vgl_homg_operators_2d.hxx

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// This is core/vgl/algo/vgl_homg_operators_2d.hxx
#ifndef vgl_homg_operators_2d_hxx_
#define vgl_homg_operators_2d_hxx_
//:
// \file

#include <iostream>
#include <limits>
#include <cmath>
#include "vgl_homg_operators_2d.h"

#ifdef _MSC_VER
#  include <vcl_msvc_warnings.h>
#endif
#include <cassert>
#include <vgl/vgl_homg_line_2d.h>
#include <vgl/vgl_homg_point_2d.h>
#include <vgl/vgl_point_2d.h>
#include <vgl/vgl_conic.h>
#include <vgl/vgl_box_2d.h>
#include <vgl/vgl_homg.h>

#include <vnl/vnl_vector_fixed.h>
#include <vnl/vnl_matrix.h>
#include <vnl/vnl_matrix_fixed.h>
#include <vnl/vnl_math.h>
#include <vnl/algo/vnl_scatter_3x3.h> // used in most_orthogonal_vector()
#include <vnl/algo/vnl_real_eigensystem.h> // used for conic intersection
#include <vnl/vnl_diag_matrix.h>  // used for conic intersection

//-----------------------------------------------------------------------------

template <class T>
inline static vgl_homg_line_2d<T> cross(const vgl_homg_point_2d<T>& p1,
                                        const vgl_homg_point_2d<T>& p2)
{
  return vgl_homg_line_2d<T> (p1.y() * p2.w() - p1.w() * p2.y(),
                              p1.w() * p2.x() - p1.x() * p2.w(),
                              p1.x() * p2.y() - p1.y() * p2.x());
}

template <class T>
inline static vgl_homg_point_2d<T> cross(const vgl_homg_line_2d<T>& l1,
                                         const vgl_homg_line_2d<T>& l2)
{
  return vgl_homg_point_2d<T> (l1.b() * l2.c() - l1.c() * l2.b(),
                               l1.c() * l2.a() - l1.a() * l2.c(),
                               l1.a() * l2.b() - l1.b() * l2.a());
}

template <class T>
inline static T dot(vgl_homg_line_2d<T> const& l,
                    vgl_homg_point_2d<T> const& p)
{
  return l.a()*p.x() + l.b()*p.y() + l.c()*p.w();
}

//-----------------------------------------------------------------------------

template <class T>
vnl_vector_fixed<T,3> vgl_homg_operators_2d<T>::get_vector(vgl_homg_point_2d<T> const& p)
{
  return vnl_vector_fixed<T,3>(p.x(),p.y(),p.w());
}

template <class T>
vnl_vector_fixed<T,3> vgl_homg_operators_2d<T>::get_vector(vgl_homg_line_2d<T> const& l)
{
  return vnl_vector_fixed<T,3>(l.a(),l.b(),l.c());
}

template <class T>
vnl_vector_fixed<T,6> vgl_homg_operators_2d<T>::get_vector(vgl_conic<T> const& c)
{
  vnl_vector_fixed<T,6> v;
  v.put(0,c.a());
  v.put(1,c.b());
  v.put(2,c.c());
  v.put(3,c.d());
  v.put(4,c.e());
  v.put(5,c.f());

  return v;
}

//-----------------------------------------------------------------------------
//: Normalize vgl_homg_point_2d<T> to unit magnitude
template <class T>
void vgl_homg_operators_2d<T>::unitize(vgl_homg_point_2d<T>& a)
{
  double norm = std::sqrt (a.x()*a.x() + a.y()*a.y() + a.w()*a.w());

  if (norm == 0.0) {
    std::cerr << "vgl_homg_operators_2d<T>::unitize() -- Zero length vector\n";
    return;
  }
  norm = 1.0/norm;
  a.set(T(a.x()*norm), T(a.y()*norm), T(a.w()*norm));
}

//  DISTANCE MEASUREMENTS IN EUCLIDEAN COORDINATES

template <class T>
T
vgl_homg_operators_2d<T>::distance(const vgl_homg_point_2d<T>& point1,
                                   const vgl_homg_point_2d<T>& point2)
{
  return std::sqrt(vgl_homg_operators_2d<T>::distance_squared(point1,point2));
}

template <class T>
T
vgl_homg_operators_2d<T>::distance_squared(const vgl_homg_point_2d<T>& p1,
                                           const vgl_homg_point_2d<T>& p2)
{
  if (p1 == p2) return T(0); // quick return if possible

  if (p1.w() == 0 || p2.w() == 0) {
    std::cerr << "vgl_homg_operators_2d<T>::distance_squared() -- point at infinity\n";
    return std::numeric_limits<T>::infinity();
  }

  return vnl_math::sqr (p1.x() / p1.w() - p2.x() / p2.w()) +
         vnl_math::sqr (p1.y() / p1.w() - p2.y() / p2.w());
}

//: Get the square of the perpendicular distance to a line.
// This is just the homogeneous form of the familiar
// \f$ \frac{a x + b y + c}{\sqrt{a^2+b^2}} \f$ :
// \f[ d = \frac{(l^\top p)}{p_z\sqrt{l_x^2 + l_y^2}} \f]
// If either the point or the line are at infinity an error message is
// printed and vgl_homg::infinity is returned.
template <class T>
T
vgl_homg_operators_2d<T>::perp_dist_squared(const vgl_homg_point_2d<T>& point,
                                            const vgl_homg_line_2d<T>& line)
{
  if ((line.a()==0 && line.b()== 0) || point.w()==0) {
    std::cerr << "vgl_homg_operators_2d<T>::perp_dist_squared() -- line or point at infinity\n";
    return vgl_homg<T>::infinity;
  }

