text: #ifndef SimTK_SIMMATH_GEO_BICUBIC_BEZIER_PATCH_H_
  #define SimTK_SIMMATH_GEO_BICUBIC_BEZIER_PATCH_H_
  
  /* -------------------------------------------------------------------------- *
   *                        Simbody(tm): SimTKmath                              *
   * -------------------------------------------------------------------------- *
   * This is part of the SimTK biosimulation toolkit originating from           *
   * Simbios, the NIH National Center for Physics-Based Simulation of           *
   * Biological Structures at Stanford, funded under the NIH Roadmap for        *
   * Medical Research, grant U54 GM072970. See https://simtk.org/home/simbody.  *
   *                                                                            *
   * Portions copyright (c) 2011-12 Stanford University and the Authors.        *
   * Authors: Michael Sherman                                                   *
   * Contributors:                                                              *
   *                                                                            *
   * Licensed under the Apache License, Version 2.0 (the "License"); you may    *
   * not use this file except in compliance with the License. You may obtain a  *
   * copy of the License at http://www.apache.org/licenses/LICENSE-2.0.         *
   *                                                                            *
   * Unless required by applicable law or agreed to in writing, software        *
   * distributed under the License is distributed on an "AS IS" BASIS,          *
   * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.   *
   * See the License for the specific language governing permissions and        *
   * limitations under the License.                                             *
   * -------------------------------------------------------------------------- */
  
  /** @file
  Provides primitive operations for a single bicubic Bezier patch using either
  single or double precision. **/
  
  #include "SimTKcommon.h"
  #include "simmath/internal/common.h"
  #include "simmath/internal/Geo.h"
  #include "simmath/internal/Geo_CubicBezierCurve.h"
  
  #include <cassert>
  #include <cmath>
  #include <algorithm>
  
  namespace SimTK {
  
  
  //==============================================================================
  //                         GEO BICUBIC BEZIER PATCH
  //==============================================================================
  /** A primitive useful for computations involving a single bicubic Bezier
  patch. Note that a bicubic Bezier spline surface would not necessarily be
  composed of these, but could use the static methods here for patch 
  computations. 
  
  <h3>Theory</h3>
  The primary reference for this implementation is the book "Geometric Modeling, 
  3rd ed." by Michael E. Mortenson, Industrial Press 2006, chapter 8. We follow
  Mortenson's notation here (with some name changes) and equation numbers are 
  from the text. See CubicHermiteCurve_ and BicubicHermitePatch_ comments for 
  introductory material; here we add the Bezier description to the algebraic and 
  Hermite (geometric) forms described there.
  
  We use B for Bezier control points (rather than P), and H for Hermite 
  coefficients (rather than B). We call the Bezier basis matrix Mb (same as
  Mortenson) but call the Hermite basis matrix Mh (rather than Mf). 
  
  The 16 control points are laid out like this, matching Mortenson, with the
  Hermite coefficient matrix for comparison:
  <pre>
          [ b11 b12 b13 b14 ] u=0        [ h00  h01  w00  w01 ]    u=dp/du
      B = [ b21 b22 b23 b24 ]  .     H = [ h10  h11  w10  w11 ]    w=dp/dw
          [ b31 b32 b33 b34 ]  .         [ u00  u01  t00  t01 ]    t=d2p/dudw
          [ b41 b42 b43 b44 ] u=1        [ u10  u11  t10  t11 ]      ("twist")
            w=0   . .   w=1
  </pre>
  We store those in a 4x4 hypermatrix; subtract one from each control point
  index to get the matrix indices (sorry). 
  
