mathlib.h

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#ifndef MATHLIB_H
#define MATHLIB_H

//  Rewrite this file.
//

#include <cassert>
#include <iostream>
#include <string>
#include <vector>   
using namespace std;

#include <point.h>
#include <print.h>
#include <typedefs.h>
#include <zero.h>

#define PI 3.1415926535897932384626433


template< typename T >
void unitbound(T & val)
{
  if (val<(T)0)
  {
    val = (T)0;
    return;
  }

  if (val>(T)1.0)
    val = (T)1.0;
}

  

void integersetdiff
(
  vector<uint> & vecnot,
  vector<uint> & vec,
  uintc N
);

void mathcombination
(
  double & result,
  uintc n,
  uintc k
);



//
// arcos is computed by subtracting arcsin from Pi/2.
//

/*
template<class T>
class arcsin
{
public:

  // y = arcsin(x) by k iterations of arcsin power series.
  T& operator()(T& y, T const & x, uintc k);
//   Provide an error bound for iteration aswell <TODO>

};
*/

class crossproduct
{
public:

  template<class T>
  static void eval(T& w, T const & a, T const & b) 
  {
    w[0] = a[1]*b[2]-b[1]*a[2];
    w[1] = a[2]*b[0]-a[0]*b[2];
    w[2] = a[0]*b[1]-a[1]*b[0];
  }

  template<class T>
  static void evalxyz(T& w, T const & a, T const & b) 
  {
    w.x = a.y*b.z-b.y*a.z;
    w.y = a.z*b.x-a.x*b.z;
    w.z = a.x*b.y-a.y*b.x;
  }

};

class intervalintersection
{
public:

  template<class T>
  static boolc unordered
  (
    T const a0, 
    T const a1,
    T const b0,
    T const b1
  );

  template<class T>
  static boolc unordered
  (
    T & c0, 
    T & c1,
    T const a0, 
    T const a1,
    T const b0,
    T const b1
  );

  template<class I>
  static boolc unordered( I const * i0, I const * i1 )
    { return unordered(i0[0],i0[1],i1[0],i1[1]); }

  template<class I>
  static boolc unorderedD2( I const * box0, I const * box1 )
  {
    if (unordered(box0,box1)==false)
      return false;

    return unordered(box0+2,box1+2);
  }

};

template<typename T>
class solverInconsistent
{
public:

  static boolc d1linearequ
  (
    T & x,
    T & y,
    T const a0,
    T const a1,
    T const c
  );

  static boolc d2linearequ
  (
    T & x,
    T & y,
    T & z,
    T const a00,
    T const a01,
    T const a02,
    T const c0,
    T const a10,
    T const a11,
    T const a12,
    T const c1
  );

};


// <TODO> Is this necessary? Doc???
template<class T>
class surfaceplane
{
public:

  void operator()(T & u, T & v, T & temp, T const & w) const;

};


// <TODO> Find out what this is and fix or delete.
template< typename T >
class convexity
{
public:

  // x-axis
  T a;
  T b;
  // y-axis
  T af;
  T bf;

  // Sets 2 points which define the interval.
  convexity(T a_, T b_, T af_, T bf_)
    : a(a_), b(b_), af(af_), bf(bf_) {}
  convexity(); // costruct in bad state.

  // Given containers of values, verify the equality against the end points.
  void operator () 
  (
    bool& result,
    vector<T> const & x,
    vector<T> const & xf
  );

  // Samples n points between a and b, testing convexity equality.
  template< typename F >  // F is the function.
  void operator ()
  ( 
    bool& result,     // Is the data set convex?
    F f,              // function
    uint n,   // number of sample points
    T const a_,       // x value bound lower
    T const b_        // x value bound upper
  );

};


// <TODO> Find out if this is necessary.
/*
\brief Orthogonal projection

Client supplies orthogonal vectors v[i] and inner product.
 Calling operator solves a[i] for c.
*/
template
< 
 typename T,       // Data type
 typename V,       // vector
 typename IP,      // Inner product functional object: ()(T&,V&,V&)
 uint Dim  // Dimension 
>
class orthoproj
{
public:

