src/cxr_math.h

   1 // Copyright (C) 2021 Harry Godden (hgn)
   2
   3 #define CXR_INLINE static inline
   4 #define CXR_PIf 3.14159265358979323846264338327950288f
   5
   6 // Simple min/max replacements
   7 CXR_INLINE double cxr_minf( double a, double b )
   8 {
   9         return a < b? a: b;
  10 }
  11
  12 CXR_INLINE double cxr_maxf( double a, double b )
  13 {
  14         return a > b? a: b;
  15 }
  16
  17 CXR_INLINE int cxr_min( int a, int b )
  18 {
  19         return a < b? a: b;
  20 }
  21
  22 CXR_INLINE int cxr_max( int a, int b )
  23 {
  24         return a > b? a: b;
  25 }
  26
  27 CXR_INLINE double cxr_clampf( double v, double a, double b )
  28 {
  29    return cxr_minf( b, cxr_maxf( a, v ) );
  30 }
  31
  32 // Convert degrees to radians
  33 CXR_INLINE double cxr_rad( double deg )
  34 {
  35         return deg * CXR_PIf / 180.0f;
  36 }
  37
  38 // Vector 2
  39 // ==================================================================================================================
  40
  41 CXR_INLINE void v2_zero( v2f a )
  42 {
  43    a[0] = 0.0; a[1] = 0.0;
  44 }
  45
  46 CXR_INLINE void v2_fill( v2f a, double v )
  47 {
  48    a[0] = v; a[1] = v;
  49 }
  50
  51 CXR_INLINE void v2_copy( v2f a, v2f b )
  52 {
  53         b[0] = a[0]; b[1] = a[1];
  54 }
  55
  56 CXR_INLINE void v2_minv( v2f a, v2f b, v2f dest )
  57 {
  58         dest[0] = cxr_minf(a[0], b[0]);
  59         dest[1] = cxr_minf(a[1], b[1]);
  60 }
  61
  62 CXR_INLINE void v2_maxv( v2f a, v2f b, v2f dest )
  63 {
  64         dest[0] = cxr_maxf(a[0], b[0]);
  65         dest[1] = cxr_maxf(a[1], b[1]);
  66 }
  67
  68 CXR_INLINE void v2_sub( v2f a, v2f b, v2f d )
  69 {
  70         d[0] = a[0]-b[0]; d[1] = a[1]-b[1];
  71 }
  72
  73 CXR_INLINE double v2_cross( v2f a, v2f b )
  74 {
  75         return a[0] * b[1] - a[1] * b[0];
  76 }
  77
  78 CXR_INLINE void v2_add( v2f a, v2f b, v2f d )
  79 {
  80         d[0] = a[0]+b[0]; d[1] = a[1]+b[1];
  81 }
  82
  83 CXR_INLINE void v2_muls( v2f a, double s, v2f d )
  84 {
  85         d[0] = a[0]*s; d[1] = a[1]*s;
  86 }
  87
  88 CXR_INLINE void v2_mul( v2f a, v2f b, v2f d )
  89 {
  90         d[0] = a[0]*b[0]; d[1] = a[1]*b[1];
  91 }
  92
  93 CXR_INLINE void v2_muladds( v2f a, v2f b, double s, v2f d )
  94 {
  95         d[0] = a[0]+b[0]*s; d[1] = a[1]+b[1]*s;
  96 }
  97
  98 CXR_INLINE double v2_dot( v2f a, v2f b )
  99 {
 100         return a[0] * b[0] + a[1] * b[1];
 101 }
 102
 103 CXR_INLINE void v2_div( v2f a, v2f b, v2f d )
 104 {
 105         d[0] = a[0]/b[0]; d[1] = a[1]/b[1];
 106 }
 107
 108 CXR_INLINE double v2_length2( v2f a )
 109 {
 110         return v2_dot( a, a );
 111 }
 112
 113 CXR_INLINE double v2_length( v2f a )
 114 {
 115         return sqrt( v2_length2( a ) );
 116 }
 117
 118 CXR_INLINE double v2_dist2( v2f a, v2f b )
 119 {
 120         v2f delta;
 121         v2_sub( a, b, delta );
 122         return v2_length2( delta );
 123 }
 124
 125 CXR_INLINE double v2_dist( v2f a, v2f b )
 126 {
 127         return sqrt( v2_dist2( a, b ) );
 128 }
 129
 130 CXR_INLINE void v2_normalize( v2f a )
 131 {
 132         v2_muls( a, 1.0 / v2_length( a ), a );