  To convert between Bezier and Hermite or algebraic forms, use
  <pre>
      A = Mb B ~Mb  (although Mb is symmetric so ~Mb=Mb)
      H = (Mh^-1 Mb) B ~(Mh^-1 Mb)
      B = (Mb^-1 Mh) H ~(Mb^-1 Mh)
  </pre>
  where the parenthesized (very simple) matrices are given explicitly in the 
  Theory section for CubicBezierCurve_. The results are:
  <pre>
      [    b11         b14           3(b12-b11)             3(b14-b13)     ]
  H = [    b41         b44           3(b42-b41)             3(b44-b43)     ]
      [ 3(b21-b11)  3(b24-b14)  9(b22-b21-{b12-b11})  9({b24-b14}+b13-b23) ]
      [ 3(b41-b31)  3(b44-b34)  9({b42-b41}+b31-b32)  9({b44-b34}+b33-b43) ]
  </pre>
  The braces in the twist terms mark common subexpressions, so the
  conversion from Bezier to Hermite takes 3x28=84 flops (all entries are 
  3-vectors).
  <pre>
      [   h00            h00+w00/3               h01-w01/3          h01    ]
  B = [h00+u00/3 {h00+u00/3}+w00/3+t00/9 {h01+u01/3}-w01/3-t01/9 h01+u01/3 ]
      [h10-u10/3 {h10-u10/3}+w10/3-t10/9 {h11-u11/3}-w11/3+t11/9 h11-u11/3 ]
      [   h10            h10+w10/3               h11-w11/3          h11    ] 
  </pre>
  Exploiting the common subexpressions in braces, conversion from Hermite to 
  Bezier takes 3x32=96 flops. **/
  template <class P>
  class Geo::BicubicBezierPatch_ {
  typedef P               RealP;
  typedef Vec<3,RealP>    Vec3P;
  
  public:
  /** Construct an uninitialized patch; control points will be garbage. **/
  BicubicBezierPatch_() {}
  
  /** Construct a bicubic Bezier patch using the given control points B. See the
  class documentation for how this matrix is defined. If instead you
  have algebraic or Hermite coefficients, you can first convert them to Bezier
  control points with the calcBFromA() or calcHFromA() provided here. **/
  explicit BicubicBezierPatch_(const Mat<4,4,Vec3P>& controlPoints) 
  :   B(controlPoints) {} 
  
  
  /** Evaluate a point P(u,w) on this patch given values for the
  parameters u and w in [0,1]. Values outside this range are permitted but do 
  not lie on the patch. Cost is 123 flops. **/
  Vec3P evalP(RealP u, RealP w) const {return evalPUsingB(B,u,w);}
  
  /** Evaluate the tangents Pu=dP/du, Pw=dP/dw on this patch given values for the
  parameters u and w in [0,1]. Values outside this range are permitted but do not
  lie on the curve segment. Cost is 248 flops. **/
  void evalP1(RealP u, RealP w, Vec3P& Pu, Vec3P& Pw) const 
  {   return evalP1UsingB(B,u,w,Pu,Pw); }
  
  /** Evaluate the second derivatives Puu=d2P/du2, Pww=d2P/dw2, and cross
  derivative Puw=Pwu=d2P/dudw on this patch given values for the parameters u 
  and w in [0,1]. Values outside this range are permitted but do not lie on the 
  curve segment. Cost is 363 flops. **/
  void evalP2(RealP u, RealP w, Vec3P& Puu, Vec3P& Puw, Vec3P& Pww) const 
  {   evalP2UsingB(B,u,w,Puu,Puw,Pww); }
  
  /** Evaluate the third derivatives Puuu=d3P/du3, Pwww=d3P/dw3, and cross
  derivatives Puuw=Pwuu=Puwu=d3P/du2dw and Puww=Pwwu=Pwuw=d3P/dudw2 on this patch 
  given values for the parameters u and w in [0,1]. Cost is 468 flops. All
  higher derivatives of a cubic patch are zero. **/
  void evalP3(RealP u, RealP w, Vec3P& Puuu, Vec3P& Puuw, 
                                Vec3P& Puww, Vec3P& Pwww) const 
  {   evalP3UsingB(B,u,w,Puuu,Puuw,Puww,Pwww); }
  
  /** Return a reference to the Bezier control points B that are
  stored in this object. See the documentation for this class to see how the
  returned matrix of control points is defined. **/
  const Mat<4,4,Vec3P>& getControlPoints() const {return B;}
  /** Return a writable reference to the Bezier control points B that are
  stored in this object. See the documentation for this class to see how the
  returned matrix of control points is defined. **/
  Mat<4,4,Vec3P>& updControlPoints() {return B;}
  /** Calculate the algebraic coefficients A from the stored Bezier control
  points. See the documentation for BicubicHermitePatch_ to see how the
  returned matrix of coefficients is defined. Cost is 240 flops. **/
  Mat<4,4,Vec3P> calcAlgebraicCoefficients() const {return calcAFromB(B);}
  /** Calculate the Hermite coefficients H from the stored 
  Bezier control points. See the documentation for this class to see how the
  returned matrix of coefficients is defined. Cost is 84 flops. **/
  Mat<4,4,Vec3P> calcHermiteCoefficients() const {return calcHFromB(B);}
  
