Developer Documentation

Algorithms.hh
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41
42
43
44 #ifndef GEO_ALGORITHMS_HH
45 #define GEO_ALGORITHMS_HH
46
47
48 //== INCLUDES =================================================================
49
50 #include <cfloat>
51 #include <ACG/Math/VectorT.hh>
52 #include <vector>
53 #include <iostream>
54
55 #include "../Math/Matrix3x3T.hh"
56
57
58 namespace ACG {
59 namespace Geometry {
60
61
62 //== 3D STUFF =================================================================
63
64
65
67 template<typename Scalar>
68 bool
69 circumCenter( const VectorT<Scalar,3>& _v0,
70 const VectorT<Scalar,3>& _v1,
71 const VectorT<Scalar,3>& _v2,
72 const VectorT<Scalar,3>& _v3,
73 VectorT<Scalar,3>& _result );
74
75
77 template<typename Scalar>
78 Scalar
79 circumRadiusSquared( const VectorT<Scalar,3>& _v0,
80 const VectorT<Scalar,3>& _v1,
81 const VectorT<Scalar,3>& _v2,
82 const VectorT<Scalar,3>& _v3 )
83 {
84 VectorT<Scalar,3> cc;
85 return circumCenter(_v0, _v1, _v2, _v3, cc) ? (cc-_v0).sqrnorm() : FLT_MAX;
86 }
87
88
90 template<typename Scalar>
91 Scalar
92 circumRadius( const VectorT<Scalar,3>& _v0,
93 const VectorT<Scalar,3>& _v1,
94 const VectorT<Scalar,3>& _v2,
95 const VectorT<Scalar,3>& _v3 )
96 {
97 return sqrt(circumRadiusSquared(_v0, _v1, _v2, _v3));
98 }
99
112 template<typename Scalar>
113 bool edgeConvexPolygonIntersection(std::vector<VectorT<Scalar,3> > _polygon_points,
114 VectorT<Scalar,3> _v0,
115 VectorT<Scalar,3> _v1,
116 VectorT<Scalar,3> &_result);
117
118
133 template<typename Scalar>
134 bool
135 rotationOfTwoVectors( const VectorT<Scalar,3>& _v0,
136 const VectorT<Scalar,3>& _v1,
137 VectorT<Scalar,3>& _axis,
138 Scalar& _angle,
139 bool _degree = true);
140
141
150 template <typename Scalar>
151 VectorT<Scalar,3>
152 perpendicular( const VectorT<Scalar,3>& _v );
153
154
171 template<typename Vec>
172 bool
173 triangleIntersection( const Vec& _o,
174 const Vec& _dir,
175 const Vec& _v0,
176 const Vec& _v1,
177 const Vec& _v2,
178 typename Vec::value_type& _t,
179 typename Vec::value_type& _u,
180 typename Vec::value_type& _v );
181
182
195 template<typename Vec>
196 bool
197 axisAlignedBBIntersection( const Vec& _o,
198 const Vec& _dir,
199 const Vec& _bbmin,
200 const Vec& _bbmax,
201 typename Vec::value_type& _t0,
202 typename Vec::value_type& _t1 );
203
204
205 //== 2D STUFF =================================================================
206
208 template<typename Scalar>
209 Scalar
210 pointLineOrientation( const VectorT<Scalar,2>& _p,
211 const VectorT<Scalar,2>& _v0,
212 const VectorT<Scalar,2>& _v1 )
213 {
214 VectorT<Scalar,2> d1(_p-_v0), d2(_v1-_v0);
215 return (d1[0]*d2[1]-d1[1]*d2[0]);
216 }
217
218
220 template<typename Scalar>
221 bool
222 isCCW( const VectorT<Scalar,2>& _v0,
223 const VectorT<Scalar,2>& _v1,
224 const VectorT<Scalar,2>& _v2 )
225 {
226 return ( pointLineOrientation(_v0, _v1, _v2) < Scalar(0.0) );
227 }
228
229
231 template<typename Scalar>
232 bool
233 isCW( const VectorT<Scalar,2>& _v0,
234 const VectorT<Scalar,2>& _v1,
235 const VectorT<Scalar,2>& _v2 )
236 {
237 return ( pointLineOrientation(_v0, _v1, _v2) > Scalar(0.0) );
238 }
239
240
242 template<typename Scalar>
243 bool
244 lineIntersection( const VectorT<Scalar,2>& _v0,
245 const VectorT<Scalar,2>& _v1,
246 const VectorT<Scalar,2>& _v2,
247 const VectorT<Scalar,2>& _v3,
248 Scalar& _t1,
249 Scalar& _t2 );
250
251
252 //===========================================================================
255 //===========================================================================
256
257
266 template<class Vec>
267 typename Vec::value_type
268 distPointLineSquared( const Vec& _p,
269 const Vec& _v0,
270 const Vec& _v1,