  T numerator = vnl_math::sqr (dot(line,point) / point.w());
  if (numerator == 0) return T(0); // efficiency
  T denominator = line.a()*line.a() + line.b()*line.b();

  return numerator / denominator;
}

//  ANGLES

//-----------------------------------------------------------------------------
//: Get the anticlockwise angle between a line and the \a x axis.
template <class T>
double
vgl_homg_operators_2d<T>::line_angle(const vgl_homg_line_2d<T>& line)
{
  return std::atan2 (line.b(), line.a());
}

//-----------------------------------------------------------------------------
//: Get the 0 to pi/2 angle between two lines
template <class T>
double
vgl_homg_operators_2d<T>::abs_angle(const vgl_homg_line_2d<T>& line1,
                                    const vgl_homg_line_2d<T>& line2)
{
  double angle1 = line_angle(line1);
  double angle2 = line_angle(line2);

  double diff = angle2 - angle1;
  if (diff >  vnl_math::pi_over_2) diff -= vnl_math::pi;
  if (diff < -vnl_math::pi_over_2) diff += vnl_math::pi;

  return std::fabs(diff);
}

//-----------------------------------------------------------------------------
//
//: Get the angle between two lines, a number between -PI and PI.
// Although homogeneous coordinates are
// only defined up to scale, here it is assumed that a line with homogeneous
// coordinates (m) is at 180 degrees to a line (-m), and this is why the term
// "oriented_line" is used.  However, the overall scale (apart from sign) is
// not significant.
template <class T>
double
vgl_homg_operators_2d<T>::angle_between_oriented_lines(const vgl_homg_line_2d<T>& line1,
                                                       const vgl_homg_line_2d<T>& line2)
{
  return std::fmod(line_angle(line2)-line_angle(line1), vnl_math::twopi);
}

//  JOINS/INTERSECTIONS

//-----------------------------------------------------------------------------
//
//: Get the line through two points (the cross-product).
//

template <class T>
vgl_homg_line_2d<T>
vgl_homg_operators_2d<T>::join(const vgl_homg_point_2d<T>& p1,
                               const vgl_homg_point_2d<T>& p2)
{
  return cross(p1,p2);
}

//-----------------------------------------------------------------------------
//
//: Get the line through two points (the cross-product).
// In this case, we assume that the points are oriented, and ensure the
// cross is computed with positive point omegas.
template <class T>
vgl_homg_line_2d<T>
vgl_homg_operators_2d<T>::join_oriented(const vgl_homg_point_2d<T>&p1,
                                        const vgl_homg_point_2d<T>&p2)
{
  int s1 = p1.w() < 0;
  int s2 = p2.w() < 0;

  if (s1 ^ s2)
    return cross(p2,p1);
  else
    return cross(p1,p2);
}

//-----------------------------------------------------------------------------
//
//: Get the intersection point of two lines (the cross-product).
template <class T>
vgl_homg_point_2d<T>
vgl_homg_operators_2d<T>::intersection(const vgl_homg_line_2d<T>& l1,
                                       const vgl_homg_line_2d<T>& l2)
{
  return cross(l1,l2);
}

//-----------------------------------------------------------------------------
//
//: Get the perpendicular line to line which passes through point.
// Params are line \f$(a,b,c)\f$ and point \f$(x,y,1)\f$.
// Then the cross product of \f$(x,y,1)\f$ and the line's direction \f$(a,b,0)\f$,
// called \f$(p,q,r)\f$ satisfies
// - \f$ap+bq=0\f$ (perpendicular condition) and
// - \f$px+qy+r=0\f$ (incidence condition).
template <class T>
vgl_homg_line_2d<T>
vgl_homg_operators_2d<T>::perp_line_through_point(const vgl_homg_line_2d<T>& l,
                                                  const vgl_homg_point_2d<T>& p)
{
  vgl_homg_point_2d<T> d(l.a(), l.b(), 0);
  return cross(d, p);
}

//-----------------------------------------------------------------------------
//
//: Get the perpendicular projection of point onto line.
template <class T>
vgl_homg_point_2d<T>
vgl_homg_operators_2d<T>::perp_projection(const vgl_homg_line_2d<T>& line,
                                          const vgl_homg_point_2d<T>& point)
{
  vgl_homg_line_2d<T> perpline = perp_line_through_point (line, point);
  vgl_homg_point_2d<T> answer = cross(line, perpline);
  return answer;
}

//: Return the midpoint of the line joining two homogeneous points
//  When one of the points is at infinity, that point is returned.
//  When both points are at infinity, the invalid point (0,0,0) is returned.
template <class T>
vgl_homg_point_2d<T>
vgl_homg_operators_2d<T>::midpoint(const vgl_homg_point_2d<T>& p1,
                                   const vgl_homg_point_2d<T>& p2)
{
  T x = p1.x() * p2.w() + p2.x() * p1.w();
  T y = p1.y() * p2.w() + p2.y() * p1.w();
  T w = p1.w() * p2.w() + p2.w() * p1.w();

  return vgl_homg_point_2d<T>(x,y,w);
}

//  FITTING

// - Kanatani sect 2.2.2.
template <class T>
vnl_vector_fixed<T,3>
vgl_homg_operators_2d<T>::most_orthogonal_vector(const std::vector<vgl_homg_line_2d<T> >& inpoints)
{
  vnl_scatter_3x3<T> scatter_matrix;

  for (typename std::vector<vgl_homg_line_2d<T> >::const_iterator i = inpoints.begin();
       i != inpoints.end(); ++i)
    scatter_matrix.add_outer_product(get_vector(*i));