  /** Return the u=0 boundary curve as a Bezier curve segment. The control points
  are just the first row of the patch control points B so no computation is 
  required. **/
  CubicBezierCurve_<P> getBoundaryCurveU0() const 
  {   return CubicBezierCurve_<P>(B[0]); } // b11 b12 b13 b14
  /** Return the u=1 boundary curve as a Bezier curve segment. The control points
  are just the last row of the patch control points B so no computation is 
  required. **/
  CubicBezierCurve_<P> getBoundaryCurveU1() const 
  {   return CubicBezierCurve_<P>(B[3]); } // b41 b42 b43 b44
  /** Return the w=0 boundary curve as a Bezier curve segment. The control points
  are just the first column of the patch control points B so no computation is 
  required. **/
  CubicBezierCurve_<P> getBoundaryCurveW0() const 
  {   return CubicBezierCurve_<P>(B(0)); } // b11 b21 b31 b41
  /** Return the w=1 boundary curve as a Bezier curve segment. The control points
  are just the last column of the patch control points B so no computation is 
  required. **/
  CubicBezierCurve_<P> getBoundaryCurveW1() const 
  {   return CubicBezierCurve_<P>(B(3)); } // b14 b24 b34 b44
  
  /** Given a particular value u0 for patch coordinate u, create a cubic Bezier
  curve segment P(w)=P(u0,w) for the isoparametric curve along the patch at fixed
  u=u0. Cost is 93 flops. **/
  CubicBezierCurve_<P> calcIsoCurveU(RealP u0) const 
  {   return calcIsoCurveU(B, u0); }
  /** Given a particular value w0 for patch coordinate w, create a cubic Bezier
  curve segment P(u)=P(u,w0) for the isoparametric curve along the patch at fixed
  w=w0. Cost is 93 flops. **/
  CubicBezierCurve_<P> calcIsoCurveW(RealP w0) const 
  {   return calcIsoCurveW(B, w0); }
  
  /** Split this patch into two along the u direction, along an isoparametric 
  curve of constant u=t such that 0 < t < 1. On return, \a patch0 coincides with
  the u=0..t subpatch, and \a patch1 coincides with the u=t..1 subpatch. Each of 
  the new patches is reparameterized so that its u parameter goes from 0 to 1, 
  and the w direction remains parameterized as before. This method is only 
  allowed for tol <= t <= 1-tol where tol is the default tolerance for this 
  precision. 
  
  @param[in]  u       The u parameter of splitting isoparametric curve
                      (tol <= u <= 1-tol).
  @param[out] patch0 Subpatch covering domain [0..u,0..1] (contains 0,0 corner).
  @param[out] patch1 Subpatch covering domain [u..1,0..1] (contains 1,1 corner).
  
  Cost is 180 flops, or 120 flops if u=1/2. **/
  void splitU(RealP u, BicubicBezierPatch_<P>& patch0, 
                       BicubicBezierPatch_<P>& patch1) const {
      const RealP tol = getDefaultTol<RealP>();
  	#ifdef SimTK_REAL_IS_ADOUBLE
      SimTK_ERRCHK1(tol <= u && u <= 1-tol, "Geo::BicubicBezierPatch::splitU()",
          "Can't split patch at parameter u=%g; it is either out of range or"
          " too close to an edge.", Recorder(u).getValue());
  	#else
  		SimTK_ERRCHK1(tol <= u && u <= 1 - tol, "Geo::BicubicBezierPatch::splitU()",
  			"Can't split patch at parameter u=%g; it is either out of range or"
  			" too close to an edge.", (double)u);
  	#endif
      CubicBezierCurve_<P> l,h;
      if (u==RealP(0.5)) // bisecting
          for (int i=0; i<4; ++i) {
              CubicBezierCurve_<P>(B(i)).bisect(l,h);
              patch0.B(i) = l.getControlPoints(); 
              patch1.B(i) = h.getControlPoints();
          }
      else 
          for (int i=0; i<4; ++i) {
              CubicBezierCurve_<P>(B(i)).split(u,l,h);
              patch0.B(i) = l.getControlPoints(); 
              patch1.B(i) = h.getControlPoints();
          }
  }
  