271 Vec* _min_v=0);
272
273
274
286 template<class Vec>
287 typename Vec::value_type
288 distPointLine( const Vec& _p,
289 const Vec& _v0,
290 const Vec& _v1,
291 Vec* _min_v=0 )
292 { return sqrt(distPointLineSquared(_p, _v0, _v1, _min_v)); }
293
294
296 template <class Vec>
297 typename Vec::value_type
298 distPointTriangleSquared( const Vec& _p,
299 const Vec& _v0,
300 const Vec& _v1,
301 const Vec& _v2,
302 Vec& _nearestPoint );
303
315 template <class Vec>
316 typename Vec::value_type
317 distPointTriangleSquaredStable( const Vec& _p,
318 const Vec& _v0,
319 const Vec& _v1,
320 const Vec& _v2,
321 Vec& _nearestPoint );
322
324 template <class Vec>
325 typename Vec::value_type
326 distPointTriangle( const Vec& _p,
327 const Vec& _v0,
328 const Vec& _v1,
329 const Vec& _v2,
330 Vec& _nearestPoint )
331 { return sqrt(distPointTriangleSquared(_p, _v0, _v1, _v2, _nearestPoint)); }
332
343 template <class Vec>
344 typename Vec::value_type
345 distPointTriangleStable( const Vec& _p,
346 const Vec& _v0,
347 const Vec& _v1,
348 const Vec& _v2,
349 Vec& _nearestPoint )
350 { return sqrt(distPointTriangleSquaredStable(_p, _v0, _v1, _v2, _nearestPoint)); }
351
360 template < typename VectorT , typename ValueT >
361 inline
362 ValueT
363 distPointPlane(const VectorT& _porigin,
364 const VectorT& _pnormal,
365 const VectorT& _point);
366
367
370 //===========================================================================
373 //===========================================================================
374
376 template<typename Scalar>
377 Scalar
378 distLineLineSquared( const VectorT<Scalar,3>& _v00,
379 const VectorT<Scalar,3>& _v01,
380 const VectorT<Scalar,3>& _v10,
381 const VectorT<Scalar,3>& _v11,
382 VectorT<Scalar,3>* _min_v0=0,
383 VectorT<Scalar,3>* _min_v1=0,
384 bool _fastApprox=false );
385
386
388 template<typename Scalar>
389 Scalar
390 distLineLine( const VectorT<Scalar,3>& _v00,
391 const VectorT<Scalar,3>& _v01,
392 const VectorT<Scalar,3>& _v10,
393 const VectorT<Scalar,3>& _v11,
394 VectorT<Scalar,3>* _min_v0=0,
395 VectorT<Scalar,3>* _min_v1=0 )
396 {
397 return sqrt(distLineLineSquared(_v00, _v01, _v10, _v11,
398 _min_v0, _min_v1));
399 }
400
404 //===========================================================================
407 //===========================================================================
408
409
416 template < typename VectorT >
417 VectorT projectToEdge(const VectorT& _start ,
418 const VectorT& _end ,
419 const VectorT& _point );
420
421
428 template < typename VectorT >
429 inline
430 VectorT
431 projectToPlane(const VectorT& _porigin,
432 const VectorT& _pnormal,
433 const VectorT& _point);
434
437 //===========================================================================
440 //===========================================================================
441
446 template<typename Scalar>
448 bool
449 baryCoord( const VectorT<Scalar,2>& _p,
450 const VectorT<Scalar,2>& _u,
451 const VectorT<Scalar,2>& _v,
452 const VectorT<Scalar,2>& _w,
453 VectorT<Scalar,3>& _result );
454
455
457 template<typename Scalar>
458 bool
459 isInTriangle( const VectorT<Scalar,2>& _p,
460 const VectorT<Scalar,2>& _u,
461 const VectorT<Scalar,2>& _v,
462 const VectorT<Scalar,2>& _w )
463 {
464 VectorT<Scalar,3> bary;
465 if (baryCoord(_p, _u, _v, _w, bary))
466 return ((bary[0]>0.0) && (bary[1]>0.0) && (bary[2]>0.0));
467 else {
468 std::cerr << "point in triangle error\n";
469 return false;
470 }
471 }
472
473 template<typename Scalar>
474 bool
475 circumCenter( const VectorT<Scalar,2>& _v0,
476 const VectorT<Scalar,2>& _v1,
477 const VectorT<Scalar,2>& _v2,
478 VectorT<Scalar,2>& _result );
479
