opennurbs_math.h

//
// Copyright (c) 1993-2022 Robert McNeel & Associates. All rights reserved.
// OpenNURBS, Rhinoceros, and Rhino3D are registered trademarks of Robert
// McNeel & Associates.
//
// THIS SOFTWARE IS PROVIDED "AS IS" WITHOUT EXPRESS OR IMPLIED WARRANTY.
// ALL IMPLIED WARRANTIES OF FITNESS FOR ANY PARTICULAR PURPOSE AND OF
// MERCHANTABILITY ARE HEREBY DISCLAIMED.
//				
// For complete openNURBS copyright information see <http://www.opennurbs.org>.
//
////////////////////////////////////////////////////////////////

#if !defined(ON_MATH_INC_)
#define ON_MATH_INC_

/*
Returns:
  True (1) if nan compares and arithmetic agree with IEEE-754.
  (nan != nan) is true.
  (nan op x) and (x op nan) is false for op ==, <, >, <=, and >=.
  (nan op x) and (x op nan) is nan for op == +, -, *, and /.
  False (0) otherwise.
*/
ON_DECL
bool ON_PassesNanTest();

class ON_3dVector;
class ON_Interval;
class ON_Line;
class ON_Arc;
class ON_Plane;

/*
Description:
  Class for carefully adding long list of numbers.
*/
class ON_CLASS ON_Sum
{
public:

  /*
  Description:
    Calls ON_Sum::Begin(x)
  */
  void operator=(double x);

  /*
  Description:
    Calls ON_Sum::Plus(x);
  */
  void operator+=(double x);

  /*
  Description:
    Calls ON_Sum::Plus(-x);
  */
  void operator-=(double x);

  /*
  Description:
    Creates a sum that is ready to be used.
  */
  ON_Sum();

  /*
  Description:
    If a sum is being used more than once, call Begin()
    before starting each sum.
  Parameters:
    starting_value - [in] Initial value of sum.
  */
  void Begin( double starting_value = 0.0 );

  /*
  Description:
    Add x to the current sum.
  Parameters:
    x - [in] value to add to the current sum.
  */
  void Plus( double x );

  /*
  Description:
    Add x to the current sum.
  Parameters:
    x - [in] value to add to the current sum.
    dx - [in] symmetric uncertainty in x.
         (true value is in the range x-dx to x+dx
  */
  void Plus( double x, double dx );

  /*
  Description:
    Calculates the total sum.   
  Parameters:
    error_estimate - [out] if not nullptr, the returned value of
       *error_estimate is an estimate of the error in the sum.
  Returns:
    Total of the sum.
  Remarks:
    You can get subtotals by mixing calls to Plus() and Total().
    In delicate sums, some precision may be lost in the final
    total if you call Total() to calculate subtotals.
  */
  double Total( double* error_estimate = nullptr );

  /*
  Returns:
    Number of summands.
  */
  int SummandCount() const;

private:
  enum {
    sum1_max_count=256,
    sum2_max_count=512,
    sum3_max_count=1024
  };
  double m_sum_err;
  double m_pos_sum;     
  double m_neg_sum;  
  
  int m_zero_count; // number of zeros added
  int m_pos_count; // number of positive numbers added
  int m_neg_count; // number of negative numbers added
  
  int m_pos_sum1_count;
  int m_pos_sum2_count;
  int m_pos_sum3_count;
  double m_pos_sum1[sum1_max_count];
  double m_pos_sum2[sum2_max_count];
  double m_pos_sum3[sum3_max_count];
  
  int m_neg_sum1_count;
  int m_neg_sum2_count;
  int m_neg_sum3_count;
  double m_neg_sum1[sum1_max_count];
  double m_neg_sum2[sum2_max_count];
  double m_neg_sum3[sum3_max_count];

  double SortAndSum( int, double* );
};

/*
Description:
  Abstract function with an arbitrary number of parameters
  and values.  ON_Evaluator is used to pass functions to
  local solvers.
*/
class ON_CLASS ON_Evaluator
{
public:

  /*
  Description:
    Construction of the class for a function that takes
    parameter_count input functions and returns
    value_count values.  If the domain is infinite, pass
    a nullptr for the domain[] and periodic[] arrays.  If
    the domain is finite, pass a domain[] array with
    parameter_count increasing intervals.  If one or more of
    the parameters is periodic, pass the fundamental domain
    in the domain[] array and a true in the periodic[] array.
  Parameters:
    parameter_count - [in] >= 1.  Number of input parameters
    value_count - [in] >= 1.  Number of output values.
    domain - [in] If not nullptr, then this is an array
                  of parameter_count increasing intervals
                  that defines the domain of the function.
    periodic - [in] if not nullptr, then this is an array of 
                parameter_count bools where b[i] is true if
                the i-th parameter is periodic.  Valid 
                increasing finite domains must be specified
                when this parameter is not nullptr.
  */
  ON_Evaluator( 
    int parameter_count,
    int value_count,
    const ON_Interval* domain,
    const bool* periodic
    );

  virtual ~ON_Evaluator();
  
  /*
  Description:
    Evaluate the function that takes m_parameter_count parameters
    and returns a m_value_count dimensional point.
  Parameters:
    parameters - [in] array of m_parameter_count evaluation parameters
    values - [out] array of m_value_count function values
    jacobian - [out] If nullptr, simply evaluate the value of the function.
                     If not nullptr, this is the jacobian of the function.
                     jacobian[i][j] = j-th partial of the i-th value
                     0 <= i < m_value_count,
                     0 <= j < m_parameter_count
                     If not nullptr, then all the memory for the
                     jacobian is allocated, you just need to fill
                     in the answers.
  Example:
    If f(u,v) = square of the distance from a fixed point P to a 
    surface evaluated at (u,v), then

          values[0] = (S-P)o(S-P)
          jacobian[0] = ( 2*(Du o (S-P)), 2*(Dv o (S-P)) )

    where S, Du, Dv = surface point and first partials evaluated
    at u=parameters[0], v = parameters[1].

