| 1 |
#ifndef lint |
| 2 |
static const char RCSid[] = "$Id: vect.c,v 1.3 2003/02/28 20:11:30 greg Exp $"; |
| 3 |
#endif |
| 4 |
#include <math.h> |
| 5 |
#include <string.h> |
| 6 |
#include "vect3ds.h" |
| 7 |
|
| 8 |
#ifndef M_PI |
| 9 |
#define M_PI 3.14159265358979323846 |
| 10 |
#endif |
| 11 |
#define PI ((double)M_PI) |
| 12 |
|
| 13 |
#define EPSILON 1e-6 |
| 14 |
|
| 15 |
void adjoint (Matrix mat); |
| 16 |
double det4x4 (Matrix mat); |
| 17 |
double det3x3 (double a1, double a2, double a3, double b1, double b2, |
| 18 |
double b3, double c1, double c2, double c3); |
| 19 |
double det2x2 (double a, double b, double c, double d); |
| 20 |
|
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|
| 22 |
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; |
| 27 |
} |
| 28 |
|
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|
| 30 |
void vect_copy (Vector v1, Vector v2) |
| 31 |
{ |
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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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|
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|
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int vect_equal (Vector v1, Vector v2) |
| 39 |
{ |
| 40 |
if (v1[X] == v2[X] && v1[Y] == v2[Y] && v1[Z] == v2[Z]) |
| 41 |
return 1; |
| 42 |
else |
| 43 |
return 0; |
| 44 |
} |
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|
| 46 |
|
| 47 |
void vect_add (Vector v1, Vector v2, Vector v3) |
| 48 |
{ |
| 49 |
v1[X] = v2[X] + v3[X]; |
| 50 |
v1[Y] = v2[Y] + v3[Y]; |
| 51 |
v1[Z] = v2[Z] + v3[Z]; |
| 52 |
} |
| 53 |
|
| 54 |
|
| 55 |
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]; |
| 60 |
} |
| 61 |
|
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|
| 63 |
void vect_scale (Vector v1, Vector v2, float k) |
| 64 |
{ |
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v1[X] = k * v2[X]; |
| 66 |
v1[Y] = k * v2[Y]; |
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v1[Z] = k * v2[Z]; |
| 68 |
} |
| 69 |
|
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|
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float vect_mag (Vector v) |
| 72 |
{ |
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float mag = sqrt(v[X]*v[X] + v[Y]*v[Y] + v[Z]*v[Z]); |
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|
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return mag; |
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} |
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|
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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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|
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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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|
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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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|
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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]); |
| 97 |
v1[Y] = (v2[Z] * v3[X]) - (v2[X] * v3[Z]); |
| 98 |
v1[Z] = (v2[X] * v3[Y]) - (v2[Y] * v3[X]); |
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} |
| 100 |
|
| 101 |
void vect_min (Vector v1, Vector v2, Vector v3) |
| 102 |
{ |
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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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|
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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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|
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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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|
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mag1 = vect_mag(v1); |
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mag2 = vect_mag(v2); |
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|
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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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|