482 //===========================================================================
485 //===========================================================================
486
489 template<typename Scalar>
490 bool
491 baryCoord( const VectorT<Scalar,3>& _p,
492 const VectorT<Scalar,3>& _u,
493 const VectorT<Scalar,3>& _v,
494 const VectorT<Scalar,3>& _w,
495 VectorT<Scalar,3>& _result );
496
497
499 template<typename Scalar>
500 bool
501 minSphere(const VectorT<Scalar,3>& _v0,
502 const VectorT<Scalar,3>& _v1,
503 const VectorT<Scalar,3>& _v2,
504 VectorT<Scalar,3>& _center,
505 Scalar& _radius);
506
507
509 template<typename Scalar>
510 Scalar
511 minRadiusSquared( const VectorT<Scalar,3>& _v0,
512 const VectorT<Scalar,3>& _v1,
513 const VectorT<Scalar,3>& _v2 );
514
515
517 template<typename Scalar>
518 Scalar
519 minRadius( const VectorT<Scalar,3>& _v0,
520 const VectorT<Scalar,3>& _v1,
521 const VectorT<Scalar,3>& _v2 )
522 {
523 return sqrt(minRadiusSquared(_v0, _v1, _v2));
524 }
525
526
528 template<typename Scalar>
529 bool
530 circumCenter( const VectorT<Scalar,3>& _v0,
531 const VectorT<Scalar,3>& _v1,
532 const VectorT<Scalar,3>& _v2,
533 VectorT<Scalar,3>& _result );
534
535
537 template<typename Scalar>
538 Scalar
539 circumRadiusSquared( const VectorT<Scalar,3>& _v0,
540 const VectorT<Scalar,3>& _v1,
541 const VectorT<Scalar,3>& _v2 );
542
543
545 template<typename Scalar>
546 Scalar
547 circumRadius( const VectorT<Scalar,3>& _v0,
548 const VectorT<Scalar,3>& _v1,
549 const VectorT<Scalar,3>& _v2 )
550 {
551 return sqrt(circumRadiusSquared(_v0, _v1, _v2));
552 }
553
561 template<class VectorT>
562 int isObtuse(const VectorT& _p0,
563 const VectorT& _p1,
564 const VectorT& _p2 );
565
568 //===========================================================================
571 //===========================================================================
572
573
580 template <class Vec>
581 typename Vec::value_type
582 triangleAreaSquared( const Vec& _v0,
583 const Vec& _v1,
584 const Vec& _v2 );
585
586
593 template <class Vec>
594 typename Vec::value_type
595 triangleArea( const Vec& _v0,
596 const Vec& _v1,
597 const Vec& _v2 )
598 {
599 return sqrt(triangleAreaSquared(_v0,_v1,_v2));
600 }
601
602
609 template <typename Scalar, int N>
610 Scalar
611 aspectRatio( const VectorT<Scalar, N>& _v0,
612 const VectorT<Scalar, N>& _v1,
613 const VectorT<Scalar, N>& _v2 );
614
621 template <typename Scalar, int N>
622 Scalar
623 roundness( const VectorT<Scalar, N>& _v0,
624 const VectorT<Scalar, N>& _v1,
625 const VectorT<Scalar, N>& _v2 );
626
629 template<typename Vector>
630 Vector closestPointLineSegment(Vector x, Vector p1, Vector p2) {
631 const auto delta = ((p2-p1)|(x-p1)) / (p2-p1).sqrnorm();
632 //std::cout << "\x1b[32mdelta = " << delta << "\x1b[0m" << std::endl;
633 if (delta <= 0) {
634 //std::cout << "\x1b[43mdelta <= 0\x1b[0m" << std::endl;
635 return p1;
636 } else if (delta >= 1) {
637 //std::cout << "\x1b[43mdelta >= 1\x1b[0m" << std::endl;
638 return p2;
639 } else if (delta != delta) { // p1 = p2 or x = p1
640 //std::cout << "\x1b[43mdelta != delta\x1b[0m" << std::endl;
641 return (x-p1).sqrnorm() < (x-p2).sqrnorm() ? p1 : p2;
642 } else {
643 //std::cout << "\x1b[43mdelta \\in [0, 1]\x1b[0m" << std::endl;
644 return (1 - delta) * p1 + delta * p2;
645 }
646 };
647
648 template<typename Vector>
649 Vector closestPointTri(Vector p, Vector a, Vector b, Vector c) {
650 constexpr double thresh = 1e-8;
651
652 const auto n = ((b - a) % (c - a)); // normalization unnecessary
653
654 if ((b-a).sqrnorm() < thresh || (c-a).sqrnorm() < thresh || n.sqrnorm() < thresh) {
655 //std::cout << "\x1b[42mDegenerate case.\x1b[0m" << std::endl;
656 // Degenerate triangle. Find distance to longest segment.