    If the function takes 3 parameters, say (x,y,z), and returns
    two values, say f(x,y,z) and g(z,y,z), then

          values[0] = f(x,y,z)
          values[1] = g(x,y,z)

          jacobian[0] = (DfDx, DfDy, DfDz)
          jacobian[1] = (DgDx, DgDy, DgDz)

    where dfx denotes the first partial of f with respect to x.

  Returns:
    0 = unable to evaluate
    1 = successful evaluation
    2 = found answer, terminate search
  */
  virtual int Evaluate(
       const double* parameters,
       double* values,
       double** jacobian
       ) = 0;

  /*
  Description:
    OPTIONAL ability to evaluate the hessian in the case when 
    m_value_count is one.  If your function has more that
    one value or it is not feasible to evaluate the hessian,
    then do not override this function.  The default implementation
    returns -1.
  Parameters:
    parameters - [in] array of m_parameter_count evaluation parameters
    value - [out] value of the function (one double)
    gradient - [out] The gradient of the function.  This is a vector
                     of length m_parameter_count; gradient[i] is
                     the first partial of the function with respect to
                     the i-th parameter.
    hessian - [out] The hessian of the function. This is an
                    m_parameter_count x m_parameter_count 
                    symmetric matrix: hessian[i][j] is the
                    second partial of the function with respect
                    to the i-th and j-th parameters.  The evaluator
                    is responsible for filling in both the upper
                    and lower triangles.  Since the matrix is
                    symmetrix, you should do something like evaluate
                    the upper triangle and copy the values to the
                    lower triangle.
  Returns:
   -1 = Hessian evaluation not available.
    0 = unable to evaluate
    1 = successful evaluation
    2 = found answer, terminate search
  */
  virtual int EvaluateHessian(
       const double* parameters,
       double* value,
       double* gradient,
       double** hessian
       );
  
  // Number of the function's input parameters. This number
  // is >= 1 and is specified in the constructor.
  const int m_parameter_count;

  // Number of the function's output values. This number
  // is >= 1 and is specified in the constructor.
  const int m_value_count;

  /*
  Description:
    Functions can have finite or infinite domains. Finite domains
    are specified by passing the domain[] array to the constructor
    or filling in the m_domain[] member variable.  If
    m_domain.Count() == m_parameter_count > 0, then the function
    has finite domains.
  Returns:
    True if the domain of the function is finite.
  */
  bool FiniteDomain() const;

  /*
  Description:
    If a function has a periodic parameter, then the m_domain
    interval for that parameter is the fundamental domain and
    the m_bPeriodicParameter bool for that parameter is true.
    A parameter is periodic if, and only if, 
    m_domain.Count() == m_parameter_count, and 
    m_bPeriodicParameter.Count() == m_parameter_count, and
    m_bPeriodicParameter[parameter_index] is true.
  Returns:
    True if the function parameter is periodic.
  */
  bool Periodic(
    int parameter_index
    ) const;

  /*
  Description:
    If a function has a periodic parameter, then the m_domain
    interval for that parameter is the fundamental domain and
    the m_bPeriodicParameter bool for that parameter is true.
    A parameter is periodic if, and only if, 
    m_domain.Count() == m_parameter_count, and 
    m_bPeriodicParameter.Count() == m_parameter_count, and
    m_bPeriodicParameter[parameter_index] is true.
  Returns:
    The domain of the parameter.  If the domain is infinite,
    the (-1.0e300, +1.0e300) is returned.
  */
  ON_Interval Domain(
    int parameter_index
    ) const;


  // If the function has a finite domain or periodic
  // parameters, then m_domain[] is an array of 
  // m_parameter_count finite increasing intervals.
  ON_SimpleArray<ON_Interval> m_domain;

  // If the function has periodic parameters, then 
  // m_bPeriodicParameter[] is an array of m_parameter_count
  // bools.  If m_bPeriodicParameter[i] is true, then
  // the i-th parameter is periodic and m_domain[i] is 
  // the fundamental domain for that parameter.
  ON_SimpleArray<bool> m_bPeriodicParameter;

private:
  ON_Evaluator(); // prohibit default constructor
  ON_Evaluator& operator=(const ON_Evaluator&); // prohibit operator= (can't copy const members)
};

/*
Description:
  Test a double to make sure it is a valid number.
Returns:
  (x > ON_UNSET_VALUE && x < ON_UNSET_POSITIVE_VALUE)
*/
ON_DECL
bool ON_IsValid( double x );

/*
Returns:
  (x > 0.0 && x < ON_UNSET_POSITIVE_VALUE);
*/
ON_DECL
bool ON_IsValidPositiveNumber(double x);

/*
Returns:
  (x > ON_UNSET_VALUE && x < 0.0));
*/
ON_DECL
bool ON_IsValidNegativeNumber(double x);