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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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angle = (180.0/PI) * acos(cos_theta); |
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} |
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|
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return angle; |
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} |
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|
| 141 |
|
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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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|
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if (dec < 0) dec = 0; |
| 147 |
if (dec > 9) dec = 9; |
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|
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fstr[2] = '0' + dec; |
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fstr[8] = '0' + dec; |
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fstr[14] = '0' + dec; |
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|
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fstr[4] = sep; |
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fstr[10] = sep; |
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|
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fprintf (f, fstr, v[X], v[Y], v[Z]); |
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} |
| 158 |
|
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|
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/* Rotate a vector about the X, Y or Z axis */ |
| 161 |
void vect_rotate (Vector v1, Vector v2, int axis, float angle) |
| 162 |
{ |
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float cosa, sina; |
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|
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cosa = cos ((PI/180.0) * angle); |
| 166 |
sina = sin ((PI/180.0) * angle); |
| 167 |
|
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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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|
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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; |
| 180 |
|
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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; |
| 186 |
} |
| 187 |
} |
| 188 |
|
| 189 |
|
| 190 |
/* Rotate a vector about a specific axis */ |
| 191 |
void vect_axis_rotate (Vector v1, Vector v2, Vector axis, float angle) |
| 192 |
{ |
| 193 |
float cosa, sina; |
| 194 |
Matrix mat; |
| 195 |
|
| 196 |
cosa = cos ((PI/180.0) * angle); |
| 197 |
sina = sin ((PI/180.0) * angle); |
| 198 |
|
| 199 |
mat[0][0] = (axis[X] * axis[X]) + ((1.0 - (axis[X] * axis[X]))*cosa); |
| 200 |
mat[0][1] = (axis[X] * axis[Y] * (1.0 - cosa)) - (axis[Z] * sina); |
| 201 |
mat[0][2] = (axis[X] * axis[Z] * (1.0 - cosa)) + (axis[Y] * sina); |
| 202 |
mat[0][3] = 0.0; |
| 203 |
|
| 204 |
mat[1][0] = (axis[X] * axis[Y] * (1.0 - cosa)) + (axis[Z] * sina); |
| 205 |
mat[1][1] = (axis[Y] * axis[Y]) + ((1.0 - (axis[Y] * axis[Y])) * cosa); |
| 206 |
mat[1][2] = (axis[Y] * axis[Z] * (1.0 - cosa)) - (axis[X] * sina); |
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mat[1][3] = 0.0; |
| 208 |
|
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mat[2][0] = (axis[X] * axis[Z] * (1.0 - cosa)) - (axis[Y] * sina); |
| 210 |
mat[2][1] = (axis[Y] * axis[Z] * (1.0 - cosa)) + (axis[X] * sina); |
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mat[2][2] = (axis[Z] * axis[Z]) + ((1.0 - (axis[Z] * axis[Z])) * cosa); |
| 212 |
mat[2][3] = 0.0; |
| 213 |
|
| 214 |
mat[3][0] = mat[3][1] = mat[3][2] = mat[3][3] = 0.0; |
| 215 |
|
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vect_transform (v1, v2, mat); |
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} |
| 218 |
|
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|
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/* Transform the given vector */ |
| 221 |
void vect_transform (Vector v1, Vector v2, Matrix mat) |