  /** Split this patch into two along the w direction, along an isoparametric 
  curve of constant w=t such that 0 < t < 1. On return, \a patch0 coincides with
  the w=0..t subpatch, and \a patch1 coincides with the w=t..1 subpatch. Each of 
  the new patches is reparameterized so that its w parameter goes from 0 to 1, 
  and the u direction remains parameterized as before. This method is only 
  allowed for tol <= t <= 1-tol where tol is the default tolerance for this 
  precision. 
  
  @param[in]  w       The w parameter of splitting isoparametric curve
                      (tol <= w <= 1-tol).
  @param[out] patch0 Subpatch covering domain [0..1,0..w] (contains 0,0 corner).
  @param[out] patch1 Subpatch covering domain [0..1,w..1] (contains 1,1 corner).
  
  Cost is 180 flops, or 120 flops if w=1/2. **/
  void splitW(RealP w, BicubicBezierPatch_<P>& patch0, 
                       BicubicBezierPatch_<P>& patch1) const {
      const RealP tol = getDefaultTol<RealP>();
  	#ifdef SimTK_REAL_IS_ADOUBLE
  		SimTK_ERRCHK1(tol <= w && w <= 1-tol, "Geo::BicubicBezierPatch::splitW()",
  			"Can't split patch at parameter w=%g; it is either out of range or"
  			" too close to an edge.", Recorder(w).getValue());
  	#else
  		SimTK_ERRCHK1(tol <= w && w <= 1 - tol, "Geo::BicubicBezierPatch::splitW()",
  			"Can't split patch at parameter w=%g; it is either out of range or"
  			" too close to an edge.", (double)w);
  	#endif
      CubicBezierCurve_<P> l,h;
      if (w==RealP(0.5)) // bisecting
          for (int i=0; i<4; ++i) {
              CubicBezierCurve_<P>(B[i]).bisect(l,h);
              patch0.B[i] = l.getControlPoints().positionalTranspose(); 
              patch1.B[i] = h.getControlPoints().positionalTranspose();
          }
      else 
          for (int i=0; i<4; ++i) {
              CubicBezierCurve_<P>(B[i]).split(w,l,h);
              patch0.B[i] = l.getControlPoints().positionalTranspose(); 
              patch1.B[i] = h.getControlPoints().positionalTranspose();
          }
  }
  
  
  /** Split this patch into four subpatches at a particular parametric point 
  (u,w) such that 0 < u,w < 1, with each resulting subpatch covering part of the 
  original parametric domain in each direction. The subpatches are reparametrized
  to have [0..1,0..1] domains. This method is only allowed for 
  tol <= u,w <= 1-tol where tol is the default tolerance for this precision. 
  
  @param[in]  u       The u parameter of the splitting point (tol <= u <= 1-tol).
  @param[in]  w       The w parameter of the splitting point (tol <= w <= 1-tol).
  @param[out] patch00 Subpatch covering domain [0..u,0..w] (contains 0,0 corner).
  @param[out] patch01 Subpatch covering domain [0..u,w..1] (contains 0,1 corner).
  @param[out] patch10 Subpatch covering domain [u..1,0..w] (contains 1,0 corner).
  @param[out] patch11 Subpatch covering domain [u..1,w..1] (contains 1,1 corner).
  
