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Eigen  3.4.0
Homogeneous.h
1 // This file is part of Eigen, a lightweight C++ template library
2 // for linear algebra.
3 //
4 // Copyright (C) 2009-2010 Gael Guennebaud <gael.guennebaud@inria.fr>
5 //
6 // This Source Code Form is subject to the terms of the Mozilla
7 // Public License v. 2.0. If a copy of the MPL was not distributed
8 // with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
9 
10 #ifndef EIGEN_HOMOGENEOUS_H
11 #define EIGEN_HOMOGENEOUS_H
12 
13 namespace Eigen {
14 
30 namespace internal {
31 
32 template<typename MatrixType,int Direction>
33 struct traits<Homogeneous<MatrixType,Direction> >
34  : traits<MatrixType>
35 {
36  typedef typename traits<MatrixType>::StorageKind StorageKind;
37  typedef typename ref_selector<MatrixType>::type MatrixTypeNested;
38  typedef typename remove_reference<MatrixTypeNested>::type _MatrixTypeNested;
39  enum {
40  RowsPlusOne = (MatrixType::RowsAtCompileTime != Dynamic) ?
41  int(MatrixType::RowsAtCompileTime) + 1 : Dynamic,
42  ColsPlusOne = (MatrixType::ColsAtCompileTime != Dynamic) ?
43  int(MatrixType::ColsAtCompileTime) + 1 : Dynamic,
44  RowsAtCompileTime = Direction==Vertical ? RowsPlusOne : MatrixType::RowsAtCompileTime,
45  ColsAtCompileTime = Direction==Horizontal ? ColsPlusOne : MatrixType::ColsAtCompileTime,
46  MaxRowsAtCompileTime = RowsAtCompileTime,
47  MaxColsAtCompileTime = ColsAtCompileTime,
48  TmpFlags = _MatrixTypeNested::Flags & HereditaryBits,
49  Flags = ColsAtCompileTime==1 ? (TmpFlags & ~RowMajorBit)
50  : RowsAtCompileTime==1 ? (TmpFlags | RowMajorBit)
51  : TmpFlags
52  };
53 };
54 
55 template<typename MatrixType,typename Lhs> struct homogeneous_left_product_impl;
56 template<typename MatrixType,typename Rhs> struct homogeneous_right_product_impl;
57 
58 } // end namespace internal
59 
60 template<typename MatrixType,int _Direction> class Homogeneous
61  : public MatrixBase<Homogeneous<MatrixType,_Direction> >, internal::no_assignment_operator
62 {
63  public:
64 
65  typedef MatrixType NestedExpression;
66  enum { Direction = _Direction };
67 
69  EIGEN_DENSE_PUBLIC_INTERFACE(Homogeneous)
70 
71  EIGEN_DEVICE_FUNC explicit inline Homogeneous(const MatrixType& matrix)
72  : m_matrix(matrix)
73  {}
74 
75  EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR
76  inline Index rows() const EIGEN_NOEXCEPT { return m_matrix.rows() + (int(Direction)==Vertical ? 1 : 0); }
77  EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR
78  inline Index cols() const EIGEN_NOEXCEPT { return m_matrix.cols() + (int(Direction)==Horizontal ? 1 : 0); }
79 
80  EIGEN_DEVICE_FUNC const NestedExpression& nestedExpression() const { return m_matrix; }
81 
82  template<typename Rhs>
83  EIGEN_DEVICE_FUNC inline const Product<Homogeneous,Rhs>
84  operator* (const MatrixBase<Rhs>& rhs) const
85  {
86  eigen_assert(int(Direction)==Horizontal);
87  return Product<Homogeneous,Rhs>(*this,rhs.derived());
88  }
89 
90  template<typename Lhs> friend
91  EIGEN_DEVICE_FUNC inline const Product<Lhs,Homogeneous>
92  operator* (const MatrixBase<Lhs>& lhs, const Homogeneous& rhs)
93  {
94  eigen_assert(int(Direction)==Vertical);
95  return Product<Lhs,Homogeneous>(lhs.derived(),rhs);
96  }
97 
98  template<typename Scalar, int Dim, int Mode, int Options> friend
99  EIGEN_DEVICE_FUNC inline const Product<Transform<Scalar,Dim,Mode,Options>, Homogeneous >