  return scatter_matrix.minimum_eigenvector();
}

#include <vnl/algo/vnl_svd.h>

template <class T>
vnl_vector_fixed<T,3>
vgl_homg_operators_2d<T>::most_orthogonal_vector_svd(const std::vector<vgl_homg_line_2d<T> >& lines)
{
  vnl_matrix<T> D(lines.size(), 3);

  typename std::vector<vgl_homg_line_2d<T> >::const_iterator i = lines.begin();
  for (unsigned j = 0; i != lines.end(); ++i,++j)
    D.set_row(j, get_vector(*i).as_ref());

  vnl_svd<T> svd(D);
#ifdef DEBUG
  std::cout << "[movrank " << svd.W() << ']';
#endif

  return svd.nullvector();
}

//: Intersect a set of 2D lines to find the least-square point of intersection.
// This finds the point $\bf x$ that minimizes $\|\tt L \bf x\|$, where $\tt L$
// is the matrix whose rows are the lines. The implementation uses either
// vnl_scatter_3x3 (default) or vnl_svd (when at compile time
// VGL_HOMG_OPERATORS_2D_LINES_TO_POINT_USE_SVD has been set) to accumulate and
// compute the nullspace of $\tt L^\top \tt L$.
template <class T>
vgl_homg_point_2d<T>
vgl_homg_operators_2d<T>::lines_to_point(const std::vector<vgl_homg_line_2d<T> >& lines)
{
  // ho_triveccam_aspect_lines_to_point
  assert(lines.size() >= 2);

#ifdef VGL_HOMG_OPERATORS_2D_LINES_TO_POINT_USE_SVD
  vnl_vector_fixed<T,3> mov = most_orthogonal_vector_svd(lines);
#else
  vnl_vector_fixed<T,3> mov = most_orthogonal_vector(lines);
#endif
  return vgl_homg_point_2d<T>(mov[0], mov[1], mov[2]);
}

//  MISCELLANEOUS

//-----------------------------------------------------------------------------
//: Calculates the crossratio of four collinear points p1, p2, p3 and p4.
// This number is projectively invariant, and it is the coordinate of p4
// in the reference frame where p2 is the origin (coordinate 0), p3 is
// the unity (coordinate 1) and p1 is the point at infinity.
// This cross ratio is often denoted as ((p1, p2; p3, p4)) (which also
// equals ((p3, p4; p1, p2)) or ((p2, p1; p4, p3)) or ((p4, p3; p2, p1)) )
// and is calculated as
// \verbatim
//                      p1 - p3   p2 - p3      (p1-p3)(p2-p4)
//                      ------- : --------  =  --------------
//                      p1 - p4   p2 - p4      (p1-p4)(p2-p3)
// \endverbatim
//
// In principle, any single nonhomogeneous coordinate from the four points
// can be used as parameters for cross_ratio (but of course the same for all
// points). The most reliable answer will be obtained when the coordinate with
// the largest spacing is used, i.e., the one with smallest slope.
template <class T>
double vgl_homg_operators_2d<T>::cross_ratio(const vgl_homg_point_2d<T>& a,
                                             const vgl_homg_point_2d<T>& b,
                                             const vgl_homg_point_2d<T>& c,
                                             const vgl_homg_point_2d<T>& d)
{
  double x1 = a.x(), y1 = a.y(), w1 = a.w();
  double x2 = b.x(), y2 = b.y(), w2 = b.w();
  double x3 = c.x(), y3 = c.y(), w3 = c.w();
  double x4 = d.x(), y4 = d.y(), w4 = d.w();
  double x = x1 - x2; if (x<0) x = -x; // assuming a != b ;-)
  double y = y1 - y2; if (y<0) y = -y;
  double n = (x>y) ? (x1*w3-x3*w1)*(x2*w4-x4*w2) : (y1*w3-y3*w1)*(y2*w4-y4*w2);
  double m = (x>y) ? (x1*w4-x4*w1)*(x2*w3-x3*w2) : (y1*w4-y4*w1)*(y2*w3-y3*w2);
  if (n == 0 && m == 0)
    std::cerr << "cross ratio not defined: three of the given points coincide\n";
  return n/m;
}

//: Conjugate point of three given collinear points.
// If cross ratio cr is given (default: -1), the generalized conjugate point
// returned is such that ((x1,x2;x3,answer)) = cr.
template <class T>
vgl_homg_point_2d<T>
vgl_homg_operators_2d<T>::conjugate(const vgl_homg_point_2d<T>& a,
                                    const vgl_homg_point_2d<T>& b,
                                    const vgl_homg_point_2d<T>& c,
                                    double cr)
// Default for cr is -1.
{
  T x1 = a.x(), y1 = a.y(), w1 = a.w();
  T x2 = b.x(), y2 = b.y(), w2 = b.w();
  T x3 = c.x(), y3 = c.y(), w3 = c.w();
  T kx = x1*w3 - x3*w1, mx = x2*w3 - x3*w2, nx = T(kx*w2-cr*mx*w1);
  T ky = y1*w3 - y3*w1, my = y2*w3 - y3*w2, ny = T(ky*w2-cr*my*w1);
  return vgl_homg_point_2d<T>(T(x2*kx-cr*x1*mx)*ny,T(y2*ky-cr*y1*my)*nx,nx*ny);
}

//: returns the vgl_conic which has the given matrix as its matrix
// \verbatim
// [A,B,C,D,E,F] <-   [ A  B/2 D/2 ]
//                    [ B/2 C  E/2 ]
//                    [ D/2 E/2 F  ]
// \endverbatim
template <class T>
vgl_conic<T>
vgl_homg_operators_2d<T>::vgl_conic_from_matrix(vnl_matrix_fixed<T,3,3> const& mat)
{
  return vgl_conic<T>(mat[0][0], mat[1][0]+mat[0][1], mat[1][1], mat[0][2]+mat[2][0], mat[1][2]+mat[2][1], mat[2][2]);
}