/*
Returns:
  -1: a < b or a is not a nan and b is a nan
  +1: a > b or a is a nan and b is a not nan
   0: a == b or both a and b are nans
*/
ON_DECL
int ON_CompareDouble( 
  double a, 
  double b 
  );

ON_DECL
int ON_CompareDoubleArray( 
  size_t count,
  const double* a, 
  const double* b
  );

/*
Returns:
  -1: a < b or a is not a nan and b is a nan
  +1: a > b or a is a nan and b is a not nan
   0: a == b or both a and b are nans
*/
ON_DECL
int ON_CompareFloat(
  float a,
  float b
  );

ON_DECL
bool ON_IsValidFloat( float x );

ON_DECL
bool ON_IsNaNd(double x);

ON_DECL
bool ON_IsQNaNd(double x);

ON_DECL
bool ON_IsSNaNd(double x);

ON_DECL
bool ON_IsNaNf(float x);

ON_DECL
bool ON_IsQNaNf(float x);

ON_DECL
bool ON_IsSNaNf(float x);

/*
class ON_CLASS ON_TimeLimit
{
  ON_TimeLimit();
  ON_TimeLimit(ON__UINT64 time_limit_seconds);
  void SetTimeLimit(ON__UINT64 time_limit_seconds);
  bool Continue() const;
  bool IsSet() const;
private:
  ON__UINT64 m_time_limit[2];
};
*/

// The ON_IS_FINITE and ON_IS_VALID defines are much faster
// than calling ON_IsValid(), but need to be used when
// the macro expansion works.

#if   defined(ON_LITTLE_ENDIAN)

// works on little endian CPUs with IEEE doubles
#define ON_IS_FINITE(x) (0x7FF0 != (*((unsigned short*)(&x) + 3) & 0x7FF0))
#define ON_IS_INFINITE(x) (0x7FF0 == (*((unsigned short*)(&x) + 3) & 0x7FF0))

#elif defined(ON_BIG_ENDIAN)

// works on big endian CPUs with IEEE doubles
#define ON_IS_FINITE(x) (0x7FF0 != (*((unsigned short*)(&x)) & 0x7FF0))
#define ON_IS_INFINITE(x) (0x7FF0 == (*((unsigned short*)(&x)) & 0x7FF0))

#else

// Returns true if x is a finite double.  Specifically,
// _finite returns a nonzero value (true) if its argument x
// is not infinite, that is, if -INF < x < +INF. 
// It returns 0 (false) if the argument is infinite or a NaN.
//
// If you are trying to compile opennurbs on a platform
// that does not support finite(), then see if you can
// use _fpclass(), fpclass(), _isnan(), or isnan().  If
// you can't find anything, then just set this
// function to return true.

#if defined(_GNU_SOURCE)
// if you are using an older version of gcc, use finite()
//#define ON_IS_FINITE(x) (finite(x)?true:false)
#define ON_IS_FINITE(x) (isfinite(x)?true:false)
#else
#define ON_IS_FINITE(x) (_finite(x)?true:false)
#endif

#endif

#define ON_IS_FINITE_FLOAT(x) ((x) <= 3.402823466e+38F && (x) >= -3.402823466e+38F)
#define ON_IS_INFINITE_FLOAT(x) ((x) > 3.402823466e+38F || (x) < -3.402823466e+38F)

#define ON_IS_VALID(x)  ((x) > ON_UNSET_VALUE && (x) < ON_UNSET_POSITIVE_VALUE)
// March 27, 2024 - Tim
// Fix for https://mcneel.myjetbrains.com/youtrack/issue/RH-81184
// The vertexes were crap just not quite ON_UNSET_VALUE crap
#define ON_CV_COORDINATE_IS_VALID(x) (-1.0e307 < x && x < 1.0e307)
#define ON_IS_VALID_FLOAT(x)  ((x) > ON_UNSET_FLOAT && (x) < ON_UNSET_POSITIVE_FLOAT)
#define ON_IS_UNSET_DOUBLE(x) (ON_UNSET_VALUE == (x) || ON_UNSET_POSITIVE_VALUE == (x))
#define ON_IS_UNSET_FLOAT(x) (ON_UNSET_FLOAT == (x) || ON_UNSET_POSITIVE_FLOAT == (x))
#define ON_IS_NAN(x) (!((x)==(x)))

ON_DECL
float ON_ArrayDotProduct( // returns AoB
          int,           // size of arrays (can be zero)
          const float*, // A[]
          const float*  // B[]
          );

ON_DECL
void   ON_ArrayScale( 
          int,           // size of arrays (can be zero)
          float,        // a
          const float*, // A[]
          float*        // returns a*A[]
          );

ON_DECL
void   ON_Array_aA_plus_B( 
          int,           // size of arrays (can be zero)
          float,        // a
          const float*, // A[]
          const float*, // B[]
          float*        // returns a*A[] + B[]
          );

ON_DECL
double ON_ArrayDotProduct( // returns AoB
          int,           // size of arrays (can be zero)
          const double*, // A[]
          const double*  // B[]
          );

ON_DECL
double ON_ArrayDotDifference( // returns A o ( B - C )
          int,           // size of arrays (can be zero)
          const double*, // A[]
          const double*, // B[]
          const double*  // C[]
          );

ON_DECL
double ON_ArrayMagnitude( // returns sqrt(AoA)
          int,           // size of arrays (can be zero)
          const double*  // A[]
          );