657 std::array<ACG::Vec3d, 2> max_segment = {a, b};
658 double max_len = (b-a).sqrnorm();
659 if ((c-a).sqrnorm() > max_len)
660 max_segment = {a, c};
661 if ((c-b).sqrnorm() > max_len)
662 max_segment = {b, c};
663
664 // closestPointLineSegment is stable, even if the segment is super short
665 return closestPointLineSegment(p, max_segment[0], max_segment[1]);
666 }
667
668 const auto abd = Matrix3x3d::fromColumns(a-c, b-c, n).inverse() * (p - c);
669 const bool alpha = (abd[0] >= 0.0),
670 beta = (abd[1] >= 0.0),
671 gamma = (1.0-abd[0]-abd[1] >= 0.0);
672
673 if (alpha && beta && gamma) {
674 //std::cout << "\x1b[42mInside case.\x1b[0m" << std::endl;
675 // Inside triangle.
676 return abd[0] * a + abd[1] * b + (1.0 - abd[0] - abd[1]) * c;
677 } else if (!alpha) {
678 //std::cout << "\x1b[42m!alpha case.\x1b[0m" << std::endl;
679 // Closest to line segment (b, c).
680 return closestPointLineSegment(p, b, c);
681 } else if (!beta) {
682 //std::cout << "\x1b[42m!beta case.\x1b[0m" << std::endl;
683 // Closest to line segment (a, c).
684 return closestPointLineSegment(p, a, c);
685 } else if (!gamma) {
686 //std::cout << "\x1b[42m!gamma case.\x1b[0m" << std::endl;
687 // Closest to line segment (a, b).
688 return closestPointLineSegment(p, a, b);
689 } else {
690 throw std::logic_error("This cannot happen.");
691 }
692 }
693
694 //=============================================================================
695 } // namespace Geometry
696 } // namespace ACG
697 //=============================================================================
698 #if defined(INCLUDE_TEMPLATES) && !defined(GEO_ALGORITHMS_C)
699 #define GEO_ALGORITHMS_TEMPLATES
700 #include "Algorithms.cc"
701 #endif
702 //=============================================================================
703 #endif // GEO_ALGORITHMS_HH defined
704 //=============================================================================
705
ACG::Geometry::distPointLineSquared
Vec::value_type distPointLineSquared(const Vec &_p, const Vec &_v0, const Vec &_v1, Vec *_min_v)
squared distance from point _p to line segment (_v0,_v1)
Definition: Algorithms.cc:290
ACG::Geometry::pointLineOrientation
Scalar pointLineOrientation(const VectorT< Scalar, 2 > &_p, const VectorT< Scalar, 2 > &_v0, const VectorT< Scalar, 2 > &_v1)
orientation of point _p w.r.t. line through _v0,_v1 in 2D
Definition: Algorithms.hh:210
ACG::Geometry::distLineLineSquared
Scalar distLineLineSquared(const VectorT< Scalar, 3 > &_v00, const VectorT< Scalar, 3 > &_v01, const VectorT< Scalar, 3 > &_v10, const VectorT< Scalar, 3 > &_v11, VectorT< Scalar, 3 > *_min_v0, VectorT< Scalar, 3 > *_min_v1, bool _fastApprox)
squared distance of lines (_v00, _v01) and (_v10, _v11)
Definition: Algorithms.cc:486
ACG::Geometry::isCCW
bool isCCW(const VectorT< Scalar, 2 > &_v0, const VectorT< Scalar, 2 > &_v1, const VectorT< Scalar, 2 > &_v2)
are 3 vertices in counterclockwise order? in 2D
Definition: Algorithms.hh:222
ACG::Geometry::distLineLine
Scalar distLineLine(const VectorT< Scalar, 3 > &_v00, const VectorT< Scalar, 3 > &_v01, const VectorT< Scalar, 3 > &_v10, const VectorT< Scalar, 3 > &_v11, VectorT< Scalar, 3 > *_min_v0=0, VectorT< Scalar, 3 > *_min_v1=0)
distance of lines (_v00, _v01) and (_v10, _v11)
Definition: Algorithms.hh:390
ACG::Geometry::triangleAreaSquared
Vec::value_type triangleAreaSquared(const Vec &_v0, const Vec &_v1, const Vec &_v2)
return squared area of triangle (_v0, _v1, _v2)
Definition: Algorithms.cc:163
ACG::Geometry::distPointTriangleStable
Vec::value_type distPointTriangleStable(const Vec &_p, const Vec &_v0, const Vec &_v1, const Vec &_v2, Vec &_nearestPoint)
distance from point _p to triangle (_v0, _v1, _v2)
Definition: Algorithms.hh:345
ACG
Namespace providing different geometric functions concerning angles.