  Cost is 360, 420, or 540 flops depending on whether both, one, or neither 
  parameter is 1/2. **/
  void split(RealP u, RealP w, BicubicBezierPatch_<P>& patch00, 
                               BicubicBezierPatch_<P>& patch01, 
                               BicubicBezierPatch_<P>& patch10, 
                               BicubicBezierPatch_<P>& patch11) const {
      const RealP tol = getDefaultTol<RealP>();
  	#ifdef SimTK_REAL_IS_ADOUBLE
  		SimTK_ERRCHK2((tol <= u && u <= 1-tol) && (tol <= w && w <= 1-tol), 
  			"Geo::BicubicBezierPatch::split()",
  			"Can't split patch at parametric point u,w=%g,%g; it is either out of"
  			" range or too close to an edge.", Recorder(u).getValue(), Recorder(w).getValue());
  	#else
  		SimTK_ERRCHK2((tol <= u && u <= 1 - tol) && (tol <= w && w <= 1 - tol),
  			"Geo::BicubicBezierPatch::split()",
  			"Can't split patch at parametric point u,w=%g,%g; it is either out of"
  			" range or too close to an edge.", (double)u, (double)w);
  	#endif
  
      BicubicBezierPatch_<P> patch0, patch1; // results of the first split
  
      // We split once along one direction, and then twice along the other, so
      // make sure the second direction is the cheap bisecting one.
      if (u == Real(0.5)) {
          splitW(w,patch0,patch1);            // 120 or 180 flops
          patch0.splitU(u, patch00, patch10); // 120 flops
          patch1.splitU(u, patch01, patch11); // 120 flops
      } else {
          splitU(u,patch0,patch1);            // 180 flops
          patch0.splitW(w, patch00, patch01); // 120 or 180 flops
          patch1.splitW(w, patch10, patch11); // 120 or 180 flops
      }
  }
  
  /** Return a sphere that surrounds the entire patch in the 0<= u,w <=1
  range. 
  
  @bug Currently we use the fact that the patch is enclosed within the convex 
  hull of its control points and generate the minimum bounding sphere that 
  includes all control points; that is not necessarily the 
  tightest-fitting sphere. **/
  Geo::Sphere_<P> calcBoundingSphere() const 
  {   const ArrayViewConst_<Vec3P> points(&B(0,0), &B(0,0)+16); // no copy 
      return Geo::Point_<P>::calcBoundingSphere(points); }
  
  /** Return an axis-aligned bounding box (AABB) that surrounds the entire patch 
  segment in the 0<= u,w <=1 range. 
  
  @bug Currently we use the fact that the patch is enclosed within the convex 
  hull of its control points and generate the minimum bounding box that 
  includes all control points; that is not necessarily the 
  tightest-fitting box. **/
  Geo::AlignedBox_<P> calcAxisAlignedBoundingBox() const 
  {   const ArrayViewConst_<Vec3P> points(&B(0,0), &B(0,0)+16); // no copy 
      return Geo::Point_<P>::calcAxisAlignedBoundingBox(points); }
  
  /** Return an oriented bounding box (OBB) that surrounds the entire curve 
  segment in the 0<= u,w <=1 range. 
  
  @bug Currently we use the fact that the patch is enclosed within the convex 
  hull of its control points and generate an oriented bounding box that 
  includes all control points; that is not necessarily the 
  tightest-fitting box. **/
  Geo::OrientedBox_<P> calcOrientedBoundingBox() const 
  {   const ArrayViewConst_<Vec3P> points(&B(0,0), &B(0,0)+16); // no copy 
      return Geo::Point_<P>::calcOrientedBoundingBox(points); }
  
  /**@name                 Utility methods
  These static methods work with given control points. **/
  /**@{**/
  
  /** Given Bezier control points B and values for the curve parameters 
  u and w in [0..1], return the point P(u,w)=Fb(u)*B*~Fb(w) at that location, 
  where Fb is a vector of Bezier basis functions. Cost is 
  3x35+18=123 flops. **/
  static Vec3P evalPUsingB(const Mat<4,4,Vec3P>& B, RealP u, RealP w) { 
      Row<4,P> Fbu = CubicBezierCurve_<P>::calcFb(u);     // 9 flops
      Row<4,P> Fbw = CubicBezierCurve_<P>::calcFb(w);     // 9 flops
      return Fbu * B * ~Fbw;                              // 3x35 flops
  }
  
