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greg |
1.1 |
#ifndef lint |
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greg |
1.4 |
static const char RCSid[] = "$Id: vect.c,v 1.3 2003/02/28 20:11:30 greg Exp $"; |
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greg |
1.1 |
#endif |
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#include <math.h> |
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#include <string.h> |
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#include "vect.h" |
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#ifndef M_PI |
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#define M_PI 3.14159265358979323846 |
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#endif |
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gregl |
1.2 |
#define PI ((double)M_PI) |
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greg |
1.1 |
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#define EPSILON 1e-6 |
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void adjoint (Matrix mat); |
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double det4x4 (Matrix mat); |
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double det3x3 (double a1, double a2, double a3, double b1, double b2, |
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double b3, double c1, double c2, double c3); |
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double det2x2 (double a, double b, double c, double d); |
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void vect_init (Vector v, float x, float y, float z) |
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{ |
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v[X] = x; |
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v[Y] = y; |
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v[Z] = z; |
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} |
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void vect_copy (Vector v1, Vector v2) |
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{ |
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v1[X] = v2[X]; |
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v1[Y] = v2[Y]; |
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v1[Z] = v2[Z]; |
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} |
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int vect_equal (Vector v1, Vector v2) |
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{ |
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if (v1[X] == v2[X] && v1[Y] == v2[Y] && v1[Z] == v2[Z]) |
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return 1; |
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else |
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return 0; |
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} |
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void vect_add (Vector v1, Vector v2, Vector v3) |
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{ |
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v1[X] = v2[X] + v3[X]; |
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v1[Y] = v2[Y] + v3[Y]; |
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v1[Z] = v2[Z] + v3[Z]; |
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} |
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void vect_sub (Vector v1, Vector v2, Vector v3) |
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{ |
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v1[X] = v2[X] - v3[X]; |
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v1[Y] = v2[Y] - v3[Y]; |
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v1[Z] = v2[Z] - v3[Z]; |
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} |
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void vect_scale (Vector v1, Vector v2, float k) |
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{ |
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v1[X] = k * v2[X]; |
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v1[Y] = k * v2[Y]; |
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v1[Z] = k * v2[Z]; |
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} |
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float vect_mag (Vector v) |
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{ |
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float mag = sqrt(v[X]*v[X] + v[Y]*v[Y] + v[Z]*v[Z]); |
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return mag; |
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} |
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void vect_normalize (Vector v) |
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{ |
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float mag = vect_mag (v); |
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if (mag > 0.0) |
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vect_scale (v, v, 1.0/mag); |
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} |
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float vect_dot (Vector v1, Vector v2) |
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{ |
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return (v1[X]*v2[X] + v1[Y]*v2[Y] + v1[Z]*v2[Z]); |
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} |