 133 }
 134
 135 // Vector 3
 136 // ==================================================================================================================
 137
 138 CXR_INLINE void v3_zero( v3f a )
 139 {
 140         a[0] = 0.f; a[1] = 0.f; a[2] = 0.f;
 141 }
 142
 143 CXR_INLINE void v3_copy( v3f a, v3f b )
 144 {
 145         b[0] = a[0]; b[1] = a[1]; b[2] = a[2];
 146 }
 147
 148 CXR_INLINE void v3_add( v3f a, v3f b, v3f d )
 149 {
 150         d[0] = a[0]+b[0]; d[1] = a[1]+b[1]; d[2] = a[2]+b[2];
 151 }
 152
 153 CXR_INLINE void v3_sub( v3f a, v3f b, v3f d )
 154 {
 155         d[0] = a[0]-b[0]; d[1] = a[1]-b[1]; d[2] = a[2]-b[2];
 156 }
 157
 158 CXR_INLINE void v3_mul( v3f a, v3f b, v3f d )
 159 {
 160         d[0] = a[0]*b[0]; d[1] = a[1]*b[1]; d[2] = a[2]*b[2];
 161 }
 162
 163 CXR_INLINE void v3_div( v3f a, v3f b, v3f d )
 164 {
 165         d[0] = a[0]/b[0]; d[1] = a[1]/b[1];     d[2] = a[2]/b[2];
 166 }
 167
 168 CXR_INLINE void v3_muls( v3f a, double s, v3f d )
 169 {
 170         d[0] = a[0]*s; d[1] = a[1]*s; d[2] = a[2]*s;
 171 }
 172
 173 CXR_INLINE void v3_divs( v3f a, double s, v3f d )
 174 {
 175         d[0] = a[0]/s; d[1] = a[1]/s; d[2] = a[2]/s;
 176 }
 177
 178 CXR_INLINE void v3_muladds( v3f a, v3f b, double s, v3f d )
 179 {
 180         d[0] = a[0]+b[0]*s; d[1] = a[1]+b[1]*s; d[2] = a[2]+b[2]*s;
 181 }
 182
 183 CXR_INLINE double v3_dot( v3f a, v3f b )
 184 {
 185         return a[0] * b[0] + a[1] * b[1] + a[2] * b[2];
 186 }
 187
 188 CXR_INLINE void v3_cross( v3f a, v3f b, v3f d )
 189 {
 190   d[0] = a[1] * b[2] - a[2] * b[1];
 191   d[1] = a[2] * b[0] - a[0] * b[2];
 192   d[2] = a[0] * b[1] - a[1] * b[0];
 193 }
 194
 195 CXR_INLINE double v3_length2( v3f a )
 196 {
 197         return v3_dot( a, a );
 198 }
 199
 200 CXR_INLINE double v3_length( v3f a )
 201 {
 202         return sqrt( v3_length2( a ) );
 203 }
 204
 205 CXR_INLINE double v3_dist2( v3f a, v3f b )
 206 {
 207         v3f delta;
 208         v3_sub( a, b, delta );
 209         return v3_length2( delta );
 210 }
 211
 212 CXR_INLINE double v3_dist( v3f a, v3f b )
 213 {
 214         return sqrt( v3_dist2( a, b ) );
 215 }
 216
 217 CXR_INLINE void v3_normalize( v3f a )
 218 {
 219         v3_muls( a, 1.0 / v3_length( a ), a );
 220 }
 221
 222 CXR_INLINE void v3_negate( v3f a, v3f dest )
 223 {
 224    v3_muls( a, -1.0, dest );
 225 }
 226
 227 CXR_INLINE double cxr_lerpf( double a, double b, double t )
 228 {
 229         return a + t*(b-a);
 230 }
 231
 232 CXR_INLINE void v3_lerp( v3f a, v3f b, double t, v3f d )
 233 {
 234         d[0] = a[0] + t*(b[0]-a[0]);
 235         d[1] = a[1] + t*(b[1]-a[1]);
 236         d[2] = a[2] + t*(b[2]-a[2]);
 237 }
 238
 239 CXR_INLINE void v3_minv( v3f a, v3f b, v3f dest )
 240 {
 241         dest[0] = cxr_minf(a[0], b[0]);
 242         dest[1] = cxr_minf(a[1], b[1]);
 243         dest[2] = cxr_minf(a[2], b[2]);
 244 }
 245
 246 CXR_INLINE void v3_maxv( v3f a, v3f b, v3f dest )
 247 {
 248         dest[0] = cxr_maxf(a[0], b[0]);
 249         dest[1] = cxr_maxf(a[1], b[1]);
 250         dest[2] = cxr_maxf(a[2], b[2]);
 251 }
 252
 253 CXR_INLINE double v3_minf( v3f a )
 254 {
