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Coin / src / base / SbDPLine.cpp

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/*!
  \class SbDPLine SbDPLinear.h Inventor/SbDPLinear.h
  \brief The SbDPLine class represents a line in 3D space.
  \ingroup base

  SbDPLine is used by many other classes in Coin.  It provides a way of
  specifying a directed line (also known as a ray) through a specified
  point (origin) and a direction in 3D space. Note that the line is
  infinite in both directions from its definition point.

  \COIN_CLASS_EXTENSION

  \sa SbVec3d
  \since Coin 2.0
*/

// FIXME: we should _really_ have double-precision classes compatible
// with those in TGS' API, for several good reasons. So either rename
// this, make a typedef (if that is sufficient), or write a "wrapper
// class" around this with inline functions, using with TGS' name for
// it. 20020225 mortene.

#include <cassert>
#include <Inventor/SbDPLine.h>
#if COIN_DEBUG
#include <Inventor/errors/SoDebugError.h>
#endif // COIN_DEBUG

/*!
  The empty constructor does nothing. The line will be uninitialized until
  the first assignment or setValue() call.
*/
SbDPLine::SbDPLine(void)
{
}

/*!
  Constructor with \a p0 specifying the line start point and \a p1 the line
  end point. \a p0 should not be the same as \a p1, as this will lead to
  having a null vector as the direction vector, which would cause division
  by zero problems in some of the other methods on this class.
*/
SbDPLine::SbDPLine(const SbVec3d& p0, const SbVec3d& p1)
{
  this->setValue(p0, p1);
}

/*!
  Set new position and direction of the line by specifying line start
  point and end point. \a p0 should not be the same as \a p1, as this
  will lead to having a null vector as the direction vector, which
  would cause division by zero problems in some of the other methods
  on this class.
*/
void
SbDPLine::setValue(const SbVec3d& p0, const SbVec3d& p1)
{
  this->pos = p0;
  this->dir = p1 - p0;

#if COIN_DEBUG
  if(!(p0 != p1)) {
    SoDebugError::postWarning("SbDPLine::setValue",
                              "The two points defining the line is "
                              "equal => line is invalid.");
    return;
  }
#endif // COIN_DEBUG

  // we test for a null vector above, just normalize
  (void) this->dir.normalize();
}

/*!
  Set position and direction

  \since Coin 4.0
 */
void
SbDPLine::setPosDir(const SbVec3d & position, const SbVec3d & direction)
{
  this->pos = position;
  this->dir = direction;
}

/*!
  Returns the two closest points on the lines. If the lines are
  parallel, all points are equally close and we return \c FALSE. If
  the lines are not parallel, the point positions will be stored in \a
  ptOnThis and \a ptOnLine2, and we'll return \c TRUE.

  \sa getClosestPoint().
*/
SbBool
SbDPLine::getClosestPoints(const SbDPLine& line2,
                         SbVec3d& ptOnThis, SbVec3d& ptOnLine2) const
{
#if 1
  // new optimized version based on formulas from from Boyko Bantchev

  // p1 = point on line 1
  // p2 = point on line 2
  // d1 = line 1 direction
  // d2 = line 2 direction
  // q1 = closest point on line 1
  // q2 = closest point on line 2

  // The solution (q1 and q2) must be on their respective 
  // lines:
  //
  // q1 = p1 + t1 * d1                               (0)
  // q2 = p2 + t2 * d2
  //
  // we set u = p2 - p1, and get:
  //
  // q2 - q1 = u + t2*d2 - t1*d1                     (1)
  //
  // the solution line q2 - q1 is orthogonal to d1 and d2 
  // (or a null vector if the lines intersect), which yields:
  //
  // (u + t2*d2 - t1*d1) � d1 = 0                    (2)
  // (u + t2*d2 - t1*d1) � d2 = 0
  //
  // we know |d1| and |d2| == 1, and set d1 � d2 = t
  //
  // t1 - t*t2 = u � d1
  // t*t1 - t2 = u � d2
  //
  // Solve for t1, and find q1 using (0):
  //
  // t1 = (u�d1 - t * (u�d2))/ (1 - t^2)
  //
  // just find q2 by using line2.getClosestPoint(q1)