100  operator* (const Transform<Scalar,Dim,Mode,Options>& lhs, const Homogeneous& rhs)
101  {
102  eigen_assert(int(Direction)==Vertical);
103  return Product<Transform<Scalar,Dim,Mode,Options>, Homogeneous>(lhs,rhs);
104  }
105 
106  template<typename Func>
107  EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE typename internal::result_of<Func(Scalar,Scalar)>::type
108  redux(const Func& func) const
109  {
110  return func(m_matrix.redux(func), Scalar(1));
111  }
112 
113  protected:
114  typename MatrixType::Nested m_matrix;
115 };
116 
130 template<typename Derived>
131 EIGEN_DEVICE_FUNC inline typename MatrixBase<Derived>::HomogeneousReturnType
133 {
134  EIGEN_STATIC_ASSERT_VECTOR_ONLY(Derived);
135  return HomogeneousReturnType(derived());
136 }
137 
148 template<typename ExpressionType, int Direction>
149 EIGEN_DEVICE_FUNC inline Homogeneous<ExpressionType,Direction>
151 {
152  return HomogeneousReturnType(_expression());
153 }
154 
172 template<typename Derived>
173 EIGEN_DEVICE_FUNC inline const typename MatrixBase<Derived>::HNormalizedReturnType
175 {
176  EIGEN_STATIC_ASSERT_VECTOR_ONLY(Derived);
177  return ConstStartMinusOne(derived(),0,0,
178  ColsAtCompileTime==1?size()-1:1,
179  ColsAtCompileTime==1?1:size()-1) / coeff(size()-1);
180 }
181 
196 template<typename ExpressionType, int Direction>
197 EIGEN_DEVICE_FUNC inline const typename VectorwiseOp<ExpressionType,Direction>::HNormalizedReturnType
199 {
200  return HNormalized_Block(_expression(),0,0,
201  Direction==Vertical ? _expression().rows()-1 : _expression().rows(),
202  Direction==Horizontal ? _expression().cols()-1 : _expression().cols()).cwiseQuotient(
204  Direction==Vertical ? HNormalized_SizeMinusOne : 1,
205  Direction==Horizontal ? HNormalized_SizeMinusOne : 1>
206  (HNormalized_Factors(_expression(),
207  Direction==Vertical ? _expression().rows()-1:0,
208  Direction==Horizontal ? _expression().cols()-1:0,
209  Direction==Vertical ? 1 : _expression().rows(),
210  Direction==Horizontal ? 1 : _expression().cols()),
211  Direction==Vertical ? _expression().rows()-1 : 1,
212  Direction==Horizontal ? _expression().cols()-1 : 1));
213 }
214 
215 namespace internal {
216 
217 template<typename MatrixOrTransformType>
218 struct take_matrix_for_product
219 {
220  typedef MatrixOrTransformType type;
221  EIGEN_DEVICE_FUNC static const type& run(const type &x) { return x; }
222 };
223 
224 template<typename Scalar, int Dim, int Mode,int Options>
225 struct take_matrix_for_product<Transform<Scalar, Dim, Mode, Options> >
226 {
227  typedef Transform<Scalar, Dim, Mode, Options> TransformType;
229  EIGEN_DEVICE_FUNC static type run (const TransformType& x) { return x.affine(); }
230 };
231 
232 template<typename Scalar, int Dim, int Options>
233 struct take_matrix_for_product<Transform<Scalar, Dim, Projective, Options> >
234 {
235  typedef Transform<Scalar, Dim, Projective, Options> TransformType;
236  typedef typename TransformType::MatrixType type;
237  EIGEN_DEVICE_FUNC static const type& run (const TransformType& x) { return x.matrix(); }
238 };
239 
240 template<typename MatrixType,typename Lhs>
241 struct traits<homogeneous_left_product_impl<Homogeneous<MatrixType,Vertical>,Lhs> >
242 {
243  typedef typename take_matrix_for_product<Lhs>::type LhsMatrixType;
244  typedef typename remove_all<MatrixType>::type MatrixTypeCleaned;