ON_DECL
double ON_ArrayMagnitudeSquared( // returns AoA
          int,           // size of arrays (can be zero)
          const double*  // A[]
          );

ON_DECL
double ON_ArrayDistance( // returns sqrt((A-B)o(A-B))
          int,           // size of arrays (can be zero)
          const double*, // A[]
          const double*  // B[]
          );

ON_DECL
double ON_ArrayDistanceSquared( // returns (A-B)o(A-B)
          int,           // size of arrays (can be zero)
          const double*, // A[]
          const double*  // B[]
          );

ON_DECL
void   ON_ArrayScale( 
          int,           // size of arrays (can be zero)
          double,        // a
          const double*, // A[]
          double*        // returns a*A[]
          );

ON_DECL
void   ON_Array_aA_plus_B( 
          int,           // size of arrays (can be zero)
          double,        // a
          const double*, // A[]
          const double*, // B[]
          double*        // returns a*A[] + B[]
          );

ON_DECL
int    ON_SearchMonotoneArray( // find a value in an increasing array
          // returns  -1: t < array[0]
          //           i: array[i] <= t < array[i+1] ( 0 <= i < length-1 )
          //    length-1: t == array[length-1]
          //      length: t >= array[length-1]
          const double*, // array[]
          int,           // length of array
          double         // t = value to search for
          );


/* 
Description:
  Compute a binomial coefficient.
Parameters:
  i - [in]
  j - [in]
Returns:
  (i+j)!/(i!j!), if 0 <= i and 0 <= j, and 0 otherwise.
See Also:
  ON_TrinomialCoefficient()
Remarks:
  If (i+j) <= 52, this function is fast and returns the exact
  value of the binomial coefficient.  

  For (i+j) > 52, the coefficient is computed recursively using
  the formula  bc(i,j) = bc(i-1,j) + bc(i,j-1).
  For (i+j) much larger than 60, this is inefficient.
  If you need binomial coefficients for large i and j, then you
  should probably be using something like Stirling's Formula.  
  (Look up "Stirling" or "Gamma function" in a calculus book.)
*/
ON_DECL
double ON_BinomialCoefficient( 
          int i,
          int j
          );


/* 
Description:
  Compute a trinomial coefficient.
Parameters:
  i - [in]
  j - [in]
  k - [in]
Returns:
  (i+j+k)!/(i!j!k!), if 0 <= i, 0 <= j and 0<= k, and 0 otherwise.
See Also:
  ON_BinomialCoefficient()
Remarks:
  The trinomial coefficient is computed using the formula

          (i+j+k)!      (i+j+k)!       (j+k)!
          --------   =  --------   *  -------
          i! j! k!      i! (j+k)!      j! k!

                      = ON_BinomialCoefficient(i,j+k)*ON_BinomialCoefficient(j,k)
  
*/
ON_DECL
double ON_TrinomialCoefficient( 
          int i,
          int j,
          int k
          );


ON_DECL
bool ON_GetParameterTolerance(
        double, double, // domain
        double,          // parameter in domain
        double*, double* // parameter tolerance (tminus, tplus) returned here
        );


ON_DECL
bool ON_IsValidPointList(
        int,  // dim
        bool, // true for homogeneous rational points
        int,  // count
        int,  // stride
        const float*
        );

ON_DECL
bool ON_IsValidPointList(
        int,  // dim
        bool, // true for homogeneous rational points
        int,  // count
        int,  // stride
        const double*
        );

/*
Description:
  Determine if a list of points is planar.
Parameters:
  bRational - [in]
    false if the points are euclidean (x,y,z)
    true if the points are homogeneous rational (x,y,z,w)
  point_count - [in]
    number of points
  point_stride - [in]
    number of doubles between point x coordinates
    first point's x coordinate = points[0],
    second point's x coordinate = points[point_stride],...
  points - [in]
    point coordinates (3d or 4d homogeneous rational)
  boxMin - [in]
  boxMax - [in]
    optional 3d bounding box - pass nulls if not readily available
  tolerance - [in] >= 0.0
  plane_equation0 - [in]
    If you want to test for planarity in a specific plane,
    pass the plane equation in here.  If you want to find
    a plane containing the points, pass null here.
  plane_equation - [out]
    If this point is not null, then the equation of the plane
    containing the points is returned here.
Returns:
  0 - points are not coplanar to the specified tolerance
  1 - points are coplanar to the specified tolerance
  2 - points are collinear to the specified tolerance
      (in this case, plane_equation is not a unique answer)
  3 - points are coincident to the specified tolerance
      (in this case, plane_equation is not a unique answer)
*/
ON_DECL
int ON_IsPointListPlanar(
    bool bRational,
    int count,
    int stride,
    const double* points,
    const double* boxMin,
    const double* boxMax,
    double tolerance,
    ON_PlaneEquation* plane_equation
    );

ON_DECL
bool ON_IsValidPointGrid(
        int,  // dim
        bool, // true for homogeneous rational points
        int, int, // point_count0, point_count1,
        int, int, // point_stride0, point_stride1,
        const double*
        );

ON_DECL
bool ON_ReversePointList(
        int,  // dim
        bool, // true for homogeneous rational points
        int,  // count
        int,  // stride
        double*
        );

ON_DECL
bool ON_ReversePointGrid(
        int,  // dim
        bool, // true for homogeneous rational points
        int, int, // point_count0, point_count1,
        int, int, // point_stride0, point_stride1,
        double*,
        int       // dir = 0 or 1
        );