  T a[Dim];
  V v[Dim];  // Orthogonal vectors
  IP p;

  void operator()(V const & c);

};


void tetrahedronvolume
( 
  double & vol, 
  point3<double> const & p0,
  point3<double> const & p1,
  point3<double> const & p2,
  point3<double> const & p3
);


void polygonconvexarea
(
  double & area,
  vector< point2<double> > const & v
);


//<TODO> Do something with the trianglearea functions.
//       Either add to triangle class or creat an area class.

void trianglearea
(
  double & f,
  doublec x0,
  doublec y0,
  doublec x1,
  doublec y1,
  doublec x2,
  doublec y2
);

void trianglearea
(
  double & f,
  point3<double> const & p0,
  point3<double> const & p1,
  point3<double> const & p2
);


class determinant
{
public:

  template< typename T >
  static T const calcD2
  ( 
    T const a00, 
    T const a01, 
    T const a10, 
    T const a11
  )
    { return a00*a11-a01*a10; }

  template< typename T >
  static T const calcD3
  (
    T const a00, 
    T const a01, 
    T const a02, 
    T const a10, 
    T const a11,
    T const a12,
    T const a20, 
    T const a21,
    T const a22
  )
    { return 
      a00*(a11*a22-a12*a21) - 
      a01*(a10*a22-a12*a20) +
      a02*(a10*a21-a11*a20); }
};




template< typename T >
class solver
{
public:

  template < typename W >
  static boolc d2linearequ
  (  
    W & x,
    W const & a0,  
    W const & a1,  
    W const & c
  )
  {
    T det = a0.x*a1.y - a1.x*a0.y;
    if (det*det<=zero<T>::val)
      return false;

    T detInv = (T)1.0/det;

    x.x = (c.x*a1.y - c.y*a1.x)*detInv;
    x.y = (c.y*a0.x - c.x*a0.y)*detInv;

    return true;
  }

  static boolc d2linearequ
  (
    T & x,
    T & y,
    T const a00,
    T const a01,
    T const a10,
    T const a11,
    T const c0,
    T const c1
  );

};



// <TODO> Move the d2/d3 matsolve into solver.


//  Solve linear equations for x0,x1.
//
//  [  a00  a01 ] [ x0 ] = [ c0 ] 
//     a10  a11     x1       c1
//  Returns false if determinant is equal to zero.
//
// All the vectors are column vectors.
template < typename T >
boolc d2matsolve
(  
  T & x,
  T const & a0,  
  T const & a1,  
  T const & c
);

template < typename T >
boolc d2matsolve3D
(  
  T & x,
  T const & a0,  
  T const & a1,  
  T const & c
);


//  Solve linear equations for x0,x1.
//
//  [  a00  a01 ] [ x0 ] = [ c0 ] 
//     a10  a11     x1       c1
//  Returns false if determinant is equal to zero.
boolc d3matsolve
(
  double & x0,
  double & x1,
  double & x2,
  doublec a00,
  doublec a01,
  doublec a02,
  doublec a10,
  doublec a11,
  doublec a12,
  doublec a20,
  doublec a21,
  doublec a22,
  doublec c0,
  doublec c1,
  doublec c2
);

boolc d3matsolve
(
  point3<double> & x,
  point3<double> const & a0,  // Column vector.
  point3<double> const & a1,  // Column vector.
  point3<double> const & a2,  // Column vector.
  point3<double> const & c
); 



//<TODO> Integrate lineintersection into line class.

//  Solve the intersection of two lines (a,b) and (c,d)
//  a + t0(b-a) = c + t1(d-c)
template < typename T, typename D >
boolc lineintersection
(
  D & t0,
  D & t1,
  T const & a,
  T const & b,
  T const & c,
  T const & d
);

boolc solvequadratic
(
  double & t0,
  double & t1,
  doublec a,
  doublec b,
  doublec c
);




/*
  \brief Rotate a 2D point about the origin.

