSN_INLINE double dmnsn_radians(double degrees) { return degrees*atan(1.0)/45.0; } /** Convert radians to degrees. */ DMNSN_INLINE double dmnsn_degrees(double radians) { return radians*45.0/atan(1.0); } /** Return the sign of a scalar. */ DMNSN_INLINE int dmnsn_sign(double n) { if (n > 0.0) { return 1; } else if (n < 0.0) { return -1; } else { return 0; } } /* Shorthand for vector/matrix construction */ /** Construct a new vector. */ DMNSN_INLINE dmnsn_vector dmnsn_new_vector(double x, double y, double z) { dmnsn_vector v = { x, y, z }; return v; } /** Construct a new transformation matrix. */ DMNSN_INLINE dmnsn_matrix dmnsn_new_matrix(double a0, double a1, double a2, double a3, double b0, double b1, double b2, double b3, double c0, double c1, double c2, double c3) { dmnsn_matrix m = { { { a0, a1, a2, a3 }, { b0, b1, b2, b3 }, { c0, c1, c2, c3 } } }; return m; } /** Return the identity matrix. */ dmnsn_matrix dmnsn_identity_matrix(void); /** * A scale transformation. * @param[in] s A vector with components representing the scaling factor in * each axis. * @return The transformation matrix. */ dmnsn_matrix dmnsn_scale_matrix(dmnsn_vector s); /** * A translation. * @param[in] d The vector to translate by. * @return The transformation matrix. */ dmnsn_matrix dmnsn_translation_matrix(dmnsn_vector d); /** * A left-handed rotation. * @param[in] theta A vector representing an axis and angle. * @f$ axis = \vec{\theta}/|\vec{\theta}| @f$, * @f$ angle = |\vec{\theta}| @f$ * @return The transformation matrix. */ dmnsn_matrix dmnsn_rotation_matrix(dmnsn_vector theta); /** * An alignment matrix. * @param[in] from The initial vector. * @param[in] to The desired direction. * @param[in] axis1 The first axis about which to rotate. * @param[in] axis2 The second axis about which to rotate. * @return A transformation matrix that will rotate \p from to \p to. */ dmnsn_matrix dmnsn_alignment_matrix(dmnsn_vector from, dmnsn_vector to, dmnsn_vector axis1, dmnsn_vector axis2); /** * Construct a new line. * @param[in] x0 A point on the line. * @param[in] n The direction of the line. * @return The new line. */ DMNSN_INLINE dmnsn_line dmnsn_new_line(dmnsn_vector x0, dmnsn_vector n) { dmnsn_line l = { x0, n }; return l; } /** Return the bounding box which contains nothing. */ DMNSN_INLINE dmnsn_bounding_box dmnsn_zero_bounding_box(void) { dmnsn_bounding_box box = { { INFINITY, INFINITY, INFINITY }, { -INFINITY, -INFINITY, -INFINITY } }; return box; } /** Return the bounding box which contains everything. */ DMNSN_INLINE dmnsn_bounding_box dmnsn_infinite_bounding_box(void) { dmnsn_bounding_box box = { { -INFINITY, -INFINITY, -INFINITY }, { INFINITY, INFINITY, INFINITY } }; return box; } /* Vector and matrix arithmetic */ /** Negate a vector. */ DMNSN_INLINE dmnsn_vector dmnsn_vector_negate(dmnsn_vector rhs) { /* 3 negations */ dmnsn_vector v = { -rhs.x, -rhs.y, -rhs.z }; return v; } /** Add two vectors. */ DMNSN_INLINE dmnsn_vector dmnsn_vector_add(dmnsn_vector