ON_DECL
bool ON_SwapPointListCoordinates( 
        int, // count
        int, // stride
        float*,
        int, int // coordinates to swap
        );

ON_DECL
bool ON_SwapPointListCoordinates( 
        int, // count
        int, // stride
        double*,
        int, int // coordinates to swap
        );

ON_DECL
bool ON_SwapPointGridCoordinates(
        int, int, // point_count0, point_count1,
        int, int, // point_stride0, point_stride1,
        double*,
        int, int // coordinates to swap
        );

ON_DECL
bool ON_TransformPointList(
        int,  // dim
        bool, // true for homogeneous rational points
        int,  // count
        int,  // stride
        float*,
        const ON_Xform&
        );

ON_DECL
bool ON_TransformPointList(
        int,  // dim
        bool, // true for homogeneous rational points
        int,  // count
        int,  // stride
        double*,
        const ON_Xform&
        );

ON_DECL
bool ON_TransformPointGrid(
        int,      // dim
        bool,     // true for homogeneous rational points
        int, int, // point_count0, point_count1,
        int, int, // point_stride0, point_stride1,
        double*,
        const ON_Xform&
        );

ON_DECL
bool ON_TransformVectorList(
       int,  // dim
       int,  // count
       int,  // stride
       float*,
       const ON_Xform&
       );

ON_DECL
bool ON_TransformVectorList(
       int,  // dim
       int,  // count
       int,  // stride
       double*,
       const ON_Xform&
       );

/*
Parameters:
  dim - [in]
    >= 1
  is_rat - [in]
    true if the points are rational and points[dim] is the "weight"
  pointA - [in]
  pointB - [in]
    point coordinates
Returns:
  True if the input is valid and for each coordinate pair,
  |a-b| <= ON_ZERO_TOLERANCE 
  or |a-b| <= (fabs(a)+fabs(b))*ON_RELATIVE_TOLERANCE.
  False otherwise.
*/
ON_DECL
bool ON_PointsAreCoincident(
    int dim,
    bool is_rat,
    const double* pointA,
    const double* pointB
    );

/*
Description
  See ON_PointsAreCoincident() for a description of when opennurbs
  considers two points to be coincident.
Parameters:
  dim - [in]
    >= 1
  is_rat - [in]
    true if the points are rational and points[dim] is the "weight"
  point_count - [in]
    number of points >= 2
  point_stride - [in]
    >= (0 != is_rat) ? (dim+1) : dim
  points - [in]
    point coordinates
Returns:
  True if the first and last points are coincident and all other
  points in the list are coincident with the previous point.
  False if there are points that are not coincident or
  point_count < 2 or other input parameters are invalid.
*/
ON_DECL
bool ON_PointsAreCoincident(
    int dim,
    bool is_rat,
    int point_count,
    int point_stride,
    const double* points
    );

ON_DECL
int ON_ComparePoint( // returns 
                              // -1: first < second
                              //  0: first == second
                              // +1: first > second
          int dim,            // dim (>=0)
          bool israt,    // true for rational CVs
          const double* cv0,  // first CV
          const double* cv1   // second CV
          );

ON_DECL
int ON_ComparePointList( // returns 
                              // -1: first < second
                              //  0: first == second
                              // +1: first > second
          int,           // dim (>=0)
          bool,          // true for rational CVs
          int,           // count
          // first point list
          int,           // stride
          const double*, // point
          // second point list
          int,           // stride
          const double*  // point
          );

ON_DECL
bool ON_IsPointListClosed(
       int,  // dim
       bool,  // true for homogeneous rational points
       int,  // count
       int,  // stride
       const double*
       );

ON_DECL
bool ON_IsPointGridClosed(
        int,  // dim
        bool, // true for homogeneous rational points
        int, int, // point_count0, point_count1,
        int, int, // point_stride0, point_stride1,
        const double*,
        int       // dir = 0 or 1
       );


/*
Description:
  Assign a unique id to each point location.  Coincident points
  get the same id.
Parameters:
  point_dim - [in]
    2 or 3
  point_count - [in]
    >= 1
  point_stride - [in]
    number of coordinates to skip between points
    >= point_dim
  points - [in]
    The first coordinate of the i-th point is points[i*point_stride]
  first_point_id - [in]
    Initial point id.  Typically 1 or 0.
  point_ids - [out]
    If not null, then point_ids[] must be an array of length point_count
    and the ids are returned in this array.  point_ids[0] = first_point_id.
  point_id_map - [out]
    If point_id_index is not null, then it must have length point_count.
    The returned values are a permutation of (0,1,...,point_count-1) such that
    (point_ids[point_id_map[0]], ..., point_ids[point_id_map[point_count-1]])
    is an increasing list of values and point_id_map[0] = 0.
Returns:
  If input is valid, then an array of point_count point location
  ids is returned.  The i-th and j-th values in the returned array are
  equal if and only if the i-th and j-th points have the same location.
  If the input point_ids pointer was null, then the array memory is
  allocated on the heap by calling onmalloc().  If input is not valid,
  nullptr is returned.
Remarks:
  The ids are invariant under invertable transformations.  
  Specifically, if one point point set is a rotation of another, then
  the assigned ids will be the same.
*/
ON_DECL
unsigned int* ON_GetPointLocationIds(
  size_t point_dim,
  size_t point_count,
  size_t point_stride,
  const float* points,
  unsigned int first_point_id,
  unsigned int* point_ids,
  unsigned int* point_id_map
  );