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void vect_cross (Vector v1, Vector v2, Vector v3) |
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{ |
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v1[X] = (v2[Y] * v3[Z]) - (v2[Z] * v3[Y]); |
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v1[Y] = (v2[Z] * v3[X]) - (v2[X] * v3[Z]); |
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v1[Z] = (v2[X] * v3[Y]) - (v2[Y] * v3[X]); |
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} |
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void vect_min (Vector v1, Vector v2, Vector v3) |
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{ |
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v1[X] = (v2[X] < v3[X]) ? v2[X] : v3[X]; |
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v1[Y] = (v2[Y] < v3[Y]) ? v2[Y] : v3[Y]; |
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v1[Z] = (v2[Z] < v3[Z]) ? v2[Z] : v3[Z]; |
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} |
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void vect_max (Vector v1, Vector v2, Vector v3) |
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{ |
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v1[X] = (v2[X] > v3[X]) ? v2[X] : v3[X]; |
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v1[Y] = (v2[Y] > v3[Y]) ? v2[Y] : v3[Y]; |
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v1[Z] = (v2[Z] > v3[Z]) ? v2[Z] : v3[Z]; |
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} |
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/* Return the angle between two vectors */ |
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float vect_angle (Vector v1, Vector v2) |
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{ |
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float mag1, mag2, angle, cos_theta; |
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mag1 = vect_mag(v1); |
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mag2 = vect_mag(v2); |
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if (mag1 * mag2 == 0.0) |
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angle = 0.0; |
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else { |
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cos_theta = vect_dot(v1,v2) / (mag1 * mag2); |
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if (cos_theta <= -1.0) |
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angle = 180.0; |
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else if (cos_theta >= +1.0) |
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angle = 0.0; |
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else |
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gregl |
1.2 |
angle = (180.0/PI) * acos(cos_theta); |
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greg |
1.1 |
} |
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return angle; |
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} |
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void vect_print (FILE *f, Vector v, int dec, char sep) |
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{ |
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char fstr[] = "%.4f, %.4f, %.4f"; |
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| 146 |
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if (dec < 0) dec = 0; |
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if (dec > 9) dec = 9; |
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fstr[2] = '0' + dec; |
| 150 |
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fstr[8] = '0' + dec; |
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fstr[14] = '0' + dec; |
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fstr[4] = sep; |
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fstr[10] = sep; |
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fprintf (f, fstr, v[X], v[Y], v[Z]); |
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} |
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/* Rotate a vector about the X, Y or Z axis */ |
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void vect_rotate (Vector v1, Vector v2, int axis, float angle) |
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{ |
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float cosa, sina; |
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gregl |
1.2 |
cosa = cos ((PI/180.0) * angle); |
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sina = sin ((PI/180.0) * angle); |
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greg |
1.1 |
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switch (axis) { |
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case X: |
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v1[X] = v2[X]; |
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v1[Y] = v2[Y] * cosa + v2[Z] * sina; |
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v1[Z] = v2[Z] * cosa - v2[Y] * sina; |
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break; |
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case Y: |
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v1[X] = v2[X] * cosa - v2[Z] * sina; |
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v1[Y] = v2[Y]; |