  SbVec3d p1 = this->pos;
  SbVec3d p2 = line2.pos;
  
  SbVec3d d1 = this->dir;
  SbVec3d d2 = line2.dir;

  SbVec3d u = p2-p1;
  double t = d1.dot(d2);

  const double eps = 1.0e-08;
  if (t < -1.0f + eps || t > 1.0f-eps) {
    // lines are parallel
    return FALSE;
  }
  t = (u.dot(d1) - t * u.dot(d2)) / (1-t*t);
  ptOnThis = p1 + t * d1;
  ptOnLine2 = line2.getClosestPoint(ptOnThis);
  return TRUE;

#else // end of new, optimized version

#if COIN_DEBUG
  if(!(this->getDirection().length() != 0.0))
    SoDebugError::postWarning("SbDPLine::getClosestPoints",
                              "This line has no direction (zero vector).");
  if(!(line2.getDirection().length() != 0.0))
    SoDebugError::postWarning("SbDPLine::getClosestPoints",
                              "argument line has no direction (zero vector).");
#endif // COIN_DEBUG

  // Check if the lines are parallel.
  // FIXME: should probably use equals() here.
  if(line2.dir == this->dir) return FALSE;
  else if(line2.dir == -this->dir) return FALSE;


  // From the discussion on getClosestPoint(), we know that the point
  // we wish to find on a line can be expressed as:
  //
  //                  (Q1-P0)�D0
  //   Q0 = P0 + D0 * ----------
  //                     |D0|
  //
  // ...where P0 is a point on the first line, D0 is the direction
  // vector and Q1 is the "closest point" on the other line. From this
  // we get two equations with two unknowns. By substituting for
  // Q1 we get a new equation with a single unknown, Q0:
  //
  //                   (         (Q0 - P1)�D1    )
  //                   (P1 + D1 * ------------ - P0) � D0
  //                   (             |D1|        )
  //   Q0 = P0 + D0 * ------------------------------------
  //                                |D0|
  //
  // Which obviously is bloody hard (perhaps impossible?) to solve
  // analytically. Damn. Back to the pen and pencil stuff.
  //
  // Ok, new try. Since we're looking for the minimum distance between the
  // two lines, we should be able to solve it by expressing the distance
  // between the points we want to find as a parametrized function and
  // take the derivative:
  //
  //   f(t0, t1) = |Q1 - Q0| = |P1+D1*t1 - (P0+D0*t0)|
  //
  //                         (t1*D1 - P0)�D0
  // t0 can be expressed as  ---------------  which gives us
  //                               |D0|
  //
  //   f(t) = |P1 + D1*t - P0 - D0N * ((t*D1 - P0)�D0)|, t = t1
  //                                                     D0N = D0 normalized
  //                               _____________
  // ..which is eual to   f(t) = \/޲ + ߲ + �  , where �, �, and �
  // is the full expression above with the x, y, and z components
  // of the vectors.
  //
  // Since we're looking for the minimum value of the function, we can just
  // ignore the square root. We'll do the next parts of the math on a
  // general components case, since it's the same for the x, y and z parts.
  //
  // Expanding any of the �, �, or � expressions, we get this:
  //   (P1[i] - D1[i]*t - P0[i] - D0N[i]*D0[x]*D1[x]*t + D0N[i]*D0[x]*P0[x]
  //      - D0N[i]*D0[y]*D1[y]*t + D0N[i]*D0[y]*P0[y] - D0N[i]*D0[z]*D1[z]*t
  //      + D0N[i]*D0[z]*P0[z])� ,
  // where i=[x|y|z].
  //
  // Deriving this by using the chain rule (i.e. g(t)� = 2*g(t)*g'(t)), we'll
  // get this equation for finding the t yielding the minimum distance
  // between two points Q0 and Q1 on the lines:
  //
  //      -(cx*dx+cy*dy+cz*dz)
  //  t = --------------------
  //        dx� + dy� + dz�
  //
  //  di = D1[i] - D0N[i] * (D0[x]*D1[x] + D0[y]*D1[y] + D0[z]*D1[z])
  // and
  //  ci = P1[i] - P0[i] + D0N[i] * (D0[x]*P0[x] + D0[y]*P0[y] + D0[z]*P0[z])
  // where i=[x|y|z].
  //
  // Now we'll substitute t back in for t1 in   Q1 = P1 + D1*t1, which'll
  // also let us find Q0 by an invocation of getClosestPoint().
  //
  // That's it. I can't believe this took me 4 hours to complete. Code worked
  // on the first run, though. :-)
  //                                                           19980815 mortene.