lhs, dmnsn_vector rhs) { /* 3 additions */ dmnsn_vector v = { lhs.x + rhs.x, lhs.y + rhs.y, lhs.z + rhs.z }; return v; } /** Subtract two vectors. */ DMNSN_INLINE dmnsn_vector dmnsn_vector_sub(dmnsn_vector lhs, dmnsn_vector rhs) { /* 3 additions */ dmnsn_vector v = { lhs.x - rhs.x, lhs.y - rhs.y, lhs.z - rhs.z }; return v; } /** Multiply a vector by a scalar. */ DMNSN_INLINE dmnsn_vector dmnsn_vector_mul(double lhs, dmnsn_vector rhs) { /* 3 multiplications */ dmnsn_vector v = { lhs*rhs.x, lhs*rhs.y, lhs*rhs.z }; return v; } /** Divide a vector by a scalar. */ DMNSN_INLINE dmnsn_vector dmnsn_vector_div(dmnsn_vector lhs, double rhs) { /* 3 divisions */ dmnsn_vector v = { lhs.x/rhs, lhs.y/rhs, lhs.z/rhs }; return v; } /** Return the dot product of two vectors. */ DMNSN_INLINE double dmnsn_vector_dot(dmnsn_vector lhs, dmnsn_vector rhs) { /* 3 multiplications, 2 additions */ return lhs.x*rhs.x + lhs.y*rhs.y + lhs.z*rhs.z; } /** Return the cross product of two vectors. */ DMNSN_INLINE dmnsn_vector dmnsn_vector_cross(dmnsn_vector lhs, dmnsn_vector rhs) { /* 6 multiplications, 3 additions */ dmnsn_vector v = { lhs.y*rhs.z - lhs.z*rhs.y, lhs.z*rhs.x - lhs.x*rhs.z, lhs.x*rhs.y - lhs.y*rhs.x }; return v; } /** Return the projection of \p u onto \p d. */ DMNSN_INLINE dmnsn_vector dmnsn_vector_proj(dmnsn_vector u, dmnsn_vector d) { /* 1 division, 9 multiplications, 4 additions */ return dmnsn_vector_mul(dmnsn_vector_dot(u, d)/dmnsn_vector_dot(d, d), d); } /** Return the magnitude of a vector. */ DMNSN_INLINE double dmnsn_vector_norm(dmnsn_vector n) { /* 1 sqrt, 3 multiplications, 2 additions */ return sqrt(dmnsn_vector_dot(n, n)); } /** Return the direction of a vector. */ DMNSN_INLINE dmnsn_vector dmnsn_vector_normalized(dmnsn_vector n) { /* 1 sqrt, 3 divisions, 3 multiplications, 2 additions */ return dmnsn_vector_div(n, dmnsn_vector_norm(n)); } /** Return the component-wise minimum of two vectors. */ DMNSN_INLINE dmnsn_vector dmnsn_vector_min(dmnsn_vector a, dmnsn_vector b) { return dmnsn_new_vector( dmnsn_min(a.x, b.x), dmnsn_min(a.y, b.y), dmnsn_min(a.z, b.z) ); } /** Return the component-wise maximum of two vectors. */ DMNSN_INLINE dmnsn_vector dmnsn_vector_max(dmnsn_vector a, dmnsn_vector b) { return dmnsn_new_vector( dmnsn_max(a.x, b.x), dmnsn_max(a.y, b.y), dmnsn_max(a.z, b.z) ); } /** Invert a matrix. */ dmnsn_matrix dmnsn_matrix_inverse(dmnsn_matrix A); /** Multiply two matricies. */ dmnsn_matrix dmnsn_matrix_mul(dmnsn_matrix lhs, dmnsn_matrix rhs); /** Transform a point by a matrix. */ DMNSN_INLINE dmnsn_vector dmnsn_transform_point(dmnsn_matrix T, dmnsn_vector v) { /* 9 multiplications, 9 additions */ dmnsn_vector r; r.x = T.n[0][0]*v.x + T.n[0][1]*v.y + T.n[0][2]*v.z + T.n[0][3]; r.y = T.n[1][0]*v.x + T.n[1][1]*v.y + T.n[1][2]*v.z + T.n[1][3]; r.z = T.n[2][0]*v.x + T.n[2][1]*v.y + T.n[2][2]*v.z + T.n[2][3]; return r; } /** Transform a direction by a matrix. */ DMNSN_INLINE dmnsn_vector dmnsn_transform_direction(dmnsn_matrix T, dmnsn_vector