//: returns 3x3 matrix containing conic coefficients.
// \verbatim
// [A,B,C,D,E,F] ->   [ A  B/2 D/2 ]
//                    [ B/2 C  E/2 ]
//                    [ D/2 E/2 F  ]
// \endverbatim
//
template <class T>
vnl_matrix_fixed<T,3,3>
vgl_homg_operators_2d<T>::matrix_from_conic(vgl_conic<T> const& c)
{
  vnl_matrix_fixed<T,3,3> mat;
  T A = c.a(), B = c.b()/2, C = c.c(), D = c.d()/2, E = c.e()/2, F = c.f();

  mat[0][0] = A; mat[0][1] = B; mat[0][2] = D;
  mat[1][0] = B; mat[1][1] = C; mat[1][2] = E;
  mat[2][0] = D; mat[2][1] = E; mat[2][2] = F;

  return mat;
}

//-------------------------------------------------------------------------
//: returns 3x3 matrix containing conic coefficients of dual conic.
// I.e., the inverse matrix (up to a scale factor) of the conic matrix.

template <class T>
vnl_matrix_fixed<T,3,3>
vgl_homg_operators_2d<T>::matrix_from_dual_conic(vgl_conic<T> const& c)
{
  vnl_matrix_fixed<T,3,3> mat;
  T A = c.a(), B = c.b()/2, C = c.c(), D = c.d()/2, E = c.e()/2, F = c.f();

  mat[0][0] = C*F-E*E; mat[0][1] = E*D-B*F; mat[0][2] = B*E-C*D;
  mat[1][0] = E*D-B*F; mat[1][1] = A*F-D*D; mat[1][2] = B*D-A*E;
  mat[2][0] = B*E-C*D; mat[2][1] = B*D-A*E; mat[2][2] = A*C-B*B;

  return mat;
}

//:
// This function is called from within intersection(vgl_conic<T>,vgl_conic<T>).
// The two conics passed to this function MUST NOT be degenerate!
template <class T>
std::list<vgl_homg_point_2d<T> >
vgl_homg_operators_2d<T>::do_intersect(vgl_conic<T> const& c1,
                                       vgl_conic<T> const& c2)
{
  if (c1==c2)
  {
    std::cerr << __FILE__
             << "Warning: the intersection of two identical conics is not a finite set of points.\n"
             << "Returning an empty list.\n";
    return std::list<vgl_homg_point_2d<T> >();
  }
  T A=c1.a(),B=c1.b(),C=c1.c(),D=c1.d(),E=c1.e(),F=c1.f();
  T a=c2.a(),b=c2.b(),c=c2.c(),d=c2.d(),e=c2.e(),f=c2.f();
  // Idea: eliminate the coefficient in x^2, solve for x in terms of y, resubstitute in the other equation.
  T ab=a*B-A*b, ac=a*C-A*c, ad=a*D-A*d, ae=a*E-A*e, af=a*F-A*f, BD=b*D+B*d;

  // If the quadratic parts of the two conics are identical (up to scale factor),
  // the intersection is the same as with a degenerate conic where one part is the line at infinity,
  // viz. the conic with as equation the (weighted) difference of the two equations;
  if ((ab==0 && ac==0)
  // If the parts without x of the two conics are identical (up to scale factor),
  // the intersection is the same as with a degenerate conic consisting of two vertical lines;
      || (ab==0 && ad==0)
  // And of course similarly for y:
      || (ab==0 && ae==0)
  // The following fix by Peter Vanroose, 28 december 2010:
  // The general idea *does not* work (at least: not easily) if two intersection points have the same y coordinate.
  // So make that a special case. It is easily detected, since in that case there is a scale factor lambda
  // such that c1-lamda*c2 decomposes into (y-y0)(x+my+n)=0. Which means that lambda=A/a, and the equation satisfies:
      || (ac*ad*ad+af*ab*ab == ab*ad*ae))
    return intersection(vgl_conic<T>(0,ab,ac,ad,ae,af),c1);

  // Back to the "general" case:
  vnl_vector_fixed<T,5> coef;
  coef(0) = ac*ac-ab*(b*C-B*c);
  coef(1) = 2*ac*ae-ab*(b*E-B*e)-BD*(a*C+A*c)+2*A*b*C*d+2*a*B*c*D;
  coef(2) = ae*ae-ab*(b*F-B*f)+ad*(c*D-C*d)-BD*(a*E+A*e)+2*a*B*e*D+2*A*b*E*d+2*ac*af;
  coef(3) = 2*ae*af-ad*(d*E-D*e)-BD*(a*F+A*f)+2*A*b*d*F+2*a*B*D*f;
  coef(4) = af*af-ad*(d*F-D*f);

  // Solutions of the fourth order equation
  //   4      3      2
  //  y  +  py  +  qy  +  ry  +  s  =  0
  // are the eigenvalues of the matrix
  // [ -p   -q   -r   -s ]
  // [  1    0    0    0 ]
  // [  0    1    0    0 ]
  // [  0    0    1    0 ]