  The point is rotated in an anticlockwise direction.
*/
class transrotate2D
{
  point2<double> r1;
  point2<double> r2;
public:

  transrotate2D(doublec theta);

  void eval(point2<double> & p);

  void eval(point2<double> & p, point2<double> const & shift);

};


void matrixmult
( 
  point3<double> & y, 
  doublec * m, 
  point3<double> const & x 
);

void matrixmult
( 
  point3<double> & y,
  point3<double> const & r0,
  point3<double> const & r1,
  point3<double> const & r2,
  point3<double> const & x
);


//<TODO> *  Integrate teh lineSegments into the line class.
//       *  Remove zero and use zero<T>::test instead.
  
boolc lineSegmentIntersection
( 
  point2<double> const & p1,
  point2<double> const & p2,
  point2<double> const & q1,
  point2<double> const & q2,
  doublec zero
);

//<TODO> Remove zero.
boolc lineIntersection
(
  double & tp,
  double & tq,
  point2<double> const & p1,
  point2<double> const & p2,
  point2<double> const & q1,
  point2<double> const & q2,
  doublec zero
);


boolc lineSegmentIntersection
(
  double & tp,
  double & tq,
  point2<double> const & p1,
  point2<double> const & p2,
  point2<double> const & q1,
  point2<double> const & q2,
  doublec zero
);
  
class circleLine
{
public:

  template< typename PT, typename PD >
  static boolc intersection2D
  ( 
    PT & p0, 
    PT & p1, 
    PD const radius, 
    PT const & q0,
    PT const & q1
  );

};



//----------------------------------------------------------
//  Implementation


template< typename PT, typename PD >
boolc circleLine::intersection2D
( 
  PT & p0, 
  PT & p1, 
  PD const radius, 
  PT const & q0,
  PT const & q1
)
{
  assert( zero<PD>::test( (q0-q1).dot()) == false );

  // Hard coded type. No templated solvequadratic.

  PT B(q1-q0);

  doublec a = B.dot();
  doublec b = q0.dot(B)*2.0;
  doublec c = q0.dot()-radius*radius;
  double t0;
  double t1;
  bool res;
  res = solvequadratic(t0,t1,a,b,c);
  if (res==false)
    return false;

  p0 = q0 + B*t0;
  p1 = q0 + B*t1;

  return true;
}

// All column implementation.
template < typename T >
boolc d2matsolve
(  
  T & x,
  T const & a0,  
  T const & a1,  
  T const & c
)
{
  doublec det = a0.x*a1.y - a1.x*a0.y;
  if (det==0.0)
    return false;

  doublec detInv = 1.0/det;

  x.x = (c.x*a1.y - c.y*a1.x)*detInv;
  x.y = (c.y*a0.x - c.x*a0.y)*detInv;

  return true;
}

template < typename T >
boolc d2matsolve3D
(  
  T & x,
  T const & a0,  
  T const & a1,  
  T const & c
)
{
  double det = a0.x*a1.y - a1.x*a0.y;
  // <TODO> make this test dependent on number.
  if (det!=0.0)
  {
    doublec detInv = 1.0/det;

    x.x = (c.x*a1.y - c.y*a1.x)*detInv;
    x.y = (c.y*a0.x - c.x*a0.y)*detInv;

    return true;
  }

  det = a0.x*a1.z - a1.x*a0.z;
  if (det!=0.0)
  {
    doublec detInv = 1.0/det;

    x.x = (c.x*a1.z - c.z*a1.x)*detInv;
    x.y = (c.z*a0.x - c.x*a0.z)*detInv;

    return true;
  }

  det = a0.y*a1.z - a1.y*a0.z;
  if (det!=0.0)
  {
    doublec detInv = 1.0/det;

    x.x = (c.y*a1.z - c.z*a1.y)*detInv;
    x.y = (c.z*a0.y - c.x*a0.z)*detInv;

    return true;
  }

  return false;
}



template < typename T, typename D >
boolc lineintersection
(
  D & t0,
  D & t1,
  T const & a,
  T const & b,
  T const & c,
  T const & d
)
{
  T A(b-a);
  T B(c-d);
  T C(c-a);