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v1[Z] = v2[Z] * cosa + v2[X] * sina; |
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break; |
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case Z: |
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v1[X] = v2[X] * cosa + v2[Y] * sina; |
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v1[Y] = v2[Y] * cosa - v2[X] * sina; |
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v1[Z] = v2[Z]; |
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break; |
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} |
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} |
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/* Rotate a vector about a specific axis */ |
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void vect_axis_rotate (Vector v1, Vector v2, Vector axis, float angle) |
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{ |
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float cosa, sina; |
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Matrix mat; |
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gregl |
1.2 |
cosa = cos ((PI/180.0) * angle); |
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sina = sin ((PI/180.0) * angle); |
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greg |
1.1 |
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mat[0][0] = (axis[X] * axis[X]) + ((1.0 - (axis[X] * axis[X]))*cosa); |
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mat[0][1] = (axis[X] * axis[Y] * (1.0 - cosa)) - (axis[Z] * sina); |
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mat[0][2] = (axis[X] * axis[Z] * (1.0 - cosa)) + (axis[Y] * sina); |
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mat[0][3] = 0.0; |
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| 204 |
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mat[1][0] = (axis[X] * axis[Y] * (1.0 - cosa)) + (axis[Z] * sina); |
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mat[1][1] = (axis[Y] * axis[Y]) + ((1.0 - (axis[Y] * axis[Y])) * cosa); |
| 206 |
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mat[1][2] = (axis[Y] * axis[Z] * (1.0 - cosa)) - (axis[X] * sina); |
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mat[1][3] = 0.0; |
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| 209 |
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mat[2][0] = (axis[X] * axis[Z] * (1.0 - cosa)) - (axis[Y] * sina); |
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mat[2][1] = (axis[Y] * axis[Z] * (1.0 - cosa)) + (axis[X] * sina); |
| 211 |
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mat[2][2] = (axis[Z] * axis[Z]) + ((1.0 - (axis[Z] * axis[Z])) * cosa); |
| 212 |
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mat[2][3] = 0.0; |
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| 214 |
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mat[3][0] = mat[3][1] = mat[3][2] = mat[3][3] = 0.0; |
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| 216 |
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vect_transform (v1, v2, mat); |
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} |
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| 220 |
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/* Transform the given vector */ |
| 221 |
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void vect_transform (Vector v1, Vector v2, Matrix mat) |
| 222 |
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{ |
| 223 |
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Vector tmp; |
| 224 |
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| 225 |
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tmp[X] = (v2[X] * mat[0][0]) + (v2[Y] * mat[1][0]) + (v2[Z] * mat[2][0]) + mat[3][0]; |
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tmp[Y] = (v2[X] * mat[0][1]) + (v2[Y] * mat[1][1]) + (v2[Z] * mat[2][1]) + mat[3][1]; |
| 227 |
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tmp[Z] = (v2[X] * mat[0][2]) + (v2[Y] * mat[1][2]) + (v2[Z] * mat[2][2]) + mat[3][2]; |
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| 229 |
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vect_copy (v1, tmp); |
| 230 |
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} |
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| 232 |
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| 233 |
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/* Create an identity matrix */ |
| 234 |
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void mat_identity (Matrix mat) |
| 235 |
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{ |
| 236 |
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int i, j; |
| 237 |
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| 238 |
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for (i = 0; i < 4; i++) |
| 239 |
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for (j = 0; j < 4; j++) |
| 240 |
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mat[i][j] = 0.0; |
| 241 |
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| 242 |
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for (i = 0; i < 4; i++) |
| 243 |
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mat[i][i] = 1.0; |
| 244 |