ON_DECL
unsigned int* ON_GetPointLocationIds(
  size_t point_dim,
  size_t point_count,
  size_t point_stride,
  const double* points,
  unsigned int first_point_id,
  unsigned int* point_ids,
  unsigned int* point_id_map
  );

ON_DECL
unsigned int* ON_GetPointLocationIds(
  size_t point_count,
  const class  ON_2fPoint* points,
  unsigned int first_point_id,
  unsigned int* point_ids,
  unsigned int* point_id_map
  );

ON_DECL
unsigned int* ON_GetPointLocationIds(
  size_t point_count,
  const class ON_3fPoint* points,
  unsigned int first_point_id,
  unsigned int* point_ids,
  unsigned int* point_id_map
  );

ON_DECL
unsigned int* ON_GetPointLocationIds(
  size_t point_count,
  const class  ON_2dPoint* points,
  unsigned int first_point_id,
  unsigned int* point_ids,
  unsigned int* point_id_map
  );

ON_DECL
unsigned int* ON_GetPointLocationIds(
  size_t point_count,
  const class ON_3dPoint* points,
  unsigned int first_point_id,
  unsigned int* point_ids,
  unsigned int* point_id_map
  );


ON_DECL
int ON_SolveQuadraticEquation( // solve a*X^2 + b*X + c = 0
        // returns 0: two distinct real roots (r0 < r1)
        //         1: one real root (r0 = r1)
        //         2: two complex conjugate roots (r0 +/- (r1)*sqrt(-1))
        //        -1: failure - a = 0, b != 0        (r0 = r1 = -c/b)
        //        -2: failure - a = 0, b  = 0 c != 0 (r0 = r1 = 0.0)
        //        -3: failure - a = 0, b  = 0 c  = 0 (r0 = r1 = 0.0)
       double, double, double, // a, b, c
       double*, double*        // roots r0 and r1 returned here
       );

/*
Description:
  Solve the cubic equation a*X^3 + b*X^2 + c*X + d = 0.
Inputs:
  a,b,c,d, polynomial coefficients ( if a==b==c== 0) then failure is returned
Returns: 
  number of real roots stored with multiplicity.
  specifically
  -1: failure (a == b == c== 0.0 case)
   0: no real roots ( a==0 and b!=0) two complex conjugate roots (r1 +/- (r2)*sqrt(-1))
   1: one real root (r1).  Either  ( a==b==0.0) or else( a!=0) and two complex conjugate 
		roots (r2 +/- (r3)*sqrt(-1))
   2: two real roots (a==0.0, b!=0.0)  *r1 <= *r2
   3: three real roots (a!=0.0) *r1 <= *r2 <= *r3
*/
ON_DECL
int ON_SolveCubicEquation(
  double a, double b, double c, double d,
  double* r1, double* r2, double* r3
);

/*
Returns:
  0: success
  <0: failure
*/
ON_DECL
int ON_SolveTriDiagonal( // solve TriDiagMatrix( a,b,c )*X = d
        int,               // dimension of d and X (>=1)
        int,               // number of equations (>=2)
        double*,           // a[n-1] = sub-diagonal (a is modified)
        const double*,     // b[n] = diagonal
        double*,           // c[n-1] = supra-diagonal
        const double*,     // d[n*dim]
        double*            // X[n*dim] = unknowns
        );

// returns rank - if rank != 2, system is under determined
// If rank = 2, then solution to 
//
//          a00*x0 + a01*x1 = b0, 
//          a10*x0 + a11*x1 = b1 
//
// is returned
ON_DECL
int ON_Solve2x2( 
        double, double,   // a00 a01 = first row of 2x2 matrix
        double, double,   // a10 a11 = second row of 2x2 matrix
        double, double,   // b0 b1
        double*, double*, // x0, x1 if not nullptr, then solution is returned here
        double*           // if not nullptr, then pivot_ratio returned here
        );

// Description:
//   Solves a system of 3 linear equations and 2 unknowns.
//
//          x*col0[0] + y*col1[0] = d0
//          x*col0[1] + y*col1[1] = d0
//          x*col0[2] + y*col1[2] = d0
//
// Parameters:
//   col0 - [in] coefficients for "x" unknown
//   col1 - [in] coefficients for "y" unknown
//   d0 - [in] constants
//   d1 - [in]
//   d2 - [in]
//   x - [out]
//   y - [out]
//   error - [out]
//   pivot_ratio - [out]
//
// Returns:
//   rank of the system.  
//   If rank != 2, system is under determined
//   If rank = 2, then the solution is
//
//         (*x)*[col0] + (*y)*[col1]
//         + (*error)*((col0 X col1)/|col0 X col1|)
//         = (d0,d1,d2).
ON_DECL
int ON_Solve3x2( 
        const double[3], // col0
        const double[3], // col1
        double,  // d0
        double,  // d1
        double,  // d2
        double*, // x
        double*, // y
        double*, // error
        double*  // pivot_ratio
        );

/* 
Description:
  Use Gauss-Jordan elimination with full pivoting to solve 
  a system of 3 linear equations and 3 unknowns(x,y,z)

        x*row0[0] + y*row0[1] + z*row0[2] = d0
        x*row1[0] + y*row1[1] + z*row1[2] = d1
        x*row2[0] + y*row2[1] + z*row2[2] = d2

Parameters:
    row0 - [in] first row of 3x3 matrix
    row1 - [in] second row of 3x3 matrix
    row2 - [in] third row of 3x3 matrix
    d0 - [in] 
    d1 - [in] 
    d2 - [in] (d0,d1,d2) right hand column of system
    x_addr - [in] first unknown
    y_addr - [in] second unknown
    z_addr - [in] third unknown
    pivot_ratio - [out] if not nullptr, the pivot ration is 
         returned here.  If the pivot ratio is "small",
         then the matrix may be singular or ill 
         conditioned. You should test the results 
         before you use them.  "Small" depends on the
         precision of the input coefficients and the
         use of the solution.  If you can't figure out
         what "small" means in your case, then you
         must check the solution before you use it.