 255         return cxr_minf( cxr_minf( a[0], a[1] ), a[2] );
 256 }
 257
 258 CXR_INLINE double v3_maxf( v3f a )
 259 {
 260         return cxr_maxf( cxr_maxf( a[0], a[1] ), a[2] );
 261 }
 262
 263 CXR_INLINE void v3_fill( v3f a, double v )
 264 {
 265         a[0] = v;
 266         a[1] = v;
 267         a[2] = v;
 268 }
 269
 270 // Vector 4
 271 // ==================================================================================================================
 272
 273 CXR_INLINE void v4_copy( v4f a, v4f b )
 274 {
 275         b[0] = a[0]; b[1] = a[1]; b[2] = a[2]; b[3] = a[3];
 276 }
 277
 278 CXR_INLINE void v4_zero( v4f a )
 279 {
 280         a[0] = 0.f; a[1] = 0.f; a[2] = 0.f; a[3] = 0.f;
 281 }
 282
 283 CXR_INLINE void v4_muls( v4f a, double s, v4f d )
 284 {
 285         d[0] = a[0]*s; d[1] = a[1]*s; d[2] = a[2]*s; d[3] = a[3]*s;
 286 }
 287
 288 // Matrix 3x3
 289 //======================================================================================================
 290
 291 CXR_INLINE void m3x3_inv_transpose( m3x3f src, m3x3f dest )
 292 {
 293         double a = src[0][0], b = src[0][1], c = src[0][2],
 294                         d = src[1][0], e = src[1][1], f = src[1][2],
 295                         g = src[2][0], h = src[2][1], i = src[2][2];
 296
 297         double det =    1.f /
 298                                         (+a*(e*i-h*f)
 299                 -b*(d*i-f*g)
 300                 +c*(d*h-e*g));
 301
 302         dest[0][0] =  (e*i-h*f)*det;
 303         dest[1][0] = -(b*i-c*h)*det;
 304         dest[2][0] =  (b*f-c*e)*det;
 305         dest[0][1] = -(d*i-f*g)*det;
 306         dest[1][1] =  (a*i-c*g)*det;
 307         dest[2][1] = -(a*f-d*c)*det;
 308         dest[0][2] =  (d*h-g*e)*det;
 309         dest[1][2] = -(a*h-g*b)*det;
 310         dest[2][2] =  (a*e-d*b)*det;
 311 }
 312
 313 CXR_INLINE void m3x3_mulv( m3x3f m, v3f v, v3f d )
 314 {
 315         v3f res;
 316
 317         res[0] = m[0][0]*v[0] + m[1][0]*v[1] + m[2][0]*v[2];
 318         res[1] = m[0][1]*v[0] + m[1][1]*v[1] + m[2][1]*v[2];
 319         res[2] = m[0][2]*v[0] + m[1][2]*v[1] + m[2][2]*v[2];
 320
 321         v3_copy( res, d );
 322 }
 323
 324 // Matrix 4x3
 325 // ==================================================================================================================
 326
 327 #define M4X3_IDENTITY   {{1.0f, 0.0f, 0.0f, },\
 328                                                                 { 0.0f, 1.0f, 0.0f, },\
 329                                                                 { 0.0f, 0.0f, 1.0f, },\
 330                                                                 { 0.0f, 0.0f, 0.0f }}
 331
 332 CXR_INLINE void m4x3_to_3x3( m4x3f a, m3x3f b )
 333 {
 334         v3_copy( a[0], b[0] );
 335         v3_copy( a[1], b[1] );
 336         v3_copy( a[2], b[2] );
 337 }
 338
 339 CXR_INLINE void m4x3_copy( m4x3f a, m4x3f b )
 340 {
 341         v3_copy( a[0], b[0] );
 342         v3_copy( a[1], b[1] );
 343         v3_copy( a[2], b[2] );
 344         v3_copy( a[3], b[3] );
 345 }
 346
 347 CXR_INLINE void m4x3_identity( m4x3f a )
 348 {
 349         m4x3f id = M4X3_IDENTITY;
 350         m4x3_copy( id, a );