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} |
| 245 |
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| 246 |
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| 247 |
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void mat_copy (Matrix mat1, Matrix mat2) |
| 248 |
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{ |
| 249 |
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int i, j; |
| 250 |
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| 251 |
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for (i = 0; i < 4; i++) |
| 252 |
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for (j = 0; j < 4; j++) |
| 253 |
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mat1[i][j] = mat2[i][j]; |
| 254 |
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} |
| 255 |
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| 256 |
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| 257 |
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/* Rotate a matrix about the X, Y or Z axis */ |
| 258 |
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void mat_rotate (Matrix mat1, Matrix mat2, int axis, float angle) |
| 259 |
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{ |
| 260 |
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Matrix mat; |
| 261 |
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float cosa, sina; |
| 262 |
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| 263 |
gregl |
1.2 |
cosa = cos ((PI/180.0) * angle); |
| 264 |
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sina = sin ((PI/180.0) * angle); |
| 265 |
greg |
1.1 |
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| 266 |
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mat_identity (mat); |
| 267 |
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| 268 |
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switch (axis) { |
| 269 |
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case X: |
| 270 |
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mat[1][1] = cosa; |
| 271 |
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mat[1][2] = sina; |
| 272 |
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mat[2][1] = -sina; |
| 273 |
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mat[2][2] = cosa; |
| 274 |
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break; |
| 275 |
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| 276 |
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case Y: |
| 277 |
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mat[0][0] = cosa; |
| 278 |
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mat[0][2] = -sina; |
| 279 |
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mat[2][0] = sina; |
| 280 |
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mat[2][2] = cosa; |
| 281 |
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break; |
| 282 |
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| 283 |
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case Z: |
| 284 |
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mat[0][0] = cosa; |
| 285 |
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mat[0][1] = sina; |
| 286 |
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mat[1][0] = -sina; |
| 287 |
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mat[1][1] = cosa; |
| 288 |
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break; |
| 289 |
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} |
| 290 |
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| 291 |
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mat_mult (mat1, mat2, mat); |
| 292 |
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} |
| 293 |
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| 294 |
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| 295 |
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void mat_axis_rotate (Matrix mat1, Matrix mat2, Vector axis, float angle) |
| 296 |
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{ |
| 297 |
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float cosa, sina; |
| 298 |
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Matrix mat; |
| 299 |
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| 300 |
gregl |
1.2 |
cosa = cos ((PI/180.0) * angle); |
| 301 |
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sina = sin ((PI/180.0) * angle); |
| 302 |
greg |
1.1 |
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| 303 |
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mat[0][0] = (axis[X] * axis[X]) + ((1.0 - (axis[X] * axis[X]))*cosa); |
| 304 |
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mat[0][1] = (axis[X] * axis[Y] * (1.0 - cosa)) - (axis[Z] * sina); |
| 305 |
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mat[0][2] = (axis[X] * axis[Z] * (1.0 - cosa)) + (axis[Y] * sina); |
| 306 |
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mat[0][3] = 0.0; |
| 307 |
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| 308 |
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mat[1][0] = (axis[X] * axis[Y] * (1.0 - cosa)) + (axis[Z] * sina); |
| 309 |
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mat[1][1] = (axis[Y] * axis[Y]) + ((1.0 - (axis[Y] * axis[Y])) * cosa); |
| 310 |