Definition: BaseObjectData.hh:68
ACG::Geometry::circumRadius
Scalar circumRadius(const VectorT< Scalar, 3 > &_v0, const VectorT< Scalar, 3 > &_v1, const VectorT< Scalar, 3 > &_v2, const VectorT< Scalar, 3 > &_v3)
return radius of circumcircle of tetrahedron (_v0,_v1,_v2,_v3)
Definition: Algorithms.hh:92
ACG::Geometry::circumRadiusSquared
Scalar circumRadiusSquared(const VectorT< Scalar, 3 > &_v0, const VectorT< Scalar, 3 > &_v1, const VectorT< Scalar, 3 > &_v2)
return squared radius of circumcircle of triangle (_v0,_v1,_v2)
Definition: Algorithms.cc:940
OpenMesh::VectorT
Definition: Vector11T.hh:83
ACG::Geometry::triangleIntersection
bool triangleIntersection(const Vec &_o, const Vec &_dir, const Vec &_v0, const Vec &_v1, const Vec &_v2, typename Vec::value_type &_t, typename Vec::value_type &_u, typename Vec::value_type &_v)
Intersect a ray and a triangle.
Definition: Algorithms.cc:1307
ACG::Geometry::projectToEdge
VectorT projectToEdge(const VectorT &_start, const VectorT &_end, const VectorT &_point)
Definition: Algorithms.cc:874
ACG::Geometry::lineIntersection
bool lineIntersection(const VectorT< Scalar, 2 > &_v0, const VectorT< Scalar, 2 > &_v1, const VectorT< Scalar, 2 > &_v2, const VectorT< Scalar, 2 > &_v3, Scalar &_t1, Scalar &_t2)
intersect two line segments (_v0,_v1) and (_v2,_v3)
Definition: Algorithms.cc:1205
ACG::Geometry::aspectRatio
Scalar aspectRatio(const VectorT< Scalar, N > &_v0, const VectorT< Scalar, N > &_v1, const VectorT< Scalar, N > &_v2)
return aspect ratio (length/height) of triangle
Definition: Algorithms.cc:1256
ACG::Geometry::edgeConvexPolygonIntersection
bool edgeConvexPolygonIntersection(std::vector< VectorT< Scalar, 3 > > _polygon_points, VectorT< Scalar, 3 > _v0, VectorT< Scalar, 3 > _v1, VectorT< Scalar, 3 > &_result)
Get intersection point of a ray and a convex polygon.
Definition: Algorithms.cc:172
ACG::Geometry::circumCenter
bool circumCenter(const VectorT< Scalar, 3 > &_v0, const VectorT< Scalar, 3 > &_v1, const VectorT< Scalar, 3 > &_v2, VectorT< Scalar, 3 > &_result)
return circumcenter of triangle (_v0,_v1,_v2)
Definition: Algorithms.cc:906
ACG::Geometry::distPointPlane
ValueT distPointPlane(const VectorT &_porigin, const VectorT &_pnormal, const VectorT &_point)
Checks the distance from a point to a plane.