245  typedef typename remove_all<LhsMatrixType>::type LhsMatrixTypeCleaned;
246  typedef typename make_proper_matrix_type<
247  typename traits<MatrixTypeCleaned>::Scalar,
248  LhsMatrixTypeCleaned::RowsAtCompileTime,
249  MatrixTypeCleaned::ColsAtCompileTime,
250  MatrixTypeCleaned::PlainObject::Options,
251  LhsMatrixTypeCleaned::MaxRowsAtCompileTime,
252  MatrixTypeCleaned::MaxColsAtCompileTime>::type ReturnType;
253 };
254 
255 template<typename MatrixType,typename Lhs>
256 struct homogeneous_left_product_impl<Homogeneous<MatrixType,Vertical>,Lhs>
257  : public ReturnByValue<homogeneous_left_product_impl<Homogeneous<MatrixType,Vertical>,Lhs> >
258 {
259  typedef typename traits<homogeneous_left_product_impl>::LhsMatrixType LhsMatrixType;
260  typedef typename remove_all<LhsMatrixType>::type LhsMatrixTypeCleaned;
261  typedef typename remove_all<typename LhsMatrixTypeCleaned::Nested>::type LhsMatrixTypeNested;
262  EIGEN_DEVICE_FUNC homogeneous_left_product_impl(const Lhs& lhs, const MatrixType& rhs)
263  : m_lhs(take_matrix_for_product<Lhs>::run(lhs)),
264  m_rhs(rhs)
265  {}
266 
267  EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR
268  inline Index rows() const EIGEN_NOEXCEPT { return m_lhs.rows(); }
269  EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR
270  inline Index cols() const EIGEN_NOEXCEPT { return m_rhs.cols(); }
271 
272  template<typename Dest> EIGEN_DEVICE_FUNC void evalTo(Dest& dst) const
273  {
274  // FIXME investigate how to allow lazy evaluation of this product when possible
275  dst = Block<const LhsMatrixTypeNested,
276  LhsMatrixTypeNested::RowsAtCompileTime,
277  LhsMatrixTypeNested::ColsAtCompileTime==Dynamic?Dynamic:LhsMatrixTypeNested::ColsAtCompileTime-1>
278  (m_lhs,0,0,m_lhs.rows(),m_lhs.cols()-1) * m_rhs;
279  dst += m_lhs.col(m_lhs.cols()-1).rowwise()
280  .template replicate<MatrixType::ColsAtCompileTime>(m_rhs.cols());
281  }
282 
283  typename LhsMatrixTypeCleaned::Nested m_lhs;
284  typename MatrixType::Nested m_rhs;
285 };
286 
287 template<typename MatrixType,typename Rhs>
288 struct traits<homogeneous_right_product_impl<Homogeneous<MatrixType,Horizontal>,Rhs> >
289 {
290  typedef typename make_proper_matrix_type<typename traits<MatrixType>::Scalar,
291  MatrixType::RowsAtCompileTime,
292  Rhs::ColsAtCompileTime,
293  MatrixType::PlainObject::Options,
294  MatrixType::MaxRowsAtCompileTime,
295  Rhs::MaxColsAtCompileTime>::type ReturnType;
296 };
297 
298 template<typename MatrixType,typename Rhs>
299 struct homogeneous_right_product_impl<Homogeneous<MatrixType,Horizontal>,Rhs>
300  : public ReturnByValue<homogeneous_right_product_impl<Homogeneous<MatrixType,Horizontal>,Rhs> >
301 {
302  typedef typename remove_all<typename Rhs::Nested>::type RhsNested;
303  EIGEN_DEVICE_FUNC homogeneous_right_product_impl(const MatrixType& lhs, const Rhs& rhs)
304  : m_lhs(lhs), m_rhs(rhs)
305  {}
306 
307  EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR inline Index rows() const EIGEN_NOEXCEPT { return m_lhs.rows(); }
308  EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR inline Index cols() const EIGEN_NOEXCEPT { return m_rhs.cols(); }
309 
310  template<typename Dest> EIGEN_DEVICE_FUNC void evalTo(Dest& dst) const
311  {
312  // FIXME investigate how to allow lazy evaluation of this product when possible
313  dst = m_lhs * Block<const RhsNested,