  if (coef(0) == 0 && coef(1) == 0) { // The equation is actually of 2nd degree
    if (coef(2) == 0 && coef(3) == 0) return std::list<vgl_homg_point_2d<T> >(); // no real solutions.
    T dis = coef(3)*coef(3)-4*coef(2)*coef(4); // discriminant
    if (dis < 0) return std::list<vgl_homg_point_2d<T> >(); // no real solutions.
    T y;
    if (coef(2) == 0) dis=0, y = - coef(4) / coef(3);
    else                     y = - coef(3) / coef(2) / 2;
    T w = y*ab+ad;
    T x = -(y*y*ac+y*ae+af);
    if (x == 0 && w == 0) x = w = 1;
    if (dis == 0) {return std::list<vgl_homg_point_2d<T> >(2,vgl_homg_point_2d<T>(x,y*w,w));}
    dis = std::sqrt(dis) / coef(2) / 2;
    std::list<vgl_homg_point_2d<T> > solutions;
    y -= dis; w = y*ab+ad; x = -(y*y*ac+y*ae+af);
    if (x == 0 && w == 0) x = w = 1;
    solutions.push_back(vgl_homg_point_2d<T>(x,y*w,w));
    y += 2*dis; w = y*ab+ad; x = -(y*y*ac+y*ae+af);
    if (x == 0 && w == 0) x = w = 1;
    solutions.push_back(vgl_homg_point_2d<T>(x,y*w,w));
    return solutions;
  }
  if (coef(0) == 0) { // The equation is actually of 3rd degree
    coef /= -coef(1);
    double data[]={coef(2),coef(3),coef(4), 1,0,0, 0,1,0};
    vnl_matrix<double> M(data,3,3);
    vnl_real_eigensystem eig(M);
    vnl_diag_matrix<std::complex<double> >  polysolutions = eig.D;
    std::list<vgl_homg_point_2d<T> > solutions;
    for (int i=0;i<3;++i)
      if (std::abs(std::imag(polysolutions(i))) < 1e-3) {// only want the real solutions
        T y = (T)std::real(polysolutions(i));
        T w = y*ab+ad;
        T x = -(y*y*ac+y*ae+af);
        if (x == 0 && w == 0) x = w = 1;
        solutions.push_back(vgl_homg_point_2d<T>(x,y*w,w));
      }
    return solutions;
  }

  coef /= -coef(0);
  double data[]={coef(1),coef(2),coef(3),coef(4), 1,0,0,0, 0,1,0,0, 0,0,1,0};
  vnl_matrix<double> M(data,4,4);
  vnl_real_eigensystem eig(M);

  vnl_diag_matrix<std::complex<double> >  polysolutions = eig.D;

  // Ignore imaginary solutions: place just the real solutions in yvals:
  std::list<T> yvals;
  for (int i=0;i<4;++i)
    if (std::abs(std::imag(polysolutions(i))) < 1e-7)
      yvals.push_back((T)std::real(polysolutions(i)));

  // These are only the solutions of the fourth order equation.
  // The solutions of the intersection of the two conics are:

  std::list<vgl_homg_point_2d<T> > solutions;

  // Special case: two or more y values of intersection points are identical:
  typename std::list<T>::const_iterator it = yvals.begin();
#if 0
  if (yvals(0) == yvals(1) || yvals(1) == yvals(2) || yvals(2) == yvals(3))
  {
  }
#endif

  for (it = yvals.begin(); it != yvals.end(); ++it) {
    T y = *it;
    T w = y*ab+ad;
    T x = -(y*y*ac+y*ae+af);
    if (x == 0 && w == 0) x = w = 1;
    solutions.push_back(vgl_homg_point_2d<T>(x,y*w,w));
  }
  return solutions;
}

//:
// This function is called from within intersection(vgl_conic<T>,vgl_homg_line_2d<T>).
// The conic passed to this function MUST NOT be degenerate!
template <class T>
std::list<vgl_homg_point_2d<T> >
vgl_homg_operators_2d<T>::do_intersect(vgl_conic<T> const& q,
                                       vgl_homg_line_2d<T> const& l)
{
  T A=q.a(), B=q.b(), C=q.c(), D=q.d(), E=q.e(), F=q.f();
  T a=l.a(), b=l.b(), c=l.c();

  if (a==0 && b==0) { // line at infinity
    if (A==0)
      return std::list<vgl_homg_point_2d<T> >(2,vgl_homg_point_2d<T>(1,0,0));
    T d = B*B-4*A*C; // discriminant
    if (d < 0) return std::list<vgl_homg_point_2d<T> >(); // no solutions
    if (d == 0)
      return std::list<vgl_homg_point_2d<T> >(2,vgl_homg_point_2d<T>(-B,2*A,0));
    d = std::sqrt(d);
    std::list<vgl_homg_point_2d<T> > v(1, vgl_homg_point_2d<T>(-B+d, 2*A, 0));
    v.push_back(vgl_homg_point_2d<T>(-B-d, 2*A, 0));
    return v;
  }
  if (a==0) { // write y in terms of w and solve for (x,w)
    T y = -c/b; B = B*y+D;
    T d = B*B-4*A*(C*y*y+E*y+F); // discriminant
    if (d < 0) return std::list<vgl_homg_point_2d<T> >(); // no solutions
    if (d == 0 && A == 0)
      return std::list<vgl_homg_point_2d<T> >(2,vgl_homg_point_2d<T>(1,0,0));
    if (d == 0)
      return std::list<vgl_homg_point_2d<T> >(2,vgl_homg_point_2d<T>(-B,y*2*A,2*A));
    if (A == 0) {
      std::list<vgl_homg_point_2d<T> > v(1, vgl_homg_point_2d<T>(1,0,0));
      v.push_back(vgl_homg_point_2d<T>(C*y*y+E*y+F, -y*B, -B));
      return v;
    }
    d = std::sqrt(d);
    std::list<vgl_homg_point_2d<T> > v(1, vgl_homg_point_2d<T>(-B+d, y*2*A, 2*A));
    v.push_back(vgl_homg_point_2d<T>(-B-d, y*2*A, 2*A));
    return v;
  }
  b /= -a; c /= -a; // now x = b*y+c*w.
  T AA = A*b*b+B*b+C;
  B = 2*A*b*c+B*c+D*b+E;
  T d = B*B-4*AA*(A*c*c+D*c+F); // discriminant
  if (d < 0) return std::list<vgl_homg_point_2d<T> >(); // no solutions
  if (d == 0 && AA == 0)
    return std::list<vgl_homg_point_2d<T> >(2,vgl_homg_point_2d<T>(b,1,0));
  if (d == 0)
    return std::list<vgl_homg_point_2d<T> >(2,vgl_homg_point_2d<T>(c*2*AA-b*B,-B,2*AA));
  if (AA == 0) {
    std::list<vgl_homg_point_2d<T> > v(1, vgl_homg_point_2d<T>(b,1,0));
    v.push_back(vgl_homg_point_2d<T>(b*A*c*c+b*D*c+b*F-c*B, A*c*c+D*c+F, -B));
    return v;
  }
  d = std::sqrt(d);
  std::list<vgl_homg_point_2d<T> > v(1, vgl_homg_point_2d<T>(c*2*AA-b*B+b*d, -B+d, 2*AA));
  v.push_back(vgl_homg_point_2d<T>(c*2*AA-b*B-b*d, -B-d, 2*AA));
  return v;
}

