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00001 // This file is part of Eigen, a lightweight C++ template library 00002 // for linear algebra. 00003 // 00004 // Copyright (C) 20010-2011 Hauke Heibel <hauke.heibel@gmail.com> 00005 // 00006 // This Source Code Form is subject to the terms of the Mozilla 00007 // Public License v. 2.0. If a copy of the MPL was not distributed 00008 // with this file, You can obtain one at http://mozilla.org/MPL/2.0/. 00009 00010 #ifndef EIGEN_SPLINE_H 00011 #define EIGEN_SPLINE_H 00012 00013 #include "SplineFwd.h" 00014 00015 namespace Eigen 00016 { 00034 template <typename _Scalar, int _Dim, int _Degree> 00035 class Spline 00036 { 00037 public: 00038 typedef _Scalar Scalar; 00039 enum { Dimension = _Dim }; 00040 enum { Degree = _Degree }; 00041 00043 typedef typename SplineTraits<Spline>::PointType PointType; 00044 00046 typedef typename SplineTraits<Spline>::KnotVectorType KnotVectorType; 00047 00049 typedef typename SplineTraits<Spline>::BasisVectorType BasisVectorType; 00050 00052 typedef typename SplineTraits<Spline>::ControlPointVectorType ControlPointVectorType; 00053 00058 Spline() 00059 : m_knots(1, (Degree==Dynamic ? 2 : 2*Degree+2)) 00060 , m_ctrls(ControlPointVectorType::Zero(2,(Degree==Dynamic ? 1 : Degree+1))) 00061 { 00062 // in theory this code can go to the initializer list but it will get pretty 00063 // much unreadable ... 00064 enum { MinDegree = (Degree==Dynamic ? 0 : Degree) }; 00065 m_knots.template segment<MinDegree+1>(0) = Array<Scalar,1,MinDegree+1>::Zero(); 00066 m_knots.template segment<MinDegree+1>(MinDegree+1) = Array<Scalar,1,MinDegree+1>::Ones(); 00067 } 00068 00074 template <typename OtherVectorType, typename OtherArrayType> 00075 Spline(const OtherVectorType& knots, const OtherArrayType& ctrls) : m_knots(knots), m_ctrls(ctrls) {} 00076 00081 template <int OtherDegree> 00082 Spline(const Spline<Scalar, Dimension, OtherDegree>& spline) : 00083 m_knots(spline.knots()), m_ctrls(spline.ctrls()) {} 00084 00088 const KnotVectorType& knots() const { return m_knots; } 00089 00093 const ControlPointVectorType& ctrls() const { return m_ctrls; } 00094 00106 PointType operator()(Scalar u) const; 00107 00120 typename SplineTraits<Spline>::DerivativeType 00121 derivatives(Scalar u, DenseIndex order) const; 00122 00128 template <int DerivativeOrder> 00129 typename SplineTraits<Spline,DerivativeOrder>::DerivativeType 00130 derivatives(Scalar u, DenseIndex order = DerivativeOrder) const; 00131 00148 typename SplineTraits<Spline>::BasisVectorType 00149 basisFunctions(Scalar u) const; 00150 00164 typename SplineTraits<Spline>::BasisDerivativeType 00165 basisFunctionDerivatives(Scalar u, DenseIndex order) const; 00166 00172 template <int DerivativeOrder> 00173 typename SplineTraits<Spline,DerivativeOrder>::BasisDerivativeType 00174 basisFunctionDerivatives(Scalar u, DenseIndex order = DerivativeOrder) const; 00175 00179 DenseIndex degree() const; 00180 00185 DenseIndex span(Scalar u) const; 00186 00190 static DenseIndex Span(typename SplineTraits<Spline>::Scalar u, DenseIndex degree, const typename SplineTraits<Spline>::KnotVectorType& knots); 00191 00204 static BasisVectorType BasisFunctions(Scalar u, DenseIndex