  /** Given Bezier control points B and values for the curve parameters 
  u and w in [0..1], return the tangents Pu(u,w)=dFb(u)*B*~Fb(w) and 
  Pw(u,w)=Fb(u)*B*~dFb(w) at 
  that location. Cost is 3x70+38=248 flops. **/
  static void evalP1UsingB(const Mat<4,4,Vec3P>& B, RealP u, RealP w,
                           Vec3P& Pu, Vec3P& Pw) {
      Row<4,P> Fbu  = CubicBezierCurve_<P>::calcFb(u);     //  9 flops
      Row<4,P> Fbw  = CubicBezierCurve_<P>::calcFb(w);     //  9 flops
      Row<4,P> dFbu = CubicBezierCurve_<P>::calcDFb(u);    // 10 flops
      Row<4,P> dFbw = CubicBezierCurve_<P>::calcDFb(w);    // 10 flops
      Pu = dFbu * B * ~Fbw;                                // 3x35
      Pw = Fbu  * B * ~dFbw;                               // 3x35
  }
  
  /** Given Bezier control points B and values for the curve parameters 
  u and w in [0..1], return the second derivatives Puu(u,w)=d2Fb(u)*B*~Fb(w),
  Puw(u,w)=dFb(u)*B*~dFb(w) and Pww(u,w)=Fb(u)*B*~d2Fb(w) at that location. 
  Cost is 3x105+48=363 flops. **/
  static void evalP2UsingB(const Mat<4,4,Vec3P>& B, RealP u, RealP w,
                           Vec3P& Puu, Vec3P& Puw, Vec3P& Pww) {
      Row<4,P> Fbu   = CubicBezierCurve_<P>::calcFb(u);     //  9 flops
      Row<4,P> Fbw   = CubicBezierCurve_<P>::calcFb(w);     //  9 flops
      Row<4,P> dFbu  = CubicBezierCurve_<P>::calcDFb(u);    // 10 flops
      Row<4,P> dFbw  = CubicBezierCurve_<P>::calcDFb(w);    // 10 flops
      Row<4,P> d2Fbu = CubicBezierCurve_<P>::calcD2Fb(u);   //  5 flops
      Row<4,P> d2Fbw = CubicBezierCurve_<P>::calcD2Fb(w);   //  5 flops
      Puu = d2Fbu * B * ~Fbw;                               // 3x35
      Puw = dFbu  * B * ~dFbw;                              // 3x35
      Pww = Fbu   * B * ~d2Fbw;                             // 3x35
  }
  
  /** Given Bezier control points B and values for the curve parameters 
  u and w in [0..1], return the third derivatives Puuu(u,w)=d3Fb(u)*B*~Fb(w),
  Puuw(u,w)=d2Fb(u)*B*~dFb(w), Puww(u,w)=dFb(u)*B*~d2Fb(w) and 
  Pwww(u,w)=Fb(u)*B*~d3Fb(w) at that location. Cost is 3x140+48=468 flops. **/
  static void evalP3UsingB(const Mat<4,4,Vec3P>& B, RealP u, RealP w,
                           Vec3P& Puuu, Vec3P& Puuw, Vec3P& Puww, Vec3P& Pwww) {
      Row<4,P> Fbu   = CubicBezierCurve_<P>::calcFb(u);     //  9 flops
      Row<4,P> Fbw   = CubicBezierCurve_<P>::calcFb(w);     //  9 flops
      Row<4,P> dFbu  = CubicBezierCurve_<P>::calcDFb(u);    // 10 flops
      Row<4,P> dFbw  = CubicBezierCurve_<P>::calcDFb(w);    // 10 flops
      Row<4,P> d2Fbu = CubicBezierCurve_<P>::calcD2Fb(u);   //  5 flops
      Row<4,P> d2Fbw = CubicBezierCurve_<P>::calcD2Fb(w);   //  5 flops
      Row<4,P> d3Fbu = CubicBezierCurve_<P>::calcD3Fb(u);   //  0 
      Row<4,P> d3Fbw = CubicBezierCurve_<P>::calcD3Fb(w);   //  0
      Puuu = d3Fbu * B * ~Fbw;                              // 3x35
      Puuw = d2Fbu * B * ~dFbw;                             // 3x35
      Puww = dFbu  * B * ~d2Fbw;                            // 3x35
      Pwww = Fbu   * B * ~d3Fbw;                            // 3x35
  }
  