//: Return the (real) intersection points of a conic and a line.
template <class T>
std::list<vgl_homg_point_2d<T> >
vgl_homg_operators_2d<T>::intersection(vgl_conic<T> const& c,
                                       vgl_homg_line_2d<T> const& l)
{
  if (c.type()==vgl_conic<T>::no_type ||
      c.type()==vgl_conic<T>::complex_parallel_lines ||
      c.type()==vgl_conic<T>::complex_intersecting_lines ||
      c.type()==vgl_conic<T>::imaginary_ellipse ||
      c.type()==vgl_conic<T>::imaginary_circle)
    return std::list<vgl_homg_point_2d<T> >(); // empty list
  // let's hope the intersection point of the two complex lines is not on the line..

  if (c.type() == vgl_conic<T>::coincident_lines) {
    vgl_homg_point_2d<T> p = intersection(l, c.components().front());
    std::list<vgl_homg_point_2d<T> > list(2, p); // intersection is *two* coincident points
    return list;
  }

  if (c.type() == vgl_conic<T>::real_intersecting_lines || c.type() == vgl_conic<T>::real_parallel_lines) {
    std::list<vgl_homg_point_2d<T> > list;
    list.push_back(intersection(l, c.components().front()));
    list.push_back(intersection(l, c.components().back()));
    return list;
  }
  return do_intersect(c, l);
}

template <class T>
std::list<vgl_homg_point_2d<T> >
vgl_homg_operators_2d<T>::intersection(vgl_conic<T> const& c1, vgl_conic<T> const& c2)
{
  if ((c1.type()==vgl_conic<T>::complex_parallel_lines ||
       c1.type()==vgl_conic<T>::complex_intersecting_lines)
      && c2.contains(c1.centre()))
    return std::list<vgl_homg_point_2d<T> >(2, c1.centre()); // double intersection point
  if ((c2.type()==vgl_conic<T>::complex_parallel_lines ||
       c2.type()==vgl_conic<T>::complex_intersecting_lines)
      && c1.contains(c2.centre()))
    return std::list<vgl_homg_point_2d<T> >(2, c2.centre()); // double intersection point
  if (c1.type() == vgl_conic<T>::no_type   ||  c2.type() == vgl_conic<T>::no_type ||
      c1.type()==vgl_conic<T>::complex_parallel_lines||c2.type()==vgl_conic<T>::complex_parallel_lines ||
      c1.type()==vgl_conic<T>::complex_intersecting_lines||c2.type()==vgl_conic<T>::complex_intersecting_lines ||
      c1.type() == vgl_conic<T>::imaginary_ellipse|| c2.type() == vgl_conic<T>::imaginary_ellipse ||
      c1.type() == vgl_conic<T>::imaginary_circle || c2.type() == vgl_conic<T>::imaginary_circle)
    return std::list<vgl_homg_point_2d<T> >(); // empty list

  if (c1.type() == vgl_conic<T>::coincident_lines ||
      c1.type() == vgl_conic<T>::real_intersecting_lines ||
      c1.type() == vgl_conic<T>::real_parallel_lines) {
    std::list<vgl_homg_point_2d<T> > l1=intersection(c2,c1.components().front());
    std::list<vgl_homg_point_2d<T> > l2=intersection(c2,c1.components().back());
    l1.insert(l1.end(), l2.begin(), l2.end()); // append l2 to l1
    return l1;
  }

  if (c2.type() == vgl_conic<T>::coincident_lines ||
      c2.type() == vgl_conic<T>::real_intersecting_lines ||
      c2.type() == vgl_conic<T>::real_parallel_lines) {
    std::list<vgl_homg_point_2d<T> > l1=intersection(c1,c2.components().front());
    std::list<vgl_homg_point_2d<T> > l2=intersection(c1,c2.components().back());
    l1.insert(l1.end(), l2.begin(), l2.end()); // append l2 to l1
    return l1;
  }