degree, const KnotVectorType& knots); 00205 00206 00207 private: 00208 KnotVectorType m_knots; 00209 ControlPointVectorType m_ctrls; 00210 }; 00211 00212 template <typename _Scalar, int _Dim, int _Degree> 00213 DenseIndex Spline<_Scalar, _Dim, _Degree>::Span( 00214 typename SplineTraits< Spline<_Scalar, _Dim, _Degree> >::Scalar u, 00215 DenseIndex degree, 00216 const typename SplineTraits< Spline<_Scalar, _Dim, _Degree> >::KnotVectorType& knots) 00217 { 00218 // Piegl & Tiller, "The NURBS Book", A2.1 (p. 68) 00219 if (u <= knots(0)) return degree; 00220 const Scalar* pos = std::upper_bound(knots.data()+degree-1, knots.data()+knots.size()-degree-1, u); 00221 return static_cast<DenseIndex>( std::distance(knots.data(), pos) - 1 ); 00222 } 00223 00224 template <typename _Scalar, int _Dim, int _Degree> 00225 typename Spline<_Scalar, _Dim, _Degree>::BasisVectorType 00226 Spline<_Scalar, _Dim, _Degree>::BasisFunctions( 00227 typename Spline<_Scalar, _Dim, _Degree>::Scalar u, 00228 DenseIndex degree, 00229 const typename Spline<_Scalar, _Dim, _Degree>::KnotVectorType& knots) 00230 { 00231 typedef typename Spline<_Scalar, _Dim, _Degree>::BasisVectorType BasisVectorType; 00232 00233 const DenseIndex p = degree; 00234 const DenseIndex i = Spline::Span(u, degree, knots); 00235 00236 const KnotVectorType& U = knots; 00237 00238 BasisVectorType left(p+1); left(0) = Scalar(0); 00239 BasisVectorType right(p+1); right(0) = Scalar(0); 00240 00241 VectorBlock<BasisVectorType,Degree>(left,1,p) = u - VectorBlock<const KnotVectorType,Degree>(U,i+1-p,p).reverse(); 00242 VectorBlock<BasisVectorType,Degree>(right,1,p) = VectorBlock<const KnotVectorType,Degree>(U,i+1,p) - u; 00243 00244 BasisVectorType N(1,p+1); 00245 N(0) = Scalar(1); 00246 for (DenseIndex j=1; j<=p; ++j) 00247 { 00248 Scalar saved = Scalar(0); 00249 for (DenseIndex r=0; r<j; r++) 00250 { 00251 const Scalar tmp = N(r)/(right(r+1)+left(j-r)); 00252 N[r] = saved + right(r+1)*tmp; 00253 saved = left(j-r)*tmp; 00254 } 00255 N(j) = saved; 00256 } 00257 return N; 00258 } 00259 00260 template <typename _Scalar, int _Dim, int _Degree> 00261 DenseIndex Spline<_Scalar, _Dim, _Degree>::degree() const 00262 { 00263 if (_Degree == Dynamic) 00264 return m_knots.size() - m_ctrls.cols() - 1; 00265 else 00266 return _Degree; 00267 } 00268 00269 template <typename _Scalar, int _Dim, int _Degree> 00270 DenseIndex Spline<_Scalar, _Dim, _Degree>::span(Scalar u) const 00271 { 00272 return Spline::Span(u, degree(), knots()); 00273 } 00274 00275 template <typename _Scalar, int _Dim, int _Degree> 00276 typename Spline<_Scalar, _Dim, _Degree>::PointType Spline<_Scalar, _Dim, _Degree>::operator()(Scalar u) const 00277 { 00278 enum { Order = SplineTraits<Spline>::OrderAtCompileTime }; 00279 00280 const DenseIndex span = this->span(u); 00281 const DenseIndex p = degree(); 00282 const BasisVectorType basis_funcs = basisFunctions(u); 00283 00284 const Replicate<BasisVectorType,Dimension,1> ctrl_weights(basis_funcs); 00285 const Block<const ControlPointVectorType,Dimension,Order> ctrl_pts(ctrls(),0,span-p,Dimension,p+1); 00286 return (ctrl_weights * ctrl_pts).rowwise().sum(); 00287 } 00288 00289 /* --------------------------------------------------------------------------------------------- */ 