geometry.cpp

This is an usefull library for Geometry in Mathmatics

{
  // determine point on line that lies at fraction dFrac of whole line
  // example: this --- 0.25 ---------  p
  // formula: this + dFrac * ( p - this ) = this - dFrac * this + dFrac * p =
  //          ( 1 - dFrac ) * this + dFrac * p
  return ( ( *this ) * ( 1.0 - dFrac ) + ( p * dFrac ) );
}

/*! This method converts a polar representation of a VecPosition into
    a Cartesian representation.

    \param dMag a double representing the polar r-coordinate, i.e. the
    distance from the point to the origin

    \param theta the angle that the polar vector makes with the xy-plane,
    i.e. the polar theta-coordinate

    \param phi the angle that the polar vector makes with the z-axis,
    i.e. the polar phi-coordinate

    \return the result of converting the given polar representation
    into a Cartesian representation thus yielding a Cartesian
    VecPosition */
VecPosition VecPosition::getVecPositionFromPolar( double dMag, AngDeg theta, AngDeg phi )
{
  double z  = dMag * sinDeg(  phi  );
  double XY = dMag * cosDeg(  phi  );
  double x  = XY   * cosDeg( theta );
  double y  = XY   * sinDeg( theta );
  return ( VecPosition( x, y, z ) );
}

/*! This method normalizes an angle. This means that the resulting
    angle lies between -180 and 180 degrees.

    \param angle the angle which must be normalized

    \return the result of normalizing the given angle */
AngDeg VecPosition::normalizeAngle( AngDeg angle )
{
  while( angle > 180.0  ) angle -= 360.0;
  while( angle < -180.0 ) angle += 360.0;

  return ( angle );
}


/*****************************************************************************/
/********************** CLASS GEOMETRY ***************************************/
/*****************************************************************************/

/*! A geometric series is one in which there is a constant ratio between each
    element and the one preceding it. This method determines the
    length of a geometric series given its first element, the sum of the
    elements in the series and the constant ratio between the elements.
    Normally: s = a + ar + ar^2 + ...  + ar^n
    Now: dSum = dFirst + dFirst*dRatio + dFirst*dRatio^2 + .. + dFist*dRatio^n
    \param dFirst first term of the series
    \param dRatio ratio with which the the first term is multiplied
    \param dSum the total sum of all the serie
    \return the length(n in above example) of the series */
double Geometry::getLengthGeomSeries( double dFirst, double dRatio, double dSum )
{
  if( dRatio < 0 )
    cerr << "(Geometry:getLengthGeomSeries): negative ratio" << endl;

  // s = a + ar + ar^2 + .. + ar^n-1 and thus sr = ar + ar^2 + .. + ar^n
  // subtract: sr - s = - a + ar^n) =>  s(1-r)/a + 1 = r^n = temp
  // log r^n / n = n log r / log r = n = length
  double temp = (dSum * ( dRatio - 1 ) / dFirst) + 1;
  if( temp <= 0 )
    return -1.0;
  return log( temp ) / log( dRatio ) ;
}

/*! A geometric series is one in which there is a constant ratio between each
    element and the one preceding it. This method determines the sum of a
    geometric series given its first element, the ratio and the number of steps
    in the series
    Normally: s = a + ar + ar^2 + ...  + ar^n
    Now: dSum = dFirst + dFirst*dRatio + ... + dFirst*dRatio^dSteps
    \param dFirst first term of the series
    \param dRatio ratio with which the the first term is multiplied
    \param dLength the number of steps to be taken into account
    \return the sum of the series */
double Geometry::getSumGeomSeries( double dFirst, double dRatio, double dLength)
{
  // s = a + ar + ar^2 + .. + ar^n-1 and thus sr = ar + ar^2 + .. + ar^n
  // subtract: s - sr = a - ar^n) =>  s = a(1-r^n)/(1-r)
  return dFirst * ( 1 - pow( dRatio, dLength ) ) / ( 1 - dRatio ) ;
}

/*! A geometric series is one in which there is a constant ratio between each
    element and the one preceding it. This method determines the sum of an
    infinite geometric series given its first element and the constant ratio
    between the elements. Note that such an infinite series will only converge
    when 0<r<1.
    Normally: s = a + ar + ar^2 + ar^3 + ....
    Now: dSum = dFirst + dFirst*dRatio + dFirst*dRatio^2...
    \param dFirst first term of the series
    \param dRatio ratio with which the the first term is multiplied
    \return the sum of the series */
double Geometry::getSumInfGeomSeries( double dFirst, double dRatio )
{
  if( dRatio > 1 )
    cerr << "(Geometry:CalcLengthGeomSeries): series does not converge" <<endl;

