curves.h

CGAL is a collaborative effort of several sites in Europe and Israel. The goal is to make the most i

  //	and returns the new (true) y-degree;
  //	It returns -2 if this is a no-op

  int contract();

  // Self-assignment
  BiPoly<NT> & operator=( const BiPoly<NT>& P);

  // Self-addition
  BiPoly<NT> & BiPoly<NT>::operator+=( BiPoly<NT>& P);
   
  // Self-subtraction
  BiPoly<NT> & BiPoly<NT>::operator-=( BiPoly<NT>& P);

  // Self-multiplication
  BiPoly<NT> & BiPoly<NT>::operator*=( BiPoly<NT>& P);
  
  // Multiply by a polynomial in X
  BiPoly<NT> & mulXpoly( Polynomial<NT> & p);

  //Multiply by a constant
  BiPoly<NT> & mulScalar( NT & c);

  // mulYpower: Multiply by Y^i (COULD be a divide if i<0)
  BiPoly<NT> & mulYpower(int s);
  
  // Divide by a polynomial in X.
  // We replace the coeffX[i] by the pseudoQuotient(coeffX[i], P)
  BiPoly<NT> & divXpoly( Polynomial<NT> & p);
  
  //Using the standard definition of pseudRemainder operation.
  //	--No optimization!
  BiPoly<NT>  pseudoRemainderY (BiPoly<NT> & Q);

  //Partial Differentiation
  //Partial Differentiation wrt Y
  BiPoly<NT> & differentiateY();

  BiPoly<NT> & differentiateX();
  BiPoly<NT> & differentiateXY(int m, int n);//m times wrt X and n times wrt Y

  //Represents the bivariate polynomial in (R[X])[Y] as a member
  //of (R[Y])[X].
  //But since our polynomials in X can only have NT coeff's thus
  // to represent the above polynomial we switch X and Y once
  // the conversion has been done.
  //NOTE: This is different from replacing X by Y which was
  //      done in the case of the constructor from a polynomial in X
  //Need to calculate resultant wrt X.
  BiPoly<NT> & convertXpoly();

  //Set Coeffecient to the polynomial passed as a parameter
  bool setCoeff(int i, Polynomial<NT> p);

  void reverse();
  Polynomial<NT> replaceYwithX();

  //Binary-power operator
  BiPoly<NT>& pow(unsigned int n);

  //Returns a Bipoly corresponding to s, which is supposed to
  //contain as place-holders the chars 'x' and 'y'.
  BiPoly<NT> getbipoly(string s);
};//BiPoly Class

  ////////////////////////////////////////////////////////
  // Helper Functions
  ////////////////////////////////////////////////////////
//Experimental version of constructor from strings containing general 
//parentheses


// zeroPinY(P)
//	checks whether a Bi-polynomial is a zero Polynomial
template <class NT>
bool zeroPinY(BiPoly<NT> & P);

// gcd(P,Q)
//   This gcd is based upon the subresultant PRS to avoid
//   exponential coeffecient growth and gcd computations, both of which 
//   are expensive since the coefficients are polynomials

template <class NT>
BiPoly<NT> gcd( BiPoly<NT>& P ,BiPoly<NT>& Q);

// resY(P,Q):
//      Resultant of Bi-Polys P and Q w.r.t. Y.
//      So the resultant is a polynomial in X
template <class NT>
Polynomial<NT>  resY( BiPoly<NT>& P ,BiPoly<NT>& Q);

// resX(P,Q):
//      Resultant of Bi-Polys P and Q w.r.t. X.
//      So the resultant is a polynomial in Y
//	We first convert P, Q to polynomials in X. Then 
// 	call resY and then turn it back into a polynomial in Y
//	QUESTION: is this last switch really necessary???
template <class NT>
BiPoly<NT>  resX( BiPoly<NT>& P ,BiPoly<NT>& Q);

//Equality operator for BiPoly
template <class NT>
bool operator==(const BiPoly<NT>& P, const BiPoly<NT>& Q);

//Addition operator for BiPoly
template <class NT>
 BiPoly<NT> operator+(const BiPoly<NT>& P, const BiPoly<NT>& Q);

//Subtraction operator for BiPoly
template <class NT>
 BiPoly<NT> operator-(const BiPoly<NT>& P, const BiPoly<NT>& Q);

//Multiplication operator for BiPoly
template <class NT>
 BiPoly<NT> operator*(const BiPoly<NT>& P, const BiPoly<NT>& Q);