  /** Given a particular value u0 for patch coordinate u, create a cubic Bezier
  curve segment P(w)=P(u0,w) for the isoparametric curve along the patch at fixed
  u=u0. Cost is 3x28+9=93 flops. **/
  static CubicBezierCurve_<P> 
  calcIsoCurveU(const Mat<4,4,Vec3P>& B, RealP u0) 
  {   const Row<4,Vec3P> Bu0 = CubicBezierCurve_<P>::calcFb(u0) * B;
      return CubicBezierCurve_<P>(Bu0); }
  
  /** Given a particular value w0 for patch coordinate w, create a cubic Bezier
  curve segment P(u)=P(u,w0) for the isoparametric curve along the patch at fixed
  w=w0. Cost is 3x28+9=93 flops. **/
  static CubicBezierCurve_<P> 
  calcIsoCurveW(const Mat<4,4,Vec3P>& B, RealP w0)
  {    const Vec<4,Vec3P> Bw0 = B * ~CubicBezierCurve_<P>::calcFb(w0);
       return CubicBezierCurve_<P>(Bw0); }
  
  /** Given the Bezier control points B, return the equivalent vector algebraic
  coefficients A. All coefficients are 3-vectors. Cost is 3x80=240 flops. **/
  static Mat<4,4,Vec3P> calcAFromB(const Mat<4,4,Vec3P>& B) {
      typedef const Vec3P& Coef;
      Coef b11=B(0,0), b12=B(0,1), b13=B(0,2), b14=B(0,3), 
           b21=B(1,0), b22=B(1,1), b23=B(1,2), b24=B(1,3), 
           b31=B(2,0), b32=B(2,1), b33=B(2,2), b34=B(2,3), 
           b41=B(3,0), b42=B(3,1), b43=B(3,2), b44=B(3,3); 
      // First calculate Mb*B:
      //       a   b   c   d
      //       e   f   g   h
      //       p   q   r   s
      //      b11 b12 b13 b14
      Vec3P a= b41-b11+3*(b21-b31), b= b42-b12+3*(b22-b32),   // 3x16 flops
            c= b43-b13+3*(b23-b33), d= b44-b14+3*(b24-b34);
      Vec3P e= 3*(b11+b31)-6*b21,   f= 3*(b12+b32)-6*b22,     // 3x16 flops
            g= 3*(b13+b33)-6*b23,   h= 3*(b14+b34)-6*b24;
      Vec3P p= 3*(b21-b11),         q= 3*(b22-b12),           // 3x8 flops
            r= 3*(b23-b13),         s= 3*(b24-b14);
  
      // Then calculate (Mb*B)*~Mb. (3x40 more flops)
      return Mat<4,4,Vec3P>
         ( d-a+3*(b-c),          3*(a+c)-6*b,       3*(b-a),        a,
           h-e+3*(f-g),          3*(e+g)-6*f,       3*(f-e),        e,
           s-p+3*(q-r),          3*(p+r)-6*q,       3*(q-p),        p,
           b14-b11+3*(b12-b13),  3*(b11+b13)-6*b12, 3*(b12-b11),   b11 );
  }
  
  /** Given the vector algebraic coefficients A, return the equivalent Bezier
  control points B. All coefficients are 3-vectors. Cost is 3x72=216 flops. **/
  static Mat<4,4,Vec3P> calcBFromA(const Mat<4,4,Vec3P>& A) {
      typedef const Vec3P& Coef;
      Coef a33=A(0,0), a32=A(0,1), a31=A(0,2), a30=A(0,3), 
           a23=A(1,0), a22=A(1,1), a21=A(1,2), a20=A(1,3), 
           a13=A(2,0), a12=A(2,1), a11=A(2,2), a10=A(2,3), 
           a03=A(3,0), a02=A(3,1), a01=A(3,2), a00=A(3,3); 
      // First calculate Mb^-1*A:
      //      a03 a02 a01 a00
      //       a   b   c   d
      //       e   f   g   h
      //       p   q   r   s
      Vec3P a=a13/3+a03, b=a12/3+a02, c=a11/3+a01, d=a10/3+a00;   // 3x8 flops
      Vec3P e=(a23+2*a13)/3+a03, f=(a22+2*a12)/3+a02,             // 3x16 flops
            g=(a21+2*a11)/3+a01, h=(a20+2*a10)/3+a00;
      Vec3P p=a33+a23+a13+a03, q=a32+a22+a12+a02,                 // 3x12 flops
            r=a31+a21+a11+a01, s=a30+a20+a10+a00;
  