 351 }
 352
 353 CXR_INLINE void m4x3_mul( m4x3f a, m4x3f b, m4x3f d )
 354 {
 355         double
 356         a00 = a[0][0], a01 = a[0][1], a02 = a[0][2],
 357         a10 = a[1][0], a11 = a[1][1], a12 = a[1][2],
 358         a20 = a[2][0], a21 = a[2][1], a22 = a[2][2],
 359         a30 = a[3][0], a31 = a[3][1], a32 = a[3][2],
 360         b00 = b[0][0], b01 = b[0][1], b02 = b[0][2],
 361         b10 = b[1][0], b11 = b[1][1], b12 = b[1][2],
 362         b20 = b[2][0], b21 = b[2][1], b22 = b[2][2],
 363         b30 = b[3][0], b31 = b[3][1], b32 = b[3][2];
 364
 365         d[0][0] = a00*b00 + a10*b01 + a20*b02;
 366         d[0][1] = a01*b00 + a11*b01 + a21*b02;
 367         d[0][2] = a02*b00 + a12*b01 + a22*b02;
 368         d[1][0] = a00*b10 + a10*b11 + a20*b12;
 369         d[1][1] = a01*b10 + a11*b11 + a21*b12;
 370         d[1][2] = a02*b10 + a12*b11 + a22*b12;
 371         d[2][0] = a00*b20 + a10*b21 + a20*b22;
 372         d[2][1] = a01*b20 + a11*b21 + a21*b22;
 373         d[2][2] = a02*b20 + a12*b21 + a22*b22;
 374         d[3][0] = a00*b30 + a10*b31 + a20*b32 + a30;
 375         d[3][1] = a01*b30 + a11*b31 + a21*b32 + a31;
 376         d[3][2] = a02*b30 + a12*b31 + a22*b32 + a32;
 377 }
 378
 379 CXR_INLINE void m4x3_mulv( m4x3f m, v3f v, v3f d )
 380 {
 381         v3f res;
 382
 383         res[0] = m[0][0]*v[0] + m[1][0]*v[1] + m[2][0]*v[2] + m[3][0];
 384         res[1] = m[0][1]*v[0] + m[1][1]*v[1] + m[2][1]*v[2] + m[3][1];
 385         res[2] = m[0][2]*v[0] + m[1][2]*v[1] + m[2][2]*v[2] + m[3][2];
 386
 387         v3_copy( res, d );
 388 }
 389
 390 // Affine transforms
 391
 392 CXR_INLINE void m4x3_translate( m4x3f m, v3f v )
 393 {
 394         v3_muladds( m[3], m[0], v[0], m[3] );
 395         v3_muladds( m[3], m[1], v[1], m[3] );
 396         v3_muladds( m[3], m[2], v[2], m[3] );
 397 }
 398
 399 CXR_INLINE void m4x3_scale( m4x3f m, double s )
 400 {
 401         v3_muls( m[0], s, m[0] );
 402         v3_muls( m[1], s, m[1] );
 403         v3_muls( m[2], s, m[2] );
 404 }
 405
 406 CXR_INLINE void m4x3_rotate_x( m4x3f m, double angle )
 407 {
 408         m4x3f t = M4X3_IDENTITY;
 409         double c, s;
 410
 411         c = cosf( angle );
 412         s = sinf( angle );
 413
 414         t[1][1] =  c;
 415         t[1][2] =  s;
 416         t[2][1] = -s;
 417         t[2][2] =  c;
 418
 419         m4x3_mul( m, t, m );
 420 }
 421
 422 CXR_INLINE void m4x3_rotate_y( m4x3f m, double angle )
 423 {
 424         m4x3f t = M4X3_IDENTITY;
 425         double c, s;
 426
 427         c = cosf( angle );
 428         s = sinf( angle );
 429
 430         t[0][0] =  c;
 431         t[0][2] = -s;
 432         t[2][0] =  s;
 433         t[2][2] =  c;
 434
 435         m4x3_mul( m, t, m );
 436 }
 437
 438 CXR_INLINE void m4x3_rotate_z( m4x3f m, double angle )
 439 {
 440         m4x3f t = M4X3_IDENTITY;
 441         double c, s;
 442
 443         c = cosf( angle );
 444         s = sinf( angle );
 445
 446         t[0][0] =  c;
 447         t[0][1] =  s;
 448         t[1][0] = -s;
 449         t[1][1] =  c;
 450
 451         m4x3_mul( m, t, m );
 452 }
 453
 454 // Warning: These functions are unoptimized..