00290 00291 template <typename SplineType, typename DerivativeType> 00292 void derivativesImpl(const SplineType& spline, typename SplineType::Scalar u, DenseIndex order, DerivativeType& der) 00293 { 00294 enum { Dimension = SplineTraits<SplineType>::Dimension }; 00295 enum { Order = SplineTraits<SplineType>::OrderAtCompileTime }; 00296 enum { DerivativeOrder = DerivativeType::ColsAtCompileTime }; 00297 00298 typedef typename SplineTraits<SplineType>::ControlPointVectorType ControlPointVectorType; 00299 typedef typename SplineTraits<SplineType,DerivativeOrder>::BasisDerivativeType BasisDerivativeType; 00300 typedef typename BasisDerivativeType::ConstRowXpr BasisDerivativeRowXpr; 00301 00302 const DenseIndex p = spline.degree(); 00303 const DenseIndex span = spline.span(u); 00304 00305 const DenseIndex n = (std::min)(p, order); 00306 00307 der.resize(Dimension,n+1); 00308 00309 // Retrieve the basis function derivatives up to the desired order... 00310 const BasisDerivativeType basis_func_ders = spline.template basisFunctionDerivatives<DerivativeOrder>(u, n+1); 00311 00312 // ... and perform the linear combinations of the control points. 00313 for (DenseIndex der_order=0; der_order<n+1; ++der_order) 00314 { 00315 const Replicate<BasisDerivativeRowXpr,Dimension,1> ctrl_weights( basis_func_ders.row(der_order) ); 00316 const Block<const ControlPointVectorType,Dimension,Order> ctrl_pts(spline.ctrls(),0,span-p,Dimension,p+1); 00317 der.col(der_order) = (ctrl_weights * ctrl_pts).rowwise().sum(); 00318 } 00319 } 00320 00321 template <typename _Scalar, int _Dim, int _Degree> 00322 typename SplineTraits< Spline<_Scalar, _Dim, _Degree> >::DerivativeType 00323 Spline<_Scalar, _Dim, _Degree>::derivatives(Scalar u, DenseIndex order) const 00324 { 00325 typename SplineTraits< Spline >::DerivativeType res; 00326 derivativesImpl(*this, u, order, res); 00327 return res; 00328 } 00329 00330 template <typename _Scalar, int _Dim, int _Degree> 00331 template <int DerivativeOrder> 00332 typename SplineTraits< Spline<_Scalar, _Dim, _Degree>, DerivativeOrder >::DerivativeType 00333 Spline<_Scalar, _Dim, _Degree>::derivatives(Scalar u, DenseIndex order) const 00334 { 00335 typename SplineTraits< Spline, DerivativeOrder >::DerivativeType res; 00336 derivativesImpl(*this, u, order, res); 00337 return res; 00338 } 00339 00340 template <typename _Scalar, int _Dim, int _Degree> 00341 typename SplineTraits< Spline<_Scalar, _Dim, _Degree> >::BasisVectorType 00342 Spline<_Scalar, _Dim, _Degree>::basisFunctions(Scalar u) const 00343 { 00344 return Spline::BasisFunctions(u, degree(), knots()); 00345 } 00346 00347 /* --------------------------------------------------------------------------------------------- */ 00348 00349 template <typename SplineType, typename DerivativeType> 00350 void basisFunctionDerivativesImpl(const SplineType& spline, typename SplineType::Scalar u, DenseIndex order, DerivativeType& N_) 00351 { 00352 enum { Order = SplineTraits<SplineType>::OrderAtCompileTime }; 00353 00354 typedef typename SplineTraits<SplineType>::Scalar Scalar; 00355 typedef typename SplineTraits<SplineType>::BasisVectorType BasisVectorType; 00356 typedef