  SbVec3d P0 = this->pos;
  SbVec3d P1 = line2.pos;
  SbVec3d D0 = this->dir;
  SbVec3d D1 = line2.dir;
  SbVec3d D0N = D0;
  
  // we warn about lines with no direction above, just normalize
  (void) D0N.normalize();

  double c[3], d[3];

  for (int i=0; i < 3; i++) {
    d[i] = (D1[i] - D0N[i]*(D0[0]*D1[0] + D0[1]*D1[1] + D0[2]*D1[2]));
    c[i] = (P1[i] - P0[i] + D0N[i]*(D0[0]*P0[0] + D0[1]*P0[1] + D0[2]*P0[2]));
  }

  double t = -(c[0]*d[0]+c[1]*d[1]+c[2]*d[2]) / (d[0]*d[0]+d[1]*d[1]+d[2]*d[2]);

  ptOnLine2 = line2.pos + line2.dir * t;
  ptOnThis = this->getClosestPoint(ptOnLine2);

  return TRUE;
#endif // old version
}

/*!
  Returns the point on the line which is closest to \a point.

  \sa getClosestPoints().
*/
SbVec3d
SbDPLine::getClosestPoint(const SbVec3d& point) const
{
  //
  //             Q      D
  //    SP x-----x------->
  //        \    |
  //         \   |
  //          \  |
  //           \ |
  //            \|
  //             x P
  //
  // P = argument point, SP = line starting point, D = line direction,
  // Q = point to find.
  //
  // Solved by:
  //                         a�b
  //             comp_b(a) = ---   , a = P-SP, b = D, comp_b(a) = |Q-SP|
  //                         |b|
  //
  //  ==>   Q = SP + comp_b(a)*D
  //                                        19980815 mortene.

  // No use warning about a zero length line here. The user will get a
  // warning when the line is constructed. Also, we don't need to
  // account for the length of the direction vector, since this->dir
  // is always normalized (or a null-vector). The result will actually
  // be correct when the line has zero length, since the line starting
  // point will then be the closest point. pederb, 2005-02-24
  return this->pos + this->dir * (point - this->pos).dot(this->dir);
}

/*!
  Return a vector representing a point on the line.
 */
const SbVec3d&
SbDPLine::getPosition(void) const
{
  return this->pos;
}

/*!
  Return a vector representing the direction of the line. The direction
  vector will always be normalized.
 */
const SbVec3d&
SbDPLine::getDirection(void) const
{
  return this->dir;
}

/*!
  Dump the state of this object to the \a file stream. Only works in
  debug version of library, method does nothing in an optimized compile.
 */
void
SbDPLine::print(FILE * fp) const
{
#if COIN_DEBUG
  fprintf( fp, "p: " );
  this->getPosition().print(fp);
  fprintf( fp, "d: " );
  this->getDirection().print(fp);
#endif // COIN_DEBUG
}