Returns:
    The rank of the 3x3 matrix (0,1,2, or 3)
    If ON_Solve3x3() is successful (returns 3), then
    the solution is returned in 
    (*x_addr, *y_addr, *z_addr)
    and *pivot_ratio = min(|pivots|)/max(|pivots|).
    If the return code is < 3, then (0,0,0) is returned
    as the "solution".

See Also:
  ON_Solve2x2
  ON_Solve3x2
  ON_Solve4x4
*/
ON_DECL
int ON_Solve3x3( 
        const double row0[3], 
        const double row1[3], 
        const double row2[3],
        double d0, 
        double d1, 
        double d2,
        double* x_addr, 
        double* y_addr, 
        double* z_addr,
        double* pivot_ratio
        );

/* 
Description:
  Use Gauss-Jordan elimination with full pivoting to solve 
  a system of 4 linear equations and 4 unknowns(x,y,z,w)

        x*row0[0] + y*row0[1] + z*row0[2] + w*row0[3] = d0
        x*row1[0] + y*row1[1] + z*row1[2] + w*row1[3] = d1
        x*row2[0] + y*row2[1] + z*row2[2] + w*row2[3] = d2
        x*row3[0] + y*row3[1] + z*row3[2] + w*row3[2] = d3

Parameters:
    row0 - [in] first row of 4x4 matrix
    row1 - [in] second row of 4x4 matrix
    row2 - [in] third row of 4x4 matrix
    row3 - [in] forth row of 4x4 matrix
    d0 - [in] 
    d1 - [in] 
    d2 - [in] 
    d3 - [in] (d0,d1,d2,d3) right hand column of system
    x_addr - [in] first unknown
    y_addr - [in] second unknown
    z_addr - [in] third unknown
    w_addr - [in] forth unknown
    pivot_ratio - [out] if not nullptr, the pivot ration is 
         returned here.  If the pivot ratio is "small",
         then the matrix may be singular or ill 
         conditioned. You should test the results 
         before you use them.  "Small" depends on the
         precision of the input coefficients and the
         use of the solution.  If you can't figure out
         what "small" means in your case, then you
         must check the solution before you use it.

Returns:
    The rank of the 4x4 matrix (0,1,2,3, or 4)
    If ON_Solve4x4() is successful (returns 4), then
    the solution is returned in 
    (*x_addr, *y_addr, *z_addr, *w_addr)
    and *pivot_ratio = min(|pivots|)/max(|pivots|).
    If the return code is < 4, then, it a solution exists,
    on is returned.  However YOU MUST CHECK THE SOLUTION
    IF THE RETURN CODE IS < 4.

See Also:
  ON_Solve2x2
  ON_Solve3x2
  ON_Solve3x3
*/
ON_DECL
int
ON_Solve4x4(
          const double row0[4], 
          const double row1[4], 
          const double row2[4],  
          const double row3[4],
          double d0, 
          double d1, 
          double d2, 
          double d3,
          double* x_addr, 
          double* y_addr, 
          double* z_addr, 
          double* w_addr,
          double* pivot_ratio
          );

/*
Description:
  Use Gauss-Jordan elimination to find a numerical 
  solution to M*X = B where M is a n x n matrix,
  B is a known n-dimensional vector and X is
  an unknown.
Parameters:
  bFullPivot - [in] if true, full pivoting is used,
    otherwise partial pivoting is used.  In rare
    cases full pivoting can produce a more accurate
    answer and never produces a less accurate answer.
    However full pivoting is slower.  If speed is an
    issue, then experiment with bFullPivot=false
    and see if it makes a difference.  Otherwise,
    set it to true.
  bNormalize - [in]
    If bNormalize is true, then the rows of the
    matrix are scaled so the sum of their squares
    is one.  This doesn't make the solution more
    accurate but in some cases it makes the pivot
    ratio more meaningful.  Set bNormalize to
    false unless you have a reason for setting it
    to true.
  n - [in] size of the matrix and vectors.
  M - [in] n x n matrix.  The values in M are
    changed as the solution is calculated.
    If you need to preserve M for future use,
    pass in a copy.
  B - [in] n-dimensional vector.  The values in
    B are changed as the solution is calculated.
    If you need to preserve B for future use,
    pass in a copy.
  X - [out] solution to M*X = B.
Returns:
  If the returned value is <= 0.0, the input matrix
  has rank < n and no solution is returned in X.
  If the returned value is > 0.0, then a solution is
  returned in X and the returned value is the ratio
  (minimum pivot)/(maximum pivot).  This value is
  called the pivot ratio and will be denoted "pr"
  the discussion below. If pr <= 1e-15, then
  M was nearly degenerate and the solution should be
  used with caution.  If an accurate solution is
  critical, then check the solution anytime pr <= 1e-10
  In general, the difference between M*X and B will be
  reasonably small.  However, when the pr is small
  there tend to be vector E, substantially different
  from zero, such that M*(X+E) - B is also reasonably
  small.
See Also:
  ON_Solve2x2
  ON_Solve3x3
  ON_Solve4x4
  ON_Solve3x2
*/
ON_DECL
double ON_SolveNxN(bool bFullPivot, bool bNormalize, int n, double* M[], double B[], double X[]);