Definition: Algorithms.cc:862
ACG::Geometry::baryCoord
bool baryCoord(const VectorT< Scalar, 3 > &_p, const VectorT< Scalar, 3 > &_u, const VectorT< Scalar, 3 > &_v, const VectorT< Scalar, 3 > &_w, VectorT< Scalar, 3 > &_result)
Definition: Algorithms.cc:83
ACG::Geometry::distPointTriangleSquaredStable
Vec::value_type distPointTriangleSquaredStable(const Vec &_p, const Vec &_v0, const Vec &_v1, const Vec &_v2, Vec &_nearestPoint)
squared distance from point _p to triangle (_v0, _v1, _v2)
Definition: Algorithms.cc:466
ACG::Geometry::distPointTriangle
Vec::value_type distPointTriangle(const Vec &_p, const Vec &_v0, const Vec &_v1, const Vec &_v2, Vec &_nearestPoint)
distance from point _p to triangle (_v0, _v1, _v2)
Definition: Algorithms.hh:326
ACG::Geometry::minSphere
bool minSphere(const VectorT< Scalar, 3 > &_v0, const VectorT< Scalar, 3 > &_v1, const VectorT< Scalar, 3 > &_v2, VectorT< Scalar, 3 > &_center, Scalar &_radius)
construct min. enclosing sphere
Definition: Algorithms.cc:1030
VectorT
ACG::Geometry::distPointLine
Vec::value_type distPointLine(const Vec &_p, const Vec &_v0, const Vec &_v1, Vec *_min_v=0)
Compute distance from point to line segment.
Definition: Algorithms.hh:288
ACG::Geometry::rotationOfTwoVectors
bool rotationOfTwoVectors(const VectorT< Scalar, 3 > &_v0, const VectorT< Scalar, 3 > &_v1, VectorT< Scalar, 3 > &_axis, Scalar &_angle, bool _degree)
Get rotation axis and signed angle of rotation between two vectors.
Definition: Algorithms.cc:963
ACG::Geometry::isCW
bool isCW(const VectorT< Scalar, 2 > &_v0, const VectorT< Scalar, 2 > &_v1, const VectorT< Scalar, 2 > &_v2)
are 3 vertices in clockwise order? in 2D
Definition: Algorithms.hh:233
OpenMesh::VectorT::sqrnorm
decltype(std::declval< S >() *std::declval< S >()) sqrnorm() const
compute squared euclidean norm
Definition: Vector11T.hh:402
ACG::Geometry::minRadius
Scalar minRadius(const VectorT< Scalar, 3 > &_v0, const VectorT< Scalar, 3 > &_v1, const VectorT< Scalar, 3 > &_v2)
return radius of min. enclosing circle of triangle (_v0,_v1,_v2)
Definition: Algorithms.hh:519
ACG::Geometry::projectToPlane
VectorT projectToPlane(const VectorT &_porigin, const VectorT &_pnormal, const VectorT &_point)
projects a point to a plane
Definition: Algorithms.cc:894
ACG::Geometry::isInTriangle
bool isInTriangle(const VectorT< Scalar, 2 > &_p, const VectorT< Scalar, 2 > &_u, const VectorT< Scalar, 2 > &_v, const VectorT< Scalar, 2 > &_w)
is point _p in triangle (_v0,_v1,_v2) in 2D
Definition: Algorithms.hh:459
ACG::Geometry::roundness
Scalar roundness(const VectorT< Scalar, N > &_v0, const VectorT< Scalar, N > &_v1, const VectorT< Scalar, N > &_v2)
return roundness of triangle: 1=equilateral, 0=colinear
Definition: Algorithms.cc:1292
ACG::Geometry::minRadiusSquared
Scalar minRadiusSquared(const VectorT< Scalar, 3 > &_v0, const VectorT< Scalar, 3 > &_v1, const VectorT< Scalar, 3 > &_v2)
return squared radius of min. enclosing circle of triangle (_v0,_v1,_v2)
Definition: Algorithms.cc:1090
ACG::Geometry::perpendicular
VectorT< Scalar, 3 > perpendicular(const VectorT< Scalar, 3 > &v)
find a vector that&#39;s perpendicular to _v
Definition: Algorithms.cc:1152
ACG::Geometry::triangleArea
Vec::value_type triangleArea(const Vec &_v0, const Vec &_v1, const Vec &_v2)
return area of triangle (_v0, _v1, _v2)
Definition: Algorithms.hh:595
ACG::Geometry::axisAlignedBBIntersection
bool axisAlignedBBIntersection(const Vec &_o, const Vec &_dir, const Vec &_bbmin, const Vec &_bbmax, typename Vec::value_type &tmin, typename Vec::value_type &tmax)
Intersect a ray and an axis aligned bounding box.
Definition: Algorithms.cc:1359
ACG::Geometry::isObtuse
int isObtuse(const VectorT &_p0, const VectorT &_p1, const VectorT &_p2)
Definition: Algorithms.cc:1003