314  RhsNested::RowsAtCompileTime==Dynamic?Dynamic:RhsNested::RowsAtCompileTime-1,
315  RhsNested::ColsAtCompileTime>
316  (m_rhs,0,0,m_rhs.rows()-1,m_rhs.cols());
317  dst += m_rhs.row(m_rhs.rows()-1).colwise()
318  .template replicate<MatrixType::RowsAtCompileTime>(m_lhs.rows());
319  }
320 
321  typename MatrixType::Nested m_lhs;
322  typename Rhs::Nested m_rhs;
323 };
324 
325 template<typename ArgType,int Direction>
326 struct evaluator_traits<Homogeneous<ArgType,Direction> >
327 {
328  typedef typename storage_kind_to_evaluator_kind<typename ArgType::StorageKind>::Kind Kind;
329  typedef HomogeneousShape Shape;
330 };
331 
332 template<> struct AssignmentKind<DenseShape,HomogeneousShape> { typedef Dense2Dense Kind; };
333 
334 
335 template<typename ArgType,int Direction>
336 struct unary_evaluator<Homogeneous<ArgType,Direction>, IndexBased>
337  : evaluator<typename Homogeneous<ArgType,Direction>::PlainObject >
338 {
339  typedef Homogeneous<ArgType,Direction> XprType;
340  typedef typename XprType::PlainObject PlainObject;
341  typedef evaluator<PlainObject> Base;
342 
343  EIGEN_DEVICE_FUNC explicit unary_evaluator(const XprType& op)
344  : Base(), m_temp(op)
345  {
346  ::new (static_cast<Base*>(this)) Base(m_temp);
347  }
348 
349 protected:
350  PlainObject m_temp;
351 };
352 
353 // dense = homogeneous
354 template< typename DstXprType, typename ArgType, typename Scalar>
355 struct Assignment<DstXprType, Homogeneous<ArgType,Vertical>, internal::assign_op<Scalar,typename ArgType::Scalar>, Dense2Dense>
356 {
357  typedef Homogeneous<ArgType,Vertical> SrcXprType;
358  EIGEN_DEVICE_FUNC static void run(DstXprType &dst, const SrcXprType &src, const internal::assign_op<Scalar,typename ArgType::Scalar> &)
359  {
360  Index dstRows = src.rows();
361  Index dstCols = src.cols();
362  if((dst.rows()!=dstRows) || (dst.cols()!=dstCols))
363  dst.resize(dstRows, dstCols);
364 
365  dst.template topRows<ArgType::RowsAtCompileTime>(src.nestedExpression().rows()) = src.nestedExpression();
366  dst.row(dst.rows()-1).setOnes();
367  }
368 };
369 
370 // dense = homogeneous
371 template< typename DstXprType, typename ArgType, typename Scalar>
372 struct Assignment<DstXprType, Homogeneous<ArgType,Horizontal>, internal::assign_op<Scalar,typename ArgType::Scalar>, Dense2Dense>
373 {
374  typedef Homogeneous<ArgType,Horizontal> SrcXprType;
375  EIGEN_DEVICE_FUNC static void run(DstXprType &dst, const SrcXprType &src, const internal::assign_op<Scalar,typename ArgType::Scalar> &)
376  {
377  Index dstRows = src.rows();
378  Index dstCols = src.cols();
379  if((dst.rows()!=dstRows) || (dst.cols()!=dstCols))
380  dst.resize(dstRows, dstCols);
381 
382  dst.template leftCols<ArgType::ColsAtCompileTime>(src.nestedExpression().cols()) = src.nestedExpression();
383  dst.col(dst.cols()-1).setOnes();
384  }
385 };
386 
387 template<typename LhsArg, typename Rhs, int ProductTag>
388 struct generic_product_impl<Homogeneous<LhsArg,Horizontal>, Rhs, HomogeneousShape, DenseShape, ProductTag>
389 {
390  template<typename Dest>
391  EIGEN_DEVICE_FUNC static void evalTo(Dest& dst, const Homogeneous<LhsArg,Horizontal>& lhs, const Rhs& rhs)
392  {
393  homogeneous_right_product_impl<Homogeneous<LhsArg,Horizontal>, Rhs>(lhs.nestedExpression(), rhs).evalTo(dst);
394  }
395 };
396 
397 template<typename Lhs,typename Rhs>
398 struct homogeneous_right_product_refactoring_helper
399 {
400  enum {