| 222 |
{ |
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Vector tmp; |
| 224 |
|
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tmp[X] = (v2[X] * mat[0][0]) + (v2[Y] * mat[1][0]) + (v2[Z] * mat[2][0]) + mat[3][0]; |
| 226 |
tmp[Y] = (v2[X] * mat[0][1]) + (v2[Y] * mat[1][1]) + (v2[Z] * mat[2][1]) + mat[3][1]; |
| 227 |
tmp[Z] = (v2[X] * mat[0][2]) + (v2[Y] * mat[1][2]) + (v2[Z] * mat[2][2]) + mat[3][2]; |
| 228 |
|
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vect_copy (v1, tmp); |
| 230 |
} |
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|
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|
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/* Create an identity matrix */ |
| 234 |
void mat_identity (Matrix mat) |
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{ |
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int i, j; |
| 237 |
|
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for (i = 0; i < 4; i++) |
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for (j = 0; j < 4; j++) |
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mat[i][j] = 0.0; |
| 241 |
|
| 242 |
for (i = 0; i < 4; i++) |
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mat[i][i] = 1.0; |
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} |
| 245 |
|
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|
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void mat_copy (Matrix mat1, Matrix mat2) |
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{ |
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int i, j; |
| 250 |
|
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for (i = 0; i < 4; i++) |
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for (j = 0; j < 4; j++) |
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mat1[i][j] = mat2[i][j]; |
| 254 |
} |
| 255 |
|
| 256 |
|
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/* Rotate a matrix about the X, Y or Z axis */ |
| 258 |
void mat_rotate (Matrix mat1, Matrix mat2, int axis, float angle) |
| 259 |
{ |
| 260 |
Matrix mat; |
| 261 |
float cosa, sina; |
| 262 |
|
| 263 |
cosa = cos ((PI/180.0) * angle); |
| 264 |
sina = sin ((PI/180.0) * angle); |
| 265 |
|
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mat_identity (mat); |
| 267 |
|
| 268 |
switch (axis) { |
| 269 |
case X: |
| 270 |
mat[1][1] = cosa; |
| 271 |
mat[1][2] = sina; |
| 272 |
mat[2][1] = -sina; |
| 273 |
mat[2][2] = cosa; |
| 274 |
break; |
| 275 |
|
| 276 |
case Y: |
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mat[0][0] = cosa; |
| 278 |
mat[0][2] = -sina; |
| 279 |
mat[2][0] = sina; |
| 280 |
mat[2][2] = cosa; |
| 281 |
break; |
| 282 |
|
| 283 |
case Z: |
| 284 |
mat[0][0] = cosa; |
| 285 |
mat[0][1] = sina; |
| 286 |
mat[1][0] = -sina; |
| 287 |
mat[1][1] = cosa; |
| 288 |
break; |
| 289 |
} |
| 290 |
|
| 291 |
mat_mult (mat1, mat2, mat); |
| 292 |
} |
| 293 |
|
| 294 |
|
| 295 |
void mat_axis_rotate (Matrix mat1, Matrix mat2, Vector axis, float angle) |
| 296 |
{ |
| 297 |
float cosa, sina; |
| 298 |
Matrix mat; |
| 299 |
|
| 300 |
cosa = cos ((PI/180.0) * angle); |
| 301 |
sina = sin ((PI/180.0) * angle); |
| 302 |
|
| 303 |
mat[0][0] = (axis[X] * axis[X]) + ((1.0 - (axis[X] * axis[X]))*cosa); |
| 304 |
mat[0][1] = (axis[X] * axis[Y] * (1.0 - cosa)) - (axis[Z] * sina); |
| 305 |
mat[0][2] = (axis[X] * axis[Z] * (1.0 - cosa)) + (axis[Y] * sina); |
| 306 |
mat[0][3] = 0.0; |
| 307 |
|
| 308 |
mat[1][0] = (axis[X] * axis[Y] * (1.0 - cosa)) + (axis[Z] * sina); |
| 309 |
mat[1][1] = (axis[Y] * axis[Y]) + ((1.0 - (axis[Y] * axis[Y])) * cosa); |
| 310 |
mat[1][2] = (axis[Y] * axis[Z] * (1.0 - cosa)) - (axis[X] * sina); |
| 311 |
mat[1][3] = 0.0; |
| 312 |
|
| 313 |
mat[2][0] = (axis[X] * axis[Z] * (1.0 - cosa)) - (axis[Y] * sina); |
| 314 |
mat[2][1] = (axis[Y] * axis[Z] * (1.0 - cosa)) + (axis[X] * sina); |
| 315 |
mat[2][2] = (axis[Z] * axis[Z]) + ((1.0 - (axis[Z] * axis[Z])) * cosa); |
| 316 |
mat[2][3] = 0.0; |
| 317 |
|
| 318 |
mat[3][0] = mat[3][1] = mat[3][2] = mat[3][3] = 0.0; |
| 319 |
|
| 320 |
mat_mult (mat1, mat2, mat); |
| 321 |
} |
| 322 |
|
| 323 |
|
| 324 |