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mat[1][2] = (axis[Y] * axis[Z] * (1.0 - cosa)) - (axis[X] * sina); |
| 311 |
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mat[1][3] = 0.0; |
| 312 |
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| 313 |
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mat[2][0] = (axis[X] * axis[Z] * (1.0 - cosa)) - (axis[Y] * sina); |
| 314 |
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mat[2][1] = (axis[Y] * axis[Z] * (1.0 - cosa)) + (axis[X] * sina); |
| 315 |
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mat[2][2] = (axis[Z] * axis[Z]) + ((1.0 - (axis[Z] * axis[Z])) * cosa); |
| 316 |
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mat[2][3] = 0.0; |
| 317 |
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| 318 |
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mat[3][0] = mat[3][1] = mat[3][2] = mat[3][3] = 0.0; |
| 319 |
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| 320 |
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mat_mult (mat1, mat2, mat); |
| 321 |
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} |
| 322 |
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| 323 |
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| 324 |
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/* mat1 <-- mat2 * mat3 */ |
| 325 |
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void mat_mult (Matrix mat1, Matrix mat2, Matrix mat3) |
| 326 |
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{ |
| 327 |
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float sum; |
| 328 |
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int i, j, k; |
| 329 |
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Matrix result; |
| 330 |
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| 331 |
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for (i = 0; i < 4; i++) { |
| 332 |
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for (j = 0; j < 4; j++) { |
| 333 |
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sum = 0.0; |
| 334 |
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| 335 |
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for (k = 0; k < 4; k++) |
| 336 |
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sum = sum + mat2[i][k] * mat3[k][j]; |
| 337 |
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| 338 |
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result[i][j] = sum; |
| 339 |
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} |
| 340 |
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} |
| 341 |
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| 342 |
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for (i = 0; i < 4; i++) |
| 343 |
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for (j = 0; j < 4; j++) |
| 344 |
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mat1[i][j] = result[i][j]; |
| 345 |
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} |
| 346 |
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| 347 |
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| 348 |
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/* |
| 349 |
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Decodes a 3x4 transformation matrix into separate scale, rotation, |
| 350 |
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translation, and shear vectors. Based on a program by Spencer W. |
| 351 |
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Thomas (Graphics Gems II) |
| 352 |
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*/ |
| 353 |
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void mat_decode (Matrix mat, Vector scale, Vector shear, Vector rotate, |
| 354 |
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Vector transl) |
| 355 |
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{ |
| 356 |
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int i; |
| 357 |
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Vector row[3], temp; |
| 358 |
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| 359 |
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for (i = 0; i < 3; i++) |
| 360 |
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transl[i] = mat[3][i]; |
| 361 |
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| 362 |
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for (i = 0; i < 3; i++) { |
| 363 |
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row[i][X] = mat[i][0]; |
| 364 |
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row[i][Y] = mat[i][1]; |
| 365 |
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row[i][Z] = mat[i][2]; |
| 366 |
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} |
| 367 |
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| 368 |
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scale[X] = vect_mag (row[0]); |
| 369 |
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vect_normalize (row[0]); |
| 370 |
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| 371 |
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shear[X] = vect_dot (row[0], row[1]); |
| 372 |
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row[1][X] = row[1][X] - shear[X]*row[0][X]; |
| 373 |
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row[1][Y] = row[1][Y] - shear[X]*row[0][Y]; |