  ////////////////////////////////////////////////////////
  //Curve Class
  //  	extends BiPoly Class
  ////////////////////////////////////////////////////////

template < class NT >
class Curve : public BiPoly<NT> {
public:
  // Colors for plotting curves

  const static int NumColors=7;
  static double red_comp(int i){
  	static double RED_COMP[] = {0.9, 0.8, 0.7, 0.6, 0.8, 0.8, 0.7};
	return RED_COMP[i % NumColors];
  }
  static double green_comp(int i){
  	static double GREEN_COMP[] = {0.5, 0.9, 0.3, 0.9, 0.7, 0.55, 0.95};
	return GREEN_COMP[i % NumColors];
  }
  static double blue_comp(int i){
  	static double BLUE_COMP[] = {0.8, 0.3, 0.8, 0.5, 0.4, 0.85, 0.35};
	return BLUE_COMP[i % NumColors];
  }

  Curve(); // zero polynomial
  
  //Curve(vp):
  //    construct from a vector of polynomials
  Curve(std::vector<Polynomial<NT> > vp);
  //	  : BiPoly<NT>(vp){
  //}
  
  //Curve(p):
  //	Converts a polynomial p(X) to a BiPoly in one of two ways:
  // 	    (1) if flag is false, the result is Y-p(X) 
  // 	    (2) if flag is true, the result is p(Y) 
  //    The default is (1) because we usually want to plot the
  //        graph of the polynomial p(X)
  Curve(Polynomial<NT> p, bool flag=false);
  //	  : BiPoly<NT>(p, flag){
  //}

  //Curve(deg, d[], C[]):
  //	Takes in a list of list of coefficients.
  //	Each cofficient list represents a polynomial in X
  //
  //  deg - ydeg of the bipoly
  //  d[] - array containing the degrees of each coefficient (i.e., X poly)
  //  C[] - list of coefficients, we use array d to select the
  //      coefficients
  Curve(int deg, int *d, NT *C);
  //	  : BiPoly<NT>(deg, d, C){
  //}

  Curve(const BiPoly<NT> &P);
  //	  : BiPoly<NT>(P){
  //}

  //Curve(n) -- the nominal y-degree is n
  Curve(int n);

  //Creates a curve from a string (no parentheses, no *, no =)
  Curve(const string & s, char myX='x', char myY='y');
  Curve(const char* s, char myX='x', char myY='y');

  /////////////////////////////////////////////////////////////////////////
  // verticalIntersections(x, vecI, aprec=0):
  //    The list vecI is passed an isolating intervals for y's such that (x,y)
  //    lies on the curve.
  //    If aprec is non-zero (!), the intervals have with < 2^{-aprec}.
  //    Return is -2 if curve equation does not depend on Y
  //    	-1 if infinitely roots at x,
  //    	0 if no roots at x
  //    	1 otherwise

  int verticalIntersections(const BigFloat & x, BFVecInterval & vI,
			    int aprec=0);
  
  // TO DO: 
  // 		horizontalIntersections(...)
  
  /////////////////////////////////////////////////////////////////////////
  // plot(eps, x1, y1, x2, y2)
  //
  // 	All parameters have defaults
  //
  //    Gives the points on the curve at resolution "eps".  Currently,
  //    eps is viewed as delta-x step size (but it could change).
  //    The display is done in the rectangale 
  //    defined by [(x1, y1), (x2, y2)].
  //    The output is written into a file in the format specified
  //    by our drawcurve function (see COREPATH/ext/graphics).
  //
  //    Heuristic: the open polygonal lines end when number of roots
  //    changes...
  //
  int  plot( BigFloat eps=0.1, BigFloat x1=-1.0,
	     BigFloat y1=-1.0, BigFloat x2=1.0, BigFloat y2=1.0, int fileNo=1);

// selfIntersections():
//   this should be another member function that lists
//   all the self-intersections of a curve
//
//  template <class NT>
//  void selfIntersections(BFVecInterval &vI){
//  ...
//  }

};// Curve class


  ////////////////////////////////////////////////////////
  // Curve helper functions
  ////////////////////////////////////////////////////////


//Xintersections(C, D, vI):
//  returns the list vI of x-ccordinates of possible intersection points.
//  Assumes that C & D are quasi-monic.(or generally aligned)
template <class NT>
void  Xintersections( Curve<NT>& P ,Curve<NT>& Q, BFVecInterval &vI);

//Yintersections(C, D, vI):
//	similar to Xintersections
template <class NT>
void  Yintersections( Curve<NT>& P ,Curve<NT>& Q, BFVecInterval &vI);

// Display Intervals
template <class NT>
void showIntervals(char* s, BFVecInterval &vI);

// Set Display Parameters
// ...

////////////////////////////////////////////////////////
// IMPLEMENTATIONS ARE FOUND IN Curves.tcc
////////////////////////////////////////////////////////
#include <CORE/poly/Curves.tcc>


CORE_END_NAMESPACE
#endif
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// END
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