  return do_intersect(c1, c2);
}

//: Return the (at most) two tangent lines that pass through p and are tangent to the conic.
// For points on the conic, exactly 1 line is returned: the tangent at that point.
// For points inside the conic, an empty list is returned.
// For points outside the conic, there are always two tangents returned.
// Found by intersecting the dual conic with the dual line.
// If the conic is degenerate, an empty list is returned, unless the point is
// on the conic (in which case the component is returned to which it belongs,
// or even both components in the exclusive case that the point is the centre).
template <class T>
std::list<vgl_homg_line_2d<T> >
vgl_homg_operators_2d<T>::tangent_from(vgl_conic<T> const& c,
                                       vgl_homg_point_2d<T> const& p)
{
  if (c.is_degenerate()) {
    if (!c.contains(p)) return std::list<vgl_homg_line_2d<T> >();
    std::list<vgl_homg_line_2d<T> > v = c.components();
    if (c.type() == vgl_conic<T>::coincident_lines || p == c.centre())
      return v;
    if (v.size() > 0 && dot(v.front(),p) == 0)
      return std::list<vgl_homg_line_2d<T> >(1,v.front());
    else if (v.size() > 1 && dot(v.back(),p) == 0)
      return std::list<vgl_homg_line_2d<T> >(1,v.back());
    else
      return std::list<vgl_homg_line_2d<T> >();
  }

  vgl_conic<T> C = c.dual_conic();
  vgl_homg_line_2d<T>  l(p.x(),p.y(),p.w()); // dual line
  std::list<vgl_homg_point_2d<T> > dualpts = intersection(C,l);
  std::list<vgl_homg_line_2d<T> > v;
  typename std::list<vgl_homg_point_2d<T> >::iterator it = dualpts.begin();
  for (; !(it == dualpts.end()); ++it)
    v.push_back(vgl_homg_line_2d<T>((*it).x(), (*it).y(), (*it).w()));
  return v;
}

//: Return the list of common tangent lines of two conics.
// This is done by finding the intersection points of the two dual conics,
// which are the duals of the common tangent lines.
// If one of the conics is degenerate, an empty list is returned.
template <class T>
std::list<vgl_homg_line_2d<T> >
vgl_homg_operators_2d<T>::common_tangents(vgl_conic<T> const& c1,
                                          vgl_conic<T> const& c2)
{
  if ((c1.type() != vgl_conic<T>::parabola && ! c1.is_central())  ||
      (c2.type() != vgl_conic<T>::parabola && ! c2.is_central()))
    return std::list<vgl_homg_line_2d<T> >();//empty list: degenerate conic has no dual

  vgl_conic<T> C1 = c1.dual_conic();
  vgl_conic<T> C2 = c2.dual_conic();
  std::list<vgl_homg_point_2d<T> > dualpts = intersection(C1,C2);
  std::list<vgl_homg_line_2d<T> > v;
  typename std::list<vgl_homg_point_2d<T> >::iterator it = dualpts.begin();
  for (; !(it == dualpts.end()); ++it)
    v.push_back(vgl_homg_line_2d<T>((*it).x(), (*it).y(), (*it).w()));
  return v;
}

//: Return the point on the conic closest to the given point
template <class T>
vgl_homg_point_2d<T>
vgl_homg_operators_2d<T>::closest_point(vgl_conic<T> const& c,
                                        vgl_homg_point_2d<T> const& pt)
{
  // First check if the point is on the curve, since this simplifies things:
  if (c.contains(pt)) return pt;

  // The nearest point must have a polar line which is orthogonal to its
  // connection line with the given point; all points with this property form
  // a certain conic  (actually an orthogonal hyperbola) :
  std::list<vgl_homg_point_2d<T> > candidates; // all intersection points
  if (pt.w() == 0) // given point is at infinity
  {
    // ==> degenerate hyperbola: line + line at infinity
    vgl_homg_line_2d<T> l(c.a()*pt.y()*2-c.b()*pt.x(),
                         -c.c()*pt.x()*2+c.b()*pt.y(),
                          c.d()*pt.y()  -c.e()*pt.x());
    candidates = intersection(c, l);
    if (candidates.size() == 0)
      return vgl_homg_point_2d<T>(0,0,0); // this cannot happen
    // just return any of the candidates, since distance makes no sense at infinity:
    else return candidates.front();
  }
  else if (c.b()==0 && c.a()==c.c()) // the given conic is a circle
  {
    // ==> degenerate hyperbola: line thru centre & point  +  line at infinity
    vgl_homg_point_2d<T> centre = c.centre();
    if (centre == pt) // in this case, any point of the circle is all right
      centre = vgl_homg_point_2d<T>(1,0,0); // take a horizontal line thru pt
    candidates = intersection(c, vgl_homg_line_2d<T>(centre,pt));
  }
  else // general case:
  {
    vgl_conic<T> conic(pt.w()*c.b(),
                       pt.w()*(c.c()-c.a())*2,
                      -pt.w()*c.b(),
                       pt.y()*c.a()*2-pt.x()*c.b()+pt.w()*c.e(),
                      -pt.x()*c.c()*2+pt.y()*c.b()-pt.w()*c.d(),
                       pt.y()*c.d()  -pt.x()*c.e());
    // Now it suffices to intersect the hyperbola with the given conic:
    candidates = intersection(c, conic);
  }
  if (candidates.size() == 0)
  {
    std::cerr << "Warning: vgl_homg_operators_2d<T>::closest_point: no intersection\n";
    return vgl_homg_point_2d<T>(0,0,0);
  }

  // And find the intersection point closest to the given location:
  typename std::list<vgl_homg_point_2d<T> >::iterator it = candidates.begin();
  vgl_homg_point_2d<T> p = (*it);
  T dist = vgl_homg_operators_2d<T>::distance_squared(*it,pt);
  for (++it; it != candidates.end(); ++it) {
    if ((*it).w() == 0) continue;
    T d = vgl_homg_operators_2d<T>::distance_squared(*it,pt);
    if (d < dist) { p = (*it); dist = d; }
  }
  return p;
}

//: Return the point on the conic closest to the given point.
//  Still return a homogeneous point, even if the argument is non-homogeneous.
template <class T>
vgl_homg_point_2d<T>
vgl_homg_operators_2d<T>::closest_point(vgl_conic<T> const& c,
                                        vgl_point_2d<T> const& pt)
{
  return closest_point(c,vgl_homg_point_2d<T>(pt));
}