401  Dim = Lhs::ColsAtCompileTime,
402  Rows = Lhs::RowsAtCompileTime
403  };
404  typedef typename Rhs::template ConstNRowsBlockXpr<Dim>::Type LinearBlockConst;
405  typedef typename remove_const<LinearBlockConst>::type LinearBlock;
406  typedef typename Rhs::ConstRowXpr ConstantColumn;
407  typedef Replicate<const ConstantColumn,Rows,1> ConstantBlock;
408  typedef Product<Lhs,LinearBlock,LazyProduct> LinearProduct;
409  typedef CwiseBinaryOp<internal::scalar_sum_op<typename Lhs::Scalar,typename Rhs::Scalar>, const LinearProduct, const ConstantBlock> Xpr;
410 };
411 
412 template<typename Lhs, typename Rhs, int ProductTag>
413 struct product_evaluator<Product<Lhs, Rhs, LazyProduct>, ProductTag, HomogeneousShape, DenseShape>
414  : public evaluator<typename homogeneous_right_product_refactoring_helper<typename Lhs::NestedExpression,Rhs>::Xpr>
415 {
416  typedef Product<Lhs, Rhs, LazyProduct> XprType;
417  typedef homogeneous_right_product_refactoring_helper<typename Lhs::NestedExpression,Rhs> helper;
418  typedef typename helper::ConstantBlock ConstantBlock;
419  typedef typename helper::Xpr RefactoredXpr;
420  typedef evaluator<RefactoredXpr> Base;
421 
422  EIGEN_DEVICE_FUNC explicit product_evaluator(const XprType& xpr)
423  : Base( xpr.lhs().nestedExpression() .lazyProduct( xpr.rhs().template topRows<helper::Dim>(xpr.lhs().nestedExpression().cols()) )
424  + ConstantBlock(xpr.rhs().row(xpr.rhs().rows()-1),xpr.lhs().rows(), 1) )
425  {}
426 };
427 
428 template<typename Lhs, typename RhsArg, int ProductTag>
429 struct generic_product_impl<Lhs, Homogeneous<RhsArg,Vertical>, DenseShape, HomogeneousShape, ProductTag>
430 {
431  template<typename Dest>
432  EIGEN_DEVICE_FUNC static void evalTo(Dest& dst, const Lhs& lhs, const Homogeneous<RhsArg,Vertical>& rhs)
433  {
434  homogeneous_left_product_impl<Homogeneous<RhsArg,Vertical>, Lhs>(lhs, rhs.nestedExpression()).evalTo(dst);
435  }
436 };
437 
438 // TODO: the following specialization is to address a regression from 3.2 to 3.3
439 // In the future, this path should be optimized.
440 template<typename Lhs, typename RhsArg, int ProductTag>
441 struct generic_product_impl<Lhs, Homogeneous<RhsArg,Vertical>, TriangularShape, HomogeneousShape, ProductTag>
442 {
443  template<typename Dest>
444  static void evalTo(Dest& dst, const Lhs& lhs, const Homogeneous<RhsArg,Vertical>& rhs)
445  {
446  dst.noalias() = lhs * rhs.eval();
447  }
448 };
449 
450 template<typename Lhs,typename Rhs>
451 struct homogeneous_left_product_refactoring_helper
452 {
453  enum {
454  Dim = Rhs::RowsAtCompileTime,
455  Cols = Rhs::ColsAtCompileTime
456  };
457  typedef typename Lhs::template ConstNColsBlockXpr<Dim>::Type LinearBlockConst;
458  typedef typename remove_const<LinearBlockConst>::type LinearBlock;
459  typedef typename Lhs::ConstColXpr ConstantColumn;
460  typedef Replicate<const ConstantColumn,1,Cols> ConstantBlock;
461  typedef Product<LinearBlock,Rhs,LazyProduct> LinearProduct;
462  typedef CwiseBinaryOp<internal::scalar_sum_op<typename Lhs::Scalar,typename Rhs::Scalar>, const LinearProduct, const ConstantBlock> Xpr;
463 };
464 
465 template<typename Lhs, typename Rhs, int ProductTag>
466 struct product_evaluator<Product<Lhs, Rhs, LazyProduct>, ProductTag, DenseShape, HomogeneousShape>
467  : public evaluator<typename homogeneous_left_product_refactoring_helper<Lhs,typename Rhs::NestedExpression>::Xpr>