| 374 |
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row[1][Z] = row[1][Z] - shear[X]*row[0][Z]; |
| 375 |
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| 376 |
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scale[Y] = vect_mag (row[1]); |
| 377 |
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vect_normalize (row[1]); |
| 378 |
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| 379 |
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if (scale[Y] != 0.0) |
| 380 |
|
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shear[X] /= scale[Y]; |
| 381 |
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| 382 |
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shear[Y] = vect_dot (row[0], row[2]); |
| 383 |
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row[2][X] = row[2][X] - shear[Y]*row[0][X]; |
| 384 |
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row[2][Y] = row[2][Y] - shear[Y]*row[0][Y]; |
| 385 |
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row[2][Z] = row[2][Z] - shear[Y]*row[0][Z]; |
| 386 |
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| 387 |
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shear[Z] = vect_dot (row[1], row[2]); |
| 388 |
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row[2][X] = row[2][X] - shear[Z]*row[1][X]; |
| 389 |
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row[2][Y] = row[2][Y] - shear[Z]*row[1][Y]; |
| 390 |
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row[2][Z] = row[2][Z] - shear[Z]*row[1][Z]; |
| 391 |
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| 392 |
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scale[Z] = vect_mag (row[2]); |
| 393 |
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vect_normalize (row[2]); |
| 394 |
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| 395 |
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if (scale[Z] != 0.0) { |
| 396 |
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shear[Y] /= scale[Z]; |
| 397 |
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shear[Z] /= scale[Z]; |
| 398 |
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} |
| 399 |
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|
| 400 |
|
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vect_cross (temp, row[1], row[2]); |
| 401 |
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if (vect_dot (row[0], temp) < 0.0) { |
| 402 |
|
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for (i = 0; i < 3; i++) { |
| 403 |
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scale[i] *= -1.0; |
| 404 |
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row[i][X] *= -1.0; |
| 405 |
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row[i][Y] *= -1.0; |
| 406 |
|
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row[i][Z] *= -1.0; |
| 407 |
|
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} |
| 408 |
|
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} |
| 409 |
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|
| 410 |
|
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if (row[0][Z] < -1.0) row[0][Z] = -1.0; |
| 411 |
|
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if (row[0][Z] > +1.0) row[0][Z] = +1.0; |
| 412 |
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| 413 |
|
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rotate[Y] = asin(-row[0][Z]); |
| 414 |
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| 415 |
|
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if (fabs(cos(rotate[Y])) > EPSILON) { |
| 416 |
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rotate[X] = atan2 (row[1][Z], row[2][Z]); |
| 417 |
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rotate[Z] = atan2 (row[0][Y], row[0][X]); |
| 418 |
|
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} |
| 419 |
|
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else { |
| 420 |
|
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rotate[X] = atan2 (row[1][X], row[1][Y]); |
| 421 |
|
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rotate[Z] = 0.0; |
| 422 |
|
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} |
| 423 |
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|
| 424 |
|
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/* Convert the rotations to degrees */ |
| 425 |
gregl |
1.2 |
rotate[X] = (180.0/PI)*rotate[X]; |
| 426 |
|
|
rotate[Y] = (180.0/PI)*rotate[Y]; |
| 427 |
|
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rotate[Z] = (180.0/PI)*rotate[Z]; |
| 428 |
greg |
1.1 |
} |
| 429 |
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| 430 |
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| 431 |
|
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/* Matrix inversion code from Graphics Gems */ |
| 432 |
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| 433 |
|
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/* mat1 <-- mat2^-1 */ |
| 434 |
|
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float mat_inv (Matrix mat1, Matrix mat2) |
| 435 |
|
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{ |
| 436 |
|
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int i, j; |
| 437 |
|