 455 CXR_INLINE void m4x3_expand_aabb_point( m4x3f m, boxf box, v3f point )
 456 {
 457         v3f v;
 458         m4x3_mulv( m, point, v );
 459
 460         v3_minv( box[0], v, box[0] );
 461         v3_maxv( box[1], v, box[1] );
 462 }
 463
 464 CXR_INLINE void box_concat( boxf a, boxf b )
 465 {
 466         v3_minv( a[0], b[0], a[0] );
 467         v3_maxv( a[1], b[1], a[1] );
 468 }
 469
 470 CXR_INLINE void box_copy( boxf a, boxf b )
 471 {
 472         v3_copy( a[0], b[0] );
 473         v3_copy( a[1], b[1] );
 474 }
 475
 476 CXR_INLINE void m4x3_transform_aabb( m4x3f m, boxf box )
 477 {
 478         v3f a; v3f b;
 479
 480         v3_copy( box[0], a );
 481         v3_copy( box[1], b );
 482         v3_fill( box[0],  INFINITY );
 483         v3_fill( box[1], -INFINITY );
 484
 485         m4x3_expand_aabb_point( m, box, a );
 486         m4x3_expand_aabb_point( m, box, (v3f){ a[0], b[1], a[2] } );
 487         m4x3_expand_aabb_point( m, box, (v3f){ b[0], a[1], a[2] } );
 488         m4x3_expand_aabb_point( m, box, (v3f){ b[0], b[1], a[2] } );
 489         m4x3_expand_aabb_point( m, box, b );
 490         m4x3_expand_aabb_point( m, box, (v3f){ a[0], b[1], b[2] } );
 491         m4x3_expand_aabb_point( m, box, (v3f){ b[0], a[1], b[2] } );
 492         m4x3_expand_aabb_point( m, box, (v3f){ b[0], b[1], b[2] } );
 493 }
 494
 495 CXR_INLINE void tri_normal( v3f p0, v3f p1, v3f p2, v3f normal )
 496 {
 497    v3f v0, v1;
 498    v3_sub( p1, p0, v0 );
 499    v3_sub( p2, p0, v1 );
 500    v3_cross( v0, v1, normal );
 501    v3_normalize( normal );
 502 }
 503
 504 CXR_INLINE void tri_to_plane( v3f a, v3f b, v3f c, v4f plane )
 505 {
 506    tri_normal( a,b,c, plane );
 507    plane[3] = v3_dot( plane, a );
 508 }
 509
 510 // TODO update this code to use normal v3_x functions
 511 CXR_INLINE void tri_to_plane1( double a[3], double b[3], double c[3], double p[4] )
 512 {
 513         double edge0[3];
 514         double edge1[3];
 515         double l;
 516
 517         edge0[0] = b[0] - a[0];
 518         edge0[1] = b[1] - a[1];
 519         edge0[2] = b[2] - a[2];
 520
 521         edge1[0] = c[0] - a[0];
 522         edge1[1] = c[1] - a[1];
 523         edge1[2] = c[2] - a[2];
 524
 525         p[0] = edge0[1] * edge1[2] - edge0[2] * edge1[1];
 526         p[1] = edge0[2] * edge1[0] - edge0[0] * edge1[2];
 527         p[2] = edge0[0] * edge1[1] - edge0[1] * edge1[0];
 528
 529         l = sqrt(p[0] * p[0] + p[1] * p[1] + p[2] * p[2]);
 530         p[3] = (p[0] * a[0] + p[1] * a[1] + p[2] * a[2]) / l;
 531
 532         p[0] = p[0] / l;
 533         p[1] = p[1] / l;
 534         p[2] = p[2] / l;
 535 }
 536
 537 CXR_INLINE int plane_intersect( double a[4], double b[4], double c[4], double p[3] )
 538 {
 539         double const epsilon = 0.001;
 540
 541         double x[3];
 542         double d;
 543
 544         x[0] = a[1] * b[2] - a[2] * b[1];