/*
Description:
Find the eigen values and eigen vectors of a real symmetric
3x3 matrix

A D F
D B E
F E C

Parameters:
A - [in] matrix entry
B - [in] matrix entry
C - [in] matrix entry
D - [in] matrix entry
E - [in] matrix entry
F - [in] matrix entry
e1 - [out] eigen value
E1 - [out] eigen vector with eigen value e1
e2 - [out] eigen value
E2 - [out] eigen vector with eigen value e2
e3 - [out] eigen value
E3 - [out] eigen vector with eigen value e3
Returns:
True if successful.
*/
ON_DECL
bool ON_Sym3x3EigenSolver(double A, double B, double C,
	double D, double E, double F,
	double* e1, ON_3dVector& E1,
	double* e2, ON_3dVector& E2,
	double* e3, ON_3dVector& E3  );

// return false if determinant is (nearly) singular
ON_DECL
bool ON_EvJacobian( 
        double, // ds o ds
        double, // ds o dt
        double, // dt o dt
        double* // jacobian = determinant ( ds_o_ds dt_o_dt / ds_o_dt ds_o_dt )
        );

/*
Description:
  Finds scalars x and y so that the component of V in the plane
  of A and B is x*A + y*B.
Parameters:
  V - [in]
  A - [in] nonzero and not parallel to B
  B - [in] nonzero and not parallel to A
  x - [out]
  y - [out]
Returns:
  1 - The rank of the problem is 2.  The decomposition is unique.
	0 - The rank less than 2.  Either there is no solution or there
			are infinitely many solutions.

See Also:
  ON_Solve2x2
*/
ON_DECL
int ON_DecomposeVector(
        const ON_3dVector& V,
        const ON_3dVector& A,
        const ON_3dVector& B,
        double* x, double* y
        );


/*
Description:
   Evaluate partial derivatives of surface unit normal
Parameters:
  ds - [in]
  dt - [in] surface first partial derivatives
  dss - [in]
  dst - [in]
  dtt - [in] surface second partial derivatives
  ns - [out]
  nt - [out] First partial derivatives of surface unit normal
             (If the Jacobian is degenerate, ns and nt are set to zero.)
Returns:
  true if Jacobian is nondegenerate
  false if Jacobian is degenerate
*/
ON_DECL
bool ON_EvNormalPartials(
        const ON_3dVector& ds,
        const ON_3dVector& dt,
        const ON_3dVector& dss,
        const ON_3dVector& dst,
        const ON_3dVector& dtt,
        ON_3dVector& ns,
        ON_3dVector& nt
        );

ON_DECL
bool 
ON_Pullback3dVector( // use to pull 3d vector back to surface parameter space
      const ON_3dVector&,   // 3d vector
      double,              // signed distance from vector location to closet point on surface
                                    // < 0 if point is below with respect to Du x Dv
      const ON_3dVector&,     // ds      surface first partials
      const ON_3dVector&,     // dt
      const ON_3dVector&,     // dss     surface 2nd partials
      const ON_3dVector&,     // dst     (used only when dist != 0)
      const ON_3dVector&,     // dtt
      ON_2dVector&            // pullback
      );

ON_DECL
bool 
ON_GetParameterTolerance(
        double,   // t0      domain
        double,   // t1 
        double,   // t       parameter in domain
        double*,  // tminus  parameter tolerance (tminus, tplus) returned here
        double*   // tplus
        );


ON_DECL
bool ON_EvNormal(
        int, // limit_dir 0=default,1=from quadrant I, 2 = from quadrant II, ...
        const ON_3dVector&, const ON_3dVector&, // first partials (Du,Dv)
        const ON_3dVector&, const ON_3dVector&, const ON_3dVector&, // optional second partials (Duu, Duv, Dvv)
        ON_3dVector& // unit normal returned here
        );

// returns false if the returned tangent is zero
ON_DECL
bool ON_EvTangent(
        const ON_3dVector&, // first derivative
        const ON_3dVector&, // second derivative
        ON_3dVector&        // Unit tangent returned here
        );

// returns false if first derivative is zero
ON_DECL
bool ON_EvCurvature(
        const ON_3dVector&, // first derivative
        const ON_3dVector&, // second derivative
        ON_3dVector&,       // Unit tangent returned here
        ON_3dVector&        // Curvature returned here
        );

// Compute derivative of curvature as well as curvature and torsion.
// Returns false if either kprime or torsion are not well defined.
ON_DECL
bool ON_EvCurvature1Der(
  const ON_3dVector& D1, // first derivative of curve
  const ON_3dVector& D2, // second derivative of curve
  const ON_3dVector& D3, // third derivative of curve
  ON_3dVector& T,       // Unit tangent returned here
  ON_3dVector& K,       // curvature vector(k*N). curvature k = K.Length() and Normal N=K.Unitize()
  double* kprime,       // first derivative of k
  double* torsion);       // torsion 


ON_DECL
bool ON_EvPrincipalCurvatures( 
        const