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float det; |
| 438 |
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|
| 439 |
|
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if (mat1 != mat2) { |
| 440 |
|
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for (i = 0; i < 4; i++) |
| 441 |
|
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for (j = 0; j < 4; j++) |
| 442 |
|
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mat1[i][j] = mat2[i][j]; |
| 443 |
|
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} |
| 444 |
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|
| 445 |
|
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det = det4x4 (mat1); |
| 446 |
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|
| 447 |
|
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if (fabs (det) < EPSILON) |
| 448 |
|
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return 0.0; |
| 449 |
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|
| 450 |
|
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adjoint (mat1); |
| 451 |
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| 452 |
|
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for (i = 0; i < 4; i++) |
| 453 |
|
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for(j = 0; j < 4; j++) |
| 454 |
|
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mat1[i][j] = mat1[i][j] / det; |
| 455 |
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|
| 456 |
|
|
return det; |
| 457 |
|
|
} |
| 458 |
|
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|
| 459 |
|
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| 460 |
|
|
void adjoint (Matrix mat) |
| 461 |
|
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{ |
| 462 |
|
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double a1, a2, a3, a4, b1, b2, b3, b4; |
| 463 |
|
|
double c1, c2, c3, c4, d1, d2, d3, d4; |
| 464 |
|
|
|
| 465 |
|
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a1 = mat[0][0]; b1 = mat[0][1]; |
| 466 |
|
|
c1 = mat[0][2]; d1 = mat[0][3]; |
| 467 |
|
|
|
| 468 |
|
|
a2 = mat[1][0]; b2 = mat[1][1]; |
| 469 |
|
|
c2 = mat[1][2]; d2 = mat[1][3]; |
| 470 |
|
|
|
| 471 |
|
|
a3 = mat[2][0]; b3 = mat[2][1]; |
| 472 |
|
|
c3 = mat[2][2]; d3 = mat[2][3]; |
| 473 |
|
|
|
| 474 |
|
|
a4 = mat[3][0]; b4 = mat[3][1]; |
| 475 |
|
|
c4 = mat[3][2]; d4 = mat[3][3]; |
| 476 |
|
|
|
| 477 |
|
|
/* row column labeling reversed since we transpose rows & columns */ |
| 478 |
|
|
mat[0][0] = det3x3 (b2, b3, b4, c2, c3, c4, d2, d3, d4); |
| 479 |
|
|
mat[1][0] = -det3x3 (a2, a3, a4, c2, c3, c4, d2, d3, d4); |
| 480 |
|
|
mat[2][0] = det3x3 (a2, a3, a4, b2, b3, b4, d2, d3, d4); |
| 481 |
|
|
mat[3][0] = -det3x3 (a2, a3, a4, b2, b3, b4, c2, c3, c4); |
| 482 |
|
|
|
| 483 |
|
|
mat[0][1] = -det3x3 (b1, b3, b4, c1, c3, c4, d1, d3, d4); |
| 484 |
|
|
mat[1][1] = det3x3 (a1, a3, a4, c1, c3, c4, d1, d3, d4); |
| 485 |
|
|
mat[2][1] = -det3x3 (a1, a3, a4, b1, b3, b4, d1, d3, d4); |
| 486 |
|
|
mat[3][1] = det3x3 (a1, a3, a4, b1, b3, b4, c1, c3, c4); |
| 487 |
|
|
|
| 488 |
|
|
mat[0][2] = det3x3 (b1, b2, b4, c1, c2, c4, d1, d2, d4); |
| 489 |
|
|
mat[1][2] = -det3x3 (a1, a2, a4, c1, c2, c4, d1, d2, d4); |
| 490 |
|
|
mat[2][2] = det3x3 (a1, a2, a4, b1, b2, b4, d1, d2, d4); |
| 491 |
|
|
mat[3][2] = -det3x3 (a1, a2, a4, b1, b2, b4, c1, c2, c4); |
| 492 |
|
|
|
| 493 |
|
|
mat[0][3] = -det3x3 (b1, b2, b3, c1, c2, c3, d1, d2, d3); |
| 494 |
|
|
mat[1][3] = det3x3 (a1, a2, a3, c1, c2, c3, d1, d2, d3); |
| 495 |
|
|
mat[2][3] = -det3x3 (a1, a2, a3, b1, b2, b3, d1, d2, d3); |
| 496 |
|
|
mat[3][3] = det3x3 (a1, a2, a3, b1, b2, b3, c1, c2, c3); |
| 497 |
|
|
} |
| 498 |
|
|
|
| 499 |
|
|
|
| 500 |
|
|
double det4x4 (Matrix mat) |
| 501 |
|
|
{ |
| 502 |
|
|
double ans; |
| 503 |
|
|
double a1, a2, a3, a4, b1, b2, b3, b4, c1, c2, c3, c4, d1, d2, d3, d4; |
| 504 |
|
|
|
| 505 |
|
|
a1 = mat[0][0]; b1 = mat[0][1]; |
| 506 |
|
|
c1 = mat[0][2]; d1 = mat[0][3]; |
| 507 |
|
|
|
| 508 |
|
|
a2 = mat[1][0]; b2 = mat[1][1]; |
| 509 |
|
|
c2 = mat[1][2]; d2 = mat[1][3]; |
| 510 |
|
|
|
| 511 |
|
|
a3 = mat[2][0]; b3 = mat[2][1]; |
| 512 |
|
|
c3 = mat[2][2]; d3 = mat[2][3]; |
| 513 |
|
|
|
| 514 |
|
|
a4 = mat[3][0]; b4 = mat[3][1]; |
| 515 |
|
|
c4 = mat[3][2]; d4 = mat[3][3]; |
| 516 |
|
|
|
| 517 |
|
|
ans = a1 * det3x3 (b2, b3, b4, c2, c3, c4, d2, d3, d4) - |
| 518 |
|
|
b1 * det3x3 (a2, a3, a4, c2, c3, c4, d2, d3, d4) + |
| 519 |
|
|
c1 * det3x3 (a2, a3, a4, b2, b3, b4, d2, d3, d4) - |
| 520 |
|
|
d1 * det3x3 (a2, a3, a4, b2, b3, b4, c2, c3, c4); |
| 521 |
|
|
|
| 522 |
|
|
return ans; |
| 523 |
|
|
} |
| 524 |
|
|
|
| 525 |
|
|
|
| 526 |
|
|
double det3x3 (double a1, double a2, double a3, double b1, double b2, |
| 527 |
|
|
double b3, double c1, double c2, double c3) |
| 528 |
|
|
{ |
| 529 |
|
|
double ans; |
| 530 |
|
|
|
| 531 |
|
|
ans = a1 * det2x2 (b2, b3, c2, c3) |
| 532 |
|
|
- b1 * det2x2 (a2, a3, c2, c3) |
| 533 |
|
|
+ c1 * det2x2 (a2, a3, b2, b3); |
| 534 |
|
|
|
| 535 |
|
|
return ans; |
| 536 |
|
|
} |
| 537 |
|
|
|
| 538 |
|
|
|
| 539 |
|
|
double det2x2 (double a, double b, double c, double d) |
| 540 |
|
|
{ |
| 541 |
|
|
double ans; |
| 542 |
|
|
ans = a * d - b * c; |
| 543 |
|
|
return ans; |
| 544 |
|
|
} |
| 545 |
|
|
|