  // s = a(1-r^n)/(1-r) with n->inf and 0<r<1 => r^n = 0
  return dFirst / ( 1 - dRatio );
}

/*! A geometric series is one in which there is a constant ratio between each
    element and the one preceding it. This method determines the first element
    of a geometric series given its element, the ratio and the number of steps
    in the series
    Normally: s = a + ar + ar^2 + ...  + ar^n
    Now: dSum = dFirst + dFirst*dRatio + ... + dFirst*dRatio^dSteps
    \param dSum sum of the series
    \param dRatio ratio with which the the first term is multiplied
    \param dLength the number of steps to be taken into account
    \return the first element (a) of a serie */
double Geometry::getFirstGeomSeries( double dSum, double dRatio, double dLength)
{
  // s = a + ar + ar^2 + .. + ar^n-1 and thus sr = ar + ar^2 + .. + ar^n
  // subtract: s - sr = a - ar^n) =>  s = a(1-r^n)/(1-r) => a = s*(1-r)/(1-r^n)
  return dSum *  ( 1 - dRatio )/( 1 - pow( dRatio, dLength ) ) ;
}

/*! A geometric series is one in which there is a constant ratio between each
    element and the one preceding it. This method determines the first element
    of an infinite geometric series given its first element and the constant
    ratio between the elements. Note that such an infinite series will only
    converge when 0<r<1.
    Normally: s = a + ar + ar^2 + ar^3 + ....
    Now: dSum = dFirst + dFirst*dRatio + dFirst*dRatio^2...
    \param dSum sum of the series
    \param dRatio ratio with which the the first term is multiplied
    \return the first term of the series */
double Geometry::getFirstInfGeomSeries( double dSum, double dRatio )
{
  if( dRatio > 1 )
    cerr << "(Geometry:getFirstInfGeomSeries):series does not converge" <<endl;

  // s = a(1-r^n)/(1-r) with r->inf and 0<r<1 => r^n = 0 => a = s ( 1 - r)
  return dSum * ( 1 - dRatio );
}

/*! This method performs the abc formula (Pythagoras' Theorem) on the given
    parameters and puts the result in *s1 en *s2. It returns the number of
    found coordinates.
    \param a a parameter in abc formula
    \param b b parameter in abc formula
    \param c c parameter in abc formula
    \param *s1 first result of abc formula
    \param *s2 second result of abc formula
    \return number of found x-coordinates */
int Geometry::abcFormula(double a, double b, double c, double *s1, double *s2)
{
  double dDiscr = b*b - 4*a*c;       // discriminant is b^2 - 4*a*c
  if (fabs(dDiscr) < EPSILON )       // if discriminant = 0
  {
    *s1 = -b / (2 * a);              //  only one solution
    return 1;
  }
  else if (dDiscr < 0)               // if discriminant < 0
    return 0;                        //  no solutions
  else                               // if discriminant > 0
  {
    dDiscr = sqrt(dDiscr);           //  two solutions
    *s1 = (-b + dDiscr ) / (2 * a);
    *s2 = (-b - dDiscr ) / (2 * a);
    return 2;
  }
}

/*****************************************************************************/
/********************* CLASS CIRCLE ******************************************/
/*****************************************************************************/

/*! This is the constructor of a circle.
    \param pos first point that defines the center of circle
    \param dR the radius of the circle
    \return circle with pos as center and radius as radius*/
Circle::Circle( VecPosition pos, double dR )
{
  setCircle( pos, dR );
}

/*! This is the constructor of a circle which initializes a circle with a
    radius of zero. */
Circle::Circle( )
{
  setCircle( VecPosition(-1000.0,-1000.0, -1000.0), 0);
}

/*! This method prints the circle information to the specified output stream
    in the following format: "c: (c_x,c_y,c_z), r: rad" where (c_x,c_y,z_z) denotes
    the center of the circle and rad the radius.
    \param os output stream to which output is written. */
void Circle::show( ostream& os)
{
  os << "c:" << m_posCenter << ", r:" << m_dRadius;
}

/*! This method sets the values of the circle.
    \param pos new center of the circle
    \param dR new radius of the circle
    ( > 0 )
    \return bool indicating whether radius was set */
bool Circle::setCircle( VecPosition pos, double dR )
{
  setCenter( pos );
  return setRadius( dR  );
}
/*! This method sets the radius of the circle.
    \param dR new radius of the circle ( > 0 )
    \return bool indicating whether radius was set */
bool Circle::setRadius( double dR )
{
  if( dR > 0 )
  {
    m_dRadius = dR;
    return true;
  }
  else
  {
    m_dRadius = 0.0;
    return false;
  }
}