 545         x[1] = a[2] * b[0] - a[0] * b[2];
 546         x[2] = a[0] * b[1] - a[1] * b[0];
 547
 548         d = x[0] * c[0] + x[1] * c[1] + x[2] * c[2];
 549
 550         if( d < epsilon && d > -epsilon ) return 0;
 551
 552         p[0] = (b[1] * c[2] - b[2] * c[1]) * -a[3];
 553         p[1] = (b[2] * c[0] - b[0] * c[2]) * -a[3];
 554         p[2] = (b[0] * c[1] - b[1] * c[0]) * -a[3];
 555
 556         p[0] += (c[1] * a[2] - c[2] * a[1]) * -b[3];
 557         p[1] += (c[2] * a[0] - c[0] * a[2]) * -b[3];
 558         p[2] += (c[0] * a[1] - c[1] * a[0]) * -b[3];
 559
 560         p[0] += (a[1] * b[2] - a[2] * b[1]) * -c[3];
 561         p[1] += (a[2] * b[0] - a[0] * b[2]) * -c[3];
 562         p[2] += (a[0] * b[1] - a[1] * b[0]) * -c[3];
 563
 564         p[0] = -p[0] / d;
 565         p[1] = -p[1] / d;
 566         p[2] = -p[2] / d;
 567
 568         return 1;
 569 }
 570
 571 CXR_INLINE void normal_to_plane( v3f normal, v3f p, v4f plane )
 572 {
 573    v3_copy( normal, plane );
 574    plane[3] = v3_dot( normal, p );
 575 }
 576
 577 CXR_INLINE double plane_polarity( double p[4], double a[3] )
 578 {
 579         return
 580         (a[0] * p[0] + a[1] * p[1] + a[2] * p[2])
 581         -(p[0]*p[3] * p[0] + p[1]*p[3] * p[1] + p[2]*p[3] * p[2])
 582         ;
 583 }
 584
 585 CXR_INLINE void plane_project_point( v4f plane, v3f a, v3f d )
 586 {
 587    v3f ref, delta;
 588    v3_muls( plane, plane[3], ref );
 589
 590    v3_sub( a, ref, delta );
 591    v3_muladds( a, plane, -v3_dot(delta,plane), d );
 592 }
 593
 594 CXR_INLINE double line_line_dist( v3f pa0, v3f pa1, v3f pb0, v3f pb1 )
 595 {
 596    v3f va, vb, n, delta;
 597    v3_sub( pa1, pa0, va );
 598    v3_sub( pb1, pb0, vb );
 599
 600    v3_cross( va, vb, n );
 601    v3_normalize( n );
 602
 603    v3_sub( pb0, pa0, delta );
 604
 605    return fabs( v3_dot( n, delta ) );
 606 }
 607
 608 CXR_INLINE double segment_segment_dist( v3f a0, v3f a1, v3f b0, v3f b1,
 609    v3f a, v3f b )
 610 {
 611    v3f r,u,v;
 612    v3_sub( b0, a0, r );
 613    v3_sub( a1, a0, u );
 614    v3_sub( b1, b0, v );
 615
 616    double ru = v3_dot( r,u ),
 617           rv = v3_dot( r,v ),
 618           uu = v3_dot( u,u ),
 619           uv = v3_dot( u,v ),
 620           vv = v3_dot( v,v );
 621
 622    double det = uu*vv - uv*uv,
 623           s,
 624           t;
 625
 626    if( det < 1e-6 *uu*vv )
 627    {
 628       s = ru/uu;
 629       t = 0.0;
 630    }
 631    else
 632    {
 633       s = (ru*vv - rv*uv)/det;
 634       t = (ru*uv - rv*uu)/det;
 635    }
 636
 637    s = cxr_clampf( s, 0.0, 1.0 );
 638    t = cxr_clampf( t, 0.0, 1.0 );
 639
 640    double S = cxr_clampf((t*uv + ru)/uu, 0.0, 1.0),
 641           T = cxr_clampf((s*uv - rv)/vv, 0.0, 1.0);
 642
 643    v3_muladds( a0, u, S, a );
 644    v3_muladds( b0, v, T, b );
 645
 646    return v3_dist( a, b );
 647 }