//: Compute the bounding box of an ellipse
// This is done by finding the tangent lines to the ellipse from the two points
// at infinity (1,0,0) and (0,1,0).
template <class T>
vgl_box_2d<T>
vgl_homg_operators_2d<T>::compute_bounding_box(vgl_conic<T> const& c)
{
  // Only ellipses have a finite bounding box:

  if (c.real_type() == "complex intersecting lines") { // a single point:
    vgl_homg_point_2d<T> pt = c.centre();
    return vgl_box_2d<T>(vgl_point_2d<T>(pt), vgl_point_2d<T>(pt));
  }

  if (c.real_type() == "invalid conic" ||
      c.real_type() == "imaginary ellipse" ||
      c.real_type() == "imaginary circle" ||
      c.real_type() == "complex parallel lines")
    return vgl_box_2d<T>(); // empty box

  if (c.real_type() == "real parallel lines" ||
      c.real_type() == "coincident lines")
  {
    // find out if these lines happen to be horizontal or vertical
    std::list<vgl_homg_line_2d<T> > l = c.components();
    if (l.front().a() == 0) // horizontal lines
      return vgl_box_2d<T>(vgl_point_2d<T>(T(1e33),-l.front().c()/l.front().b()),
                           vgl_point_2d<T>(T(-1e33),-l.back().c()/l.back().b()));
    if (l.front().b() == 0) // vertical lines
      return vgl_box_2d<T>(vgl_point_2d<T>(-l.front().c()/l.front().b(),T(1e33)),
                           vgl_point_2d<T>(-l.back().c()/l.back().b(),T(-1e33)));
    // if not, go to the general case, i.e., return "everything".
  }

  if (c.real_type() != "real ellipse" && c.real_type() != "real circle")
    return vgl_box_2d<T>(T(-1e33), T(1e33), T(-1e33), T(1e33)); // everything

  // Now for the ellipses:

  vgl_homg_point_2d<T> px (1,0,0); // point at infinity of the X axis
  vgl_homg_point_2d<T> py (0,1,0); // point at infinity of the Y axis

  std::list<vgl_homg_line_2d<T> > lx = vgl_homg_operators_2d<T>::tangent_from(c, px);
  std::list<vgl_homg_line_2d<T> > ly = vgl_homg_operators_2d<T>::tangent_from(c, py);

  T y1 = - lx.front().c() / lx.front().b(); // lx are two horizontal lines
  T y2 = - lx.back().c() / lx.back().b();
  if (y1 > y2) { T t = y1; y1 = y2; y2 = t; }
  T x1 = - ly.front().c() / ly.front().a(); // ly are two vertical lines
  T x2 = - ly.back().c() / ly.back().a();
  if (x1 > x2) { T t = x1; x1 = x2; x2 = t; }

  return vgl_box_2d<T>(x1, x2, y1, y2);
}

//: Transform a point through a 3x3 projective transformation matrix
// \relatesalso vgl_homg_point_2d
template <class T>
vgl_homg_point_2d<T> operator*(vnl_matrix_fixed<T,3,3> const& m,
                               vgl_homg_point_2d<T> const& p)
{
  return vgl_homg_point_2d<T>(m(0,0)*p.x()+m(0,1)*p.y()+m(0,2)*p.w(),
                              m(1,0)*p.x()+m(1,1)*p.y()+m(1,2)*p.w(),
                              m(2,0)*p.x()+m(2,1)*p.y()+m(2,2)*p.w());
}

//: Transform a line through a 3x3 projective transformation matrix
// \relatesalso vgl_homg_line_2d
template <class T>
vgl_homg_line_2d<T> operator*(vnl_matrix_fixed<T,3,3> const& m,
                              vgl_homg_line_2d<T> const& l)
{
  return vgl_homg_line_2d<T>(m(0,0)*l.a()+m(0,1)*l.b()+m(0,2)*l.c(),
                             m(1,0)*l.a()+m(1,1)*l.b()+m(1,2)*l.c(),
                             m(2,0)*l.a()+m(2,1)*l.b()+m(2,2)*l.c());
}

//: Return the point on the line closest to the given point
template <class T>
vgl_homg_point_2d<T> vgl_homg_operators_2d<T>::closest_point(vgl_homg_line_2d<T> const& l,
                                                             vgl_homg_point_2d<T> const& p)
{
  // Return p itself, if p lies on l:
  if (l.a()*p.x()+l.b()*p.y()+l.c()*p.w() == 0) return p;
  // Otherwise, make sure that both l and p are not at infinity:
  assert(!l.ideal()); // should not be the line at infinity
  // Line othogonal to l and through p is  bx-ay+d=0, with d = (a*py-b*px)/pw.
  vgl_homg_line_2d<T> o(l.b()*p.w(), -l.a()*p.w(), l.a()*p.y()-l.b()*p.x());
  // Finally return the intersection point of l with this orthogonal line:
  return vgl_homg_operators_2d<T>::intersection(l,o);
}

#undef VGL_HOMG_OPERATORS_2D_INSTANTIATE
#define VGL_HOMG_OPERATORS_2D_INSTANTIATE(T) \
template class vgl_homg_operators_2d<T >; \
template vgl_homg_point_2d<T > operator*(vnl_matrix_fixed<T,3,3> const& m, vgl_homg_point_2d<T > const& p); \
template vgl_homg_line_2d<T > operator*(vnl_matrix_fixed<T,3,3> const& m, vgl_homg_line_2d<T > const& p)

#endif // vgl_homg_operators_2d_hxx_