/*! This method returns the radius of the circle.
    \return radius of the circle */
double Circle::getRadius() const
{
  return m_dRadius;
}

/*! This method sets the center of the circle.
    \param pos new center of the circle
    \return bool indicating whether center was set */
bool Circle::setCenter( VecPosition pos )
{
  m_posCenter = pos;
  return true;
}

/*! This method returns the center of the circle.
    \return center of the circle */
VecPosition Circle::getCenter()
{
  return m_posCenter;
}

/*! This method returns the circumference of the circle.
    \return circumference of the circle */
double Circle::getCircumference()
{
  return 2.0*M_PI*getRadius();
}

/*! This method returns the area inside the circle.
    \return area inside the circle */
double Circle::getArea()
{
  return M_PI*getRadius()*getRadius();
}

/*! This method returns the volume of a circle.
    \return volume inside the circle */
double Circle::getVolume() const
{
  return M_PI*4/3*getRadius()*getRadius()*getRadius();
}

/*! This method returns a boolean that indicates whether 'pos' is
    located inside the circle.
 
   \param pos position of which should be checked whether it is
   located in the circle
   \param bCircle denotes wheather the check is made for a 
   circle or a sphere

   \return bool indicating whether pos lies inside the circle */
bool Circle::isInside( VecPosition pos, bool bCircle )
{
  if( bCircle )
    pos.setZ( m_posCenter.getZ() );

  return m_posCenter.getDistanceTo( pos ) < getRadius() ;
}

/*! This method returns the two possible intersection points between two
    circles. This method returns the number of solutions that were found.
    \param c circle with which intersection should be found
    \param p1 will be filled with first solution
    \param p2 will be filled with second solution
    \return number of solutions. */
int Circle::getCircleIntersectionPoints( Circle c, VecPosition *p1, VecPosition *p2)
{
    double x0, y0, r0;
    double x1, y1, r1;

    VecPosition tmpCenter = c.getCenter();
    tmpCenter.setZ(m_posCenter.getZ());
    c.setCenter( tmpCenter );

    x0 = getCenter( ).getX();
    y0 = getCenter( ).getY();
    r0 = getRadius( );
    x1 = c.getCenter( ).getX();
    y1 = c.getCenter( ).getY();
    r1 = c.getRadius( );

    double      d, dx, dy, h, a, x, y, p2_x, p2_y;

    // first calculate distance between two centers circles P0 and P1.
    dx = x1 - x0;
    dy = y1 - y0;
    d = sqrt(dx*dx + dy*dy);

    // normalize differences
    dx /= d; dy /= d;

    // a is distance between p0 and point that is the intersection point P2
    // that intersects P0-P1 and the line that crosses the two intersection
    // points P3 and P4.
    // Define two triangles: P0,P2,P3 and P1,P2,P3.
    // with distances a, h, r0 and b, h, r1 with d = a + b
    // We know a^2 + h^2 = r0^2 and b^2 + h^2 = r1^2 which then gives
    // a^2 + r1^2 - b^2 = r0^2 with d = a + b ==> a^2 + r1^2 - (d-a)^2 = r0^2
    // ==> r0^2 + d^2 - r1^2 / 2*d
    a = (r0*r0 + d*d - r1*r1) / (2.0 * d);

    // h is then a^2 + h^2 = r0^2 ==> h = sqrt( r0^2 - a^2 )
    double      arg = r0*r0 - a*a;
    h = (arg > 0.0) ? sqrt(arg) : 0.0;

    // First calculate P2
    p2_x = x0 + a * dx;
    p2_y = y0 + a * dy;

    // and finally the two intersection points
    x =  p2_x - h * dy;
    y =  p2_y + h * dx;
    p1->setVecPosition( x, y );
    x =  p2_x + h * dy;
    y =  p2_y - h * dx;
    p2->setVecPosition( x, y );

    return (arg < 0.0) ? 0 : ((arg == 0.0 ) ? 1 :  2);
}

/*! This method returns the size of the intersection area of two circles.
    \param c circle with which intersection should be determined
    \param bStrict weather to check the circles z- intersection
    \return size of the intersection area. */
double Circle::getIntersectionArea( Circle c, bool bStrict )
{
  VecPosition pos1, pos2, pos3;