typename SplineTraits<SplineType>::KnotVectorType KnotVectorType; 00357 00358 const KnotVectorType& U = spline.knots(); 00359 00360 const DenseIndex p = spline.degree(); 00361 const DenseIndex span = spline.span(u); 00362 00363 const DenseIndex n = (std::min)(p, order); 00364 00365 N_.resize(n+1, p+1); 00366 00367 BasisVectorType left = BasisVectorType::Zero(p+1); 00368 BasisVectorType right = BasisVectorType::Zero(p+1); 00369 00370 Matrix<Scalar,Order,Order> ndu(p+1,p+1); 00371 00372 double saved, temp; 00373 00374 ndu(0,0) = 1.0; 00375 00376 DenseIndex j; 00377 for (j=1; j<=p; ++j) 00378 { 00379 left[j] = u-U[span+1-j]; 00380 right[j] = U[span+j]-u; 00381 saved = 0.0; 00382 00383 for (DenseIndex r=0; r<j; ++r) 00384 { 00385 /* Lower triangle */ 00386 ndu(j,r) = right[r+1]+left[j-r]; 00387 temp = ndu(r,j-1)/ndu(j,r); 00388 /* Upper triangle */ 00389 ndu(r,j) = static_cast<Scalar>(saved+right[r+1] * temp); 00390 saved = left[j-r] * temp; 00391 } 00392 00393 ndu(j,j) = static_cast<Scalar>(saved); 00394 } 00395 00396 for (j = p; j>=0; --j) 00397 N_(0,j) = ndu(j,p); 00398 00399 // Compute the derivatives 00400 DerivativeType a(n+1,p+1); 00401 DenseIndex r=0; 00402 for (; r<=p; ++r) 00403 { 00404 DenseIndex s1,s2; 00405 s1 = 0; s2 = 1; // alternate rows in array a 00406 a(0,0) = 1.0; 00407 00408 // Compute the k-th derivative 00409 for (DenseIndex k=1; k<=static_cast<DenseIndex>(n); ++k) 00410 { 00411 double d = 0.0; 00412 DenseIndex rk,pk,j1,j2; 00413 rk = r-k; pk = p-k; 00414 00415 if (r>=k) 00416 { 00417 a(s2,0) = a(s1,0)/ndu(pk+1,rk); 00418 d = a(s2,0)*ndu(rk,pk); 00419 } 00420 00421 if (rk>=-1) j1 = 1; 00422 else j1 = -rk; 00423 00424 if (r-1 <= pk) j2 = k-1; 00425 else j2 = p-r; 00426 00427 for (j=j1; j<=j2; ++j) 00428 { 00429 a(s2,j) = (a(s1,j)-a(s1,j-1))/ndu(pk+1,rk+j); 00430 d += a(s2,j)*ndu(rk+j,pk); 00431 } 00432 00433 if (r<=pk) 00434 { 00435 a(s2,k) = -a(s1,k-1)/ndu(pk+1,r); 00436 d += a(s2,k)*ndu(r,pk); 00437 } 00438 00439 N_(k,r) = static_cast<Scalar>(d); 00440 j = s1; s1 = s2; s2 = j; // Switch rows 00441 } 00442 } 00443 00444 /* Multiply through by the correct factors */ 00445 /* (Eq. [2.9]) */ 00446 r = p; 00447 for (DenseIndex k=1; k<=static_cast<DenseIndex>(n); ++k) 00448 { 00449 for (DenseIndex j=p; j>=0; --j) N_(k,j) *= r; 00450 r *= p-k; 00451 } 00452 } 00453 00454 template <typename _Scalar, int _Dim, int _Degree> 00455 typename SplineTraits< Spline<_Scalar, _Dim, _Degree> >::BasisDerivativeType 00456 Spline<_Scalar, _Dim, _Degree>::basisFunctionDerivatives(Scalar u, DenseIndex order) const 00457 { 00458 typename SplineTraits< Spline >::BasisDerivativeType der; 00459 basisFunctionDerivativesImpl(*this, u, order, der); 00460 return der; 00461 } 00462 00463 template <typename _Scalar, int _Dim, int _Degree> 00464 template <int DerivativeOrder> 00465 typename SplineTraits< Spline<_Scalar, _Dim, _Degree>, DerivativeOrder >::BasisDerivativeType 00466 Spline<_Scalar, _Dim, _Degree>::basisFunctionDerivatives(Scalar u, DenseIndex order) const 00467 { 00468 typename SplineTraits< Spline, DerivativeOrder >::BasisDerivativeType der; 00469 basisFunctionDerivativesImpl(*this, u, order, der); 00470 return der; 00471 } 00472 } 00473 00474 #endif // EIGEN_SPLINE_H