  
      // Then calculate (Mb^-1*A)*Mb^-T (3x36 more flops)
      return Mat<4,4,Vec3P>
         ( a00,   a01/3+a00,  (a02+2*a01)/3+a00,     a03+a02+a01+a00,
            d,      c/3+d,       (b+2*c)/3+d,             a+b+c+d,
            h,      g/3+h,       (f+2*g)/3+h,             e+f+g+h,
            s,      r/3+s,       (q+2*r) /3+s,            p+q+r+s     );
  }
  
  /** Given the Bezier control points B, return the equivalent vector Hermite
  coefficients H. All coefficients are 3-vectors. Cost is 3x28=84 flops. **/
  static Mat<4,4,Vec3P> calcHFromB(const Mat<4,4,Vec3P>& B) {
      typedef const Vec3P& Coef;
      Coef b11=B(0,0), b12=B(0,1), b13=B(0,2), b14=B(0,3), 
           b21=B(1,0), b22=B(1,1), b23=B(1,2), b24=B(1,3), 
           b31=B(2,0), b32=B(2,1), b33=B(2,2), b34=B(2,3), 
           b41=B(3,0), b42=B(3,1), b43=B(3,2), b44=B(3,3); 
  
      // First calculate a few temps -- see class comments. (3x4 flops)
      Vec3P b12mb11=b12-b11, b24mb14=b24-b14, b42mb41=b42-b41, b44mb34=b44-b34;
  
      // Then calculate (Mh^-1 Mb) B ~(Mh^-1 Mb). (3x24 more flops)
      return Mat<4,4,Vec3P>
         (     b11,         b14,          3*b12mb11,           3*(b14-b13),
               b41,         b44,          3*b42mb41,           3*(b44-b43),
           3*(b21-b11),  3*b24mb14,  9*(b22-b21-b12mb11),  9*(b24mb14+b13-b23),
           3*(b41-b31),  3*b44mb34,  9*(b42mb41+b31-b32),  9*(b44mb34+b33-b43) );
  }
  
  /** Given the vector Hermite coefficients H, return the equivalent Bezier
  control points B. All coefficients are 3-vectors. Cost is 3x32=96 flops. **/
  static Mat<4,4,Vec3P> calcBFromH(const Mat<4,4,Vec3P>& H) {
      typedef const Vec3P& Coef;
      Coef h00=H(0,0), h01=H(0,1), w00=H(0,2), w01=H(0,3), 
           h10=H(1,0), h11=H(1,1), w10=H(1,2), w11=H(1,3), 
           u00=H(2,0), u01=H(2,1), t00=H(2,2), t01=H(2,3), 
           u10=H(3,0), u11=H(3,1), t10=H(3,2), t11=H(3,3); 
  
      // First calculate a few temps -- see class comments. (3x8 flops)
      Vec3P tmp00=h00+u00/3, tmp01=h01+u01/3, 
            tmp10=h10-u10/3, tmp11=h11-u11/3;
  
      // Then calculate (Mb^-1 Mh) H ~(Mb^-1 Mh). (3x24 more flops)
      return Mat<4,4,Vec3P>
         (  h00,       h00+w00/3,          h01-w01/3,       h01,
           tmp00,  tmp00+w00/3+t00/9,  tmp01-w01/3-t01/9,  tmp01,
           tmp10,  tmp10+w10/3-t10/9,  tmp11-w11/3+t11/9,  tmp11,
            h10,       h10+w10/3,          h11-w11/3,       h11  );
  }
  /**@}**/
  
  private:
  Mat<4,4,Vec3P> B;   // 16 Bezier control points; see above for definition
  };
  
  
  
  } // namespace SimTK
  
  #endif // SimTK_SIMMATH_GEO_BICUBIC_BEZIER_PATCH_H_
  