/* mat1 <-- mat2 * mat3 */ |
| 325 |
void mat_mult (Matrix mat1, Matrix mat2, Matrix mat3) |
| 326 |
{ |
| 327 |
float sum; |
| 328 |
int i, j, k; |
| 329 |
Matrix result; |
| 330 |
|
| 331 |
for (i = 0; i < 4; i++) { |
| 332 |
for (j = 0; j < 4; j++) { |
| 333 |
sum = 0.0; |
| 334 |
|
| 335 |
for (k = 0; k < 4; k++) |
| 336 |
sum = sum + mat2[i][k] * mat3[k][j]; |
| 337 |
|
| 338 |
result[i][j] = sum; |
| 339 |
} |
| 340 |
} |
| 341 |
|
| 342 |
for (i = 0; i < 4; i++) |
| 343 |
for (j = 0; j < 4; j++) |
| 344 |
mat1[i][j] = result[i][j]; |
| 345 |
} |
| 346 |
|
| 347 |
|
| 348 |
/* |
| 349 |
Decodes a 3x4 transformation matrix into separate scale, rotation, |
| 350 |
translation, and shear vectors. Based on a program by Spencer W. |
| 351 |
Thomas (Graphics Gems II) |
| 352 |
*/ |
| 353 |
void mat_decode (Matrix mat, Vector scale, Vector shear, Vector rotate, |
| 354 |
Vector transl) |
| 355 |
{ |
| 356 |
int i; |
| 357 |
Vector row[3], temp; |
| 358 |
|
| 359 |
for (i = 0; i < 3; i++) |
| 360 |
transl[i] = mat[3][i]; |
| 361 |
|
| 362 |
for (i = 0; i < 3; i++) { |
| 363 |
row[i][X] = mat[i][0]; |
| 364 |
row[i][Y] = mat[i][1]; |
| 365 |
row[i][Z] = mat[i][2]; |
| 366 |
} |
| 367 |
|
| 368 |
scale[X] = vect_mag (row[0]); |
| 369 |
vect_normalize (row[0]); |
| 370 |
|
| 371 |
shear[X] = vect_dot (row[0], row[1]); |
| 372 |
row[1][X] = row[1][X] - shear[X]*row[0][X]; |
| 373 |
row[1][Y] = row[1][Y] - shear[X]*row[0][Y]; |
| 374 |
row[1][Z] = row[1][Z] - shear[X]*row[0][Z]; |
| 375 |
|
| 376 |
scale[Y] = vect_mag (row[1]); |
| 377 |
vect_normalize (row[1]); |
| 378 |
|
| 379 |
if (scale[Y] != 0.0) |
| 380 |
shear[X] /= scale[Y]; |
| 381 |
|
| 382 |
shear[Y] = vect_dot (row[0], row[2]); |
| 383 |
row[2][X] = row[2][X] - shear[Y]*row[0][X]; |
| 384 |
row[2][Y] = row[2][Y] - shear[Y]*row[0][Y]; |
| 385 |
row[2][Z] = row[2][Z] - shear[Y]*row[0][Z]; |
| 386 |
|
| 387 |
shear[Z] = vect_dot (row[1], row[2]); |
| 388 |
row[2][X] = row[2][X] - shear[Z]*row[1][X]; |
| 389 |
row[2][Y] = row[2][Y] - shear[Z]*row[1][Y]; |
| 390 |
row[2][Z] = row[2][Z] - shear[Z]*row[1][Z]; |
| 391 |
|
| 392 |
scale[Z] = vect_mag (row[2]); |
| 393 |
vect_normalize (row[2]); |
| 394 |
|
| 395 |
if (scale[Z] != 0.0) { |
| 396 |
shear[Y] /= scale[Z]; |
| 397 |
shear[Z] /= scale[Z]; |
| 398 |
} |
| 399 |
|
| 400 |
vect_cross (temp, row[1], row[2]); |
| 401 |
if (vect_dot (row[0], temp) < 0.0) { |
| 402 |
for (i = 0; i < 3; i++) { |
| 403 |
scale[i] *= -1.0; |
| 404 |
row[i][X] *= -1.0; |
| 405 |
row[i][Y] *= -1.0; |
| 406 |
row[i][Z] *= -1.0; |
| 407 |
} |
| 408 |
} |
| 409 |
|
| 410 |
if (row[0][Z] < -1.0) row[0][Z] = -1.0; |
| 411 |
if (row[0][Z] > +1.0) row[0][Z] = +1.0; |
| 412 |
|
| 413 |
rotate[Y] = asin(-row[0][Z]); |
| 414 |
|
| 415 |
if (fabs(cos(rotate[Y])) > EPSILON) { |
| 416 |
rotate[X] = atan2 (row[1][Z], row[2][Z]); |
| 417 |
rotate[Z] = atan2 (row[0][Y], row[0][X]); |
| 418 |
} |
| 419 |
else { |
| 420 |
rotate[X] = atan2 (row[1][X], row[1][Y]); |
| 421 |
rotate[Z] = 0.0; |
| 422 |
} |
| 423 |
|
| 424 |
/* Convert the rotations to degrees */ |
| 425 |
rotate[X] = (180.0/PI)*rotate[X]; |
| 426 |
rotate[Y] = (180.0/PI)*rotate[Y]; |
| 427 |
rotate[Z] = (180.0/PI)*rotate[Z]; |
| 428 |
} |
| 429 |
|
| 430 |
|
| 431 |
/* Matrix inversion code from Graphics Gems */ |
| 432 |
|
| 433 |
/* mat1 <-- mat2^-1 */ |
| 434 |
float mat_inv (Matrix mat1, Matrix mat2) |
| 435 |
{ |
| 436 |
int i, j; |
| 437 |
float det; |
| 438 |
|
| 439 |
if (mat1 != mat2) { |
| 440 |
for (i = 0; i < 4; i++) |
| 441 |
for (j = 0; j < 4; j++) |
| 442 |
mat1[i][j] = mat2[i][j]; |
| 443 |
} |
| 444 |
|
| 445 |
det = det4x4 (mat1); |
| 446 |
|
| 447 |
if (fabs (det) < EPSILON) |
| 448 |
return 0.0; |
| 449 |
|
| 450 |
adjoint (mat1); |
| 451 |
|
| 452 |
for (i = 0; i < 4; i++) |
| 453 |
for(j = 0; j < 4; j++) |
| 454 |
mat1[i][j] = mat1[i][j] / det; |
| 455 |
|
| 456 |
return det; |
| 457 |
} |
| 458 |
|
| 459 |
|
| 460 |
void adjoint (Matrix mat) |
| 461 |
{ |
| 462 |
double a1, a2, a3, a4, b1, b2, b3, b4; |
| 463 |
double c1, c2, c3, c4, d1, d2, d3, d4; |
| 464 |
|
| 465 |
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 |
|