v) { /* 9 multiplications, 6 additions */ dmnsn_vector r; r.x = T.n[0][0]*v.x + T.n[0][1]*v.y + T.n[0][2]*v.z; r.y = T.n[1][0]*v.x + T.n[1][1]*v.y + T.n[1][2]*v.z; r.z = T.n[2][0]*v.x + T.n[2][1]*v.y + T.n[2][2]*v.z; return r; } /** * Transform a pseudovector by a matrix. * @param[in] Tinv The inverse of the transformation matrix. * @param[in] v The pseudovector to transform * @return The transformed pseudovector. */ DMNSN_INLINE dmnsn_vector dmnsn_transform_normal(dmnsn_matrix Tinv, dmnsn_vector v) { /* Multiply by the transpose of the inverse (9 multiplications, 6 additions) */ dmnsn_vector r; r.x = Tinv.n[0][0]*v.x + Tinv.n[1][0]*v.y + Tinv.n[2][0]*v.z; r.y = Tinv.n[0][1]*v.x + Tinv.n[1][1]*v.y + Tinv.n[2][1]*v.z; r.z = Tinv.n[0][2]*v.x + Tinv.n[1][2]*v.y + Tinv.n[2][2]*v.z; return r; } /** Transform a bounding box by a matrix. */ dmnsn_bounding_box dmnsn_transform_bounding_box(dmnsn_matrix T, dmnsn_bounding_box box); /** * Transform a line by a matrix. * \f$ n' = T(l.\vec{x_0} + l.\vec{n}) - T(l.\vec{x_0}) \f$, * \f$ \vec{x_0}' = T(l.\vec{x_0}) \f$ */ DMNSN_INLINE dmnsn_line dmnsn_transform_line(dmnsn_matrix T, dmnsn_line l) { /* 18 multiplications, 24 additions */ dmnsn_line ret; ret.x0 = dmnsn_transform_point(T, l.x0); ret.n = dmnsn_transform_direction(T, l.n); return ret; } /** * Return the point at \p t on a line. * The point is defined by \f$ l.\vec{x_0} + t \cdot l.\vec{n} \f$ */ DMNSN_INLINE dmnsn_vector dmnsn_line_point(dmnsn_line l, double t) { return dmnsn_vector_add(l.x0, dmnsn_vector_mul(t, l.n)); } /** Add epsilon*l.n to l.x0, to avoid self-intersections. */ DMNSN_INLINE dmnsn_line dmnsn_line_add_epsilon(dmnsn_line l) { return dmnsn_new_line( dmnsn_vector_add( l.x0, dmnsn_vector_mul(1.0e3*dmnsn_epsilon, l.n) ), l.n ); } /** Return whether \p p is within the axis-aligned bounding box. */ DMNSN_INLINE bool dmnsn_bounding_box_contains(dmnsn_bounding_box box, dmnsn_vector p) { return (p.x >= box.min.x && p.y >= box.min.y && p.z >= box.min.z) && (p.x <= box.max.x && p.y <= box.max.y && p.z <= box.max.z); } /** Return whether a bounding box is infinite. */ DMNSN_INLINE bool dmnsn_bounding_box_is_infinite(dmnsn_bounding_box box) { return box.min.x == -INFINITY; } /** Return whether a vector contains any NaN components. */ DMNSN_INLINE bool dmnsn_vector_isnan(dmnsn_vector v) { return isnan(v.x) || isnan(v.y) || isnan(v.z); } /** Return whether a matrix contains any NaN components. */ DMNSN_INLINE bool dmnsn_matrix_isnan(dmnsn_matrix m) { size_t i, j; for (i = 0; i < 3; ++i) { for (j = 0; j < 4; ++j) { if (isnan(m.n[i][j])) { return true; } } } return false; } /** Return whether a line contains any NaN entries. */ DMNSN_INLINE bool dmnsn_line_isnan(dmnsn_line l) { return dmnsn_vector_isnan(l.x0) || dmnsn_vector_isnan(l.n); } /** Return whether a bounding box has any NaN components. */ DMNSN_INLINE bool dmnsn_bounding_box_isnan(dmnsn_bounding_box box) { return dmnsn_vector_isnan(box.min) || dmnsn_vector_isnan(box.max); }