  T x;
  bool res = d2matsolve(x,A,B,C);
  t0 = x.x;
  t1 = x.y;

  return res;
}





template
< 
 typename T, 
 typename V, 
 typename IP, 
 uint Dim 
>
void orthoproj<T,V,IP,Dim>::operator()(V const & c)
{
  T t;
  for (uint i=0; i<Dim; ++i)
  {
    p(a[i],c,v[i]);
    p(t,v[i],v[i]);
    a[i] /= t;
  }
}


template<class T> template< typename F >
void convexity<T>::operator ()
( 
  bool& result,  
  F f,
  uint n,
  T const a_,
  T const b_
)
{
  if (n==0)
  {
    result=false;
    return;
  }

  a = a_;
  b = b_;

  f(af,a);
  f(bf,b);

//  cout << "af=" << af << ", bf=" << bf << endl;

  T one(1.0); //Unbelievabley  (j+1)/(n+1)*(b-a) was somewhere converting to int.

/*
// Building containers and forwarding.
  vector<T> x(n);
  vector<T> xf(n);


  for (uint j=0; j<n; ++j)
  {
    x[j] = a+(j+one)/(n+one)*(b-a);
    f(xf[j],x[j]);
    //cout << "x[" << j << "]=" << x[j] << ", xf[" << j << "]=" << xf[j] << endl;
  }
  operator()(result,x,xf);
*/

  T x;
  T xf;
  T lambda;

  for (uint j=0; j<n; ++j)
  {
    x = a+(j+one)/(n+one)*(b-a);
    f(xf,x);
//    cout << "x[" << j << "]=" << x << ", xf[" << j << "]=" << xf << endl;
    lambda = (x-b)/(a-b);
    if ( lambda*af+bf*(one-lambda) < xf )
    {
      result = false;
      return;
    }
  }
 
}


// Call appropriate operator to construct the object.
template<class T>
convexity<T>::convexity()
{}

template<class T>
void convexity<T>::operator()
(
  bool& result,
  vector<T> const & x,
  vector<T> const & xf
)
{
  result = true;

  T one(1.0);

  T lambda;

  for (uint i=0; i<x.size(); ++i)
  {
    lambda = (x[i]-b)/(a-b);
    if ( lambda*af+bf*(one-lambda) < xf[i] )
    {
      result = false;
      return;
    }
  }
}



template<class T>
void surfaceplane<T>::operator()(T & u, T & v, T & temp, T const & w) const
{
  // Right shift.
  temp[1] = w[0];
  temp[2] = w[1];
  temp[0] = w[2];

  crossproduct::eval(u,w,temp);
  crossproduct::eval(v,u,w);

  //crossprod<T>()(u,w,temp);
  //crossprod<T>()(v,u,w);
}

/*
template<class T>
T& arcsin<T>::operator()(T& y, T const & x, uintc k)
{
  T x2 = x * x;
  T a;
  a = 1.0; // Numerator pattern 1  1.3  1.3.5  1.3.5.7 ...
  T b;
  b = 2.0; // Denominator pattern 2  2.4  2.4.6  2.4.6.8 ...
  T xi = x; // Current power of x
  uint c = 2;

  // An improvement would be to calculate from the smallest
  // magnitude to the largest magnitude, minimising errors from
  // largely different magnitudes.