468 {
469  typedef Product<Lhs, Rhs, LazyProduct> XprType;
470  typedef homogeneous_left_product_refactoring_helper<Lhs,typename Rhs::NestedExpression> helper;
471  typedef typename helper::ConstantBlock ConstantBlock;
472  typedef typename helper::Xpr RefactoredXpr;
473  typedef evaluator<RefactoredXpr> Base;
474 
475  EIGEN_DEVICE_FUNC explicit product_evaluator(const XprType& xpr)
476  : Base( xpr.lhs().template leftCols<helper::Dim>(xpr.rhs().nestedExpression().rows()) .lazyProduct( xpr.rhs().nestedExpression() )
477  + ConstantBlock(xpr.lhs().col(xpr.lhs().cols()-1),1,xpr.rhs().cols()) )
478  {}
479 };
480 
481 template<typename Scalar, int Dim, int Mode,int Options, typename RhsArg, int ProductTag>
482 struct generic_product_impl<Transform<Scalar,Dim,Mode,Options>, Homogeneous<RhsArg,Vertical>, DenseShape, HomogeneousShape, ProductTag>
483 {
484  typedef Transform<Scalar,Dim,Mode,Options> TransformType;
485  template<typename Dest>
486  EIGEN_DEVICE_FUNC static void evalTo(Dest& dst, const TransformType& lhs, const Homogeneous<RhsArg,Vertical>& rhs)
487  {
488  homogeneous_left_product_impl<Homogeneous<RhsArg,Vertical>, TransformType>(lhs, rhs.nestedExpression()).evalTo(dst);
489  }
490 };
491 
492 template<typename ExpressionType, int Side, bool Transposed>
493 struct permutation_matrix_product<ExpressionType, Side, Transposed, HomogeneousShape>
494  : public permutation_matrix_product<ExpressionType, Side, Transposed, DenseShape>
495 {};
496 
497 } // end namespace internal
498 
499 } // end namespace Eigen
500 
501 #endif // EIGEN_HOMOGENEOUS_H
Expression of the product of two arbitrary matrices or vectors.
Definition: Product.h:71
Definition: Constants.h:264
HomogeneousReturnType homogeneous() const
Definition: Homogeneous.h:150
Eigen::Index Index
Definition: VectorwiseOp.h:192
Expression of the transpose of a matrix.
Definition: Transpose.h:52
const HNormalizedReturnType hnormalized() const
column or row-wise homogeneous normalization
Definition: Homogeneous.h:198
Namespace containing all symbols from the Eigen library.
Definition: Core:141
Derived & derived()
Definition: EigenBase.h:46
const unsigned int RowMajorBit
Definition: Constants.h:66
Generic expression where a coefficient-wise binary operator is applied to two expressions.
Definition: CwiseBinaryOp.h:77
const Product< MatrixDerived, PermutationDerived, AliasFreeProduct > operator*(const MatrixBase< MatrixDerived > &matrix, const PermutationBase< PermutationDerived > &permutation)
Definition: PermutationMatrix.h:515
EIGEN_DEFAULT_DENSE_INDEX_TYPE Index
The Index type as used for the API.
Definition: Meta.h:74
Expression of the multiple replication of a matrix or vector.
Definition: Replicate.h:61
Definition: Constants.h:267
Definition: Eigen_Colamd.h:50
const HNormalizedReturnType hnormalized() const
homogeneous normalization
Definition: Homogeneous.h:174
Expression of a fixed-size or dynamic-size block.
Definition: Block.h:103
EvalReturnType eval() const
Definition: DenseBase.h:407
const int Dynamic
Definition: Constants.h:22
HomogeneousReturnType homogeneous() const
Definition: Homogeneous.h:132
Base class for all dense matrices, vectors, and expressions.
Definition: MatrixBase.h:48
Represents an homogeneous transformation in a N dimensional space.
Definition: Transform.h:204
Expression of one (or a set of) homogeneous vector(s)
Definition: Homogeneous.h:60