  y = x; // First term approximation.
  for (uint i=1; i<k; ++i)
  {
    xi *= x2;
    y += xi*a/(b*(c+1));
    c += 2;
    a *= (c-1);
    b *= c;
  }

  return y;
}
*/


template<class T>
boolc intervalintersection::unordered
(
  T const a0, 
  T const a1,
  T const b0,
  T const b1
)
{
  // Order the points.
  T a[2];
  if (a0<a1)
  {
    a[0] = a0;
    a[1] = a1;
  }
  else
  {
    a[1] = a0;
    a[0] = a1;
  }
  T b[2];
  if (b0<b1)
  {
    b[0] = b0;
    b[1] = b1;
  }
  else
  {
    b[1] = b0;
    b[0] = b1;
  }

  if (a[1]<b[0])
    return false;

  if (b[1]<a[0])
    return false;

  return true;
}




template<class T>
boolc intervalintersection::unordered
(
  T & c0, 
  T & c1,
  T const a0, 
  T const a1,
  T const b0,
  T const b1
)
{
  // Order the points.
  T a[2];
  if (a0<a1)
  {
    a[0] = a0;
    a[1] = a1;
  }
  else
  {
    a[1] = a0;
    a[0] = a1;
  }
  T b[2];
  if (b0<b1)
  {
    b[0] = b0;
    b[1] = b1;
  }
  else
  {
    b[1] = b0;
    b[0] = b1;
  }

//cout << SHOW(a[0]) << " " << SHOW(a[1]) << endl;
//cout << SHOW(b[0]) << " " << SHOW(b[1]) << endl;

  if (a[1]<b[0])
    return false;

  if (b[1]<a[0])
    return false;

  // Find the intersection interval.
  if (a[0]<b[0])
  {
//cout << "a[0]<b[0]" << endl;
    if (b[1]<a[1])
    {
//cout << "b[1]<a[1]" << endl;
      c0 = b[0];
      c1 = b[1];
    }
    else
    {
      c0 = b[0];
      c1 = a[1];
    }
  }
  else
  {
//cout << "b[0]<a[0]" << endl;
    if (a[1]<b[1])
    {
      c0 = a[0];
      c1 = a[1];
    }
    else
    {
      c0 = a[0];
      c1 = b[1];
    }
  }

  return true;
}


template<typename T>
boolc solverInconsistent<T>::d1linearequ
(
  T & x,
  T & y,
  T const a0,
  T const a1,
  T const c
)
{
  if (zero<T>::test(a0))
  {
    x = 0;
    if (zero<T>::test(a1))
    {
      y = 0;
      return (c==0);
    }
    y = c/a1;
    return true;
  }

  if (zero<T>::test(a1))
  {
    y = 0;
    x = c/a0;
    return true;
  }

  x=1.0;
  y = (c-a0)/a1;
  return true;
}


template<typename T>
boolc solverInconsistent<T>::d2linearequ
(
  T & x,
  T & y,
  T & z,
  T const a00,
  T const a01,
  T const a02,
  T const c0,
  T const a10,
  T const a11,
  T const a12,
  T const c1
)
{
  if (zero<T>::test(a00))
  {
    if (zero<T>::test(a10))
    {
      x=0;
      return solver<T>::d2linearequ(y,z,a01,a02,a10,a11,c0,c1);
    }

    return d2linearequ(x,y,z,a10,a11,a12,c1,a00,a01,a02,c0);
  }

  T e[3];
  e[0] = a11-a10*a01/a00; 
  e[1] = a12-a10*a02/a00;
  e[2] = c1-a10*c0/a00;

  bool res = d1linearequ(y,z,e[0],e[1],e[2]);
  if (res==false)
    return false;

  x = (c0-a01*y-a02*z)/a00;

  return true;
}

  


template< typename T >
boolc solver<T>::d2linearequ
(
  T & x,
  T & y,
  T const a00,
  T const a01,
  T const a10,
  T const a11,
  T const c0,
  T const c1
)
{
  T det = a00*a11-a01*a10;
  if (det*det<=zero<T>::val)
    return false;

  T detInv = (T)1.0/det;
   
  x = (c0*a11 - c1*a01)*detInv;
  y = (c1*a00 - c0*a10)*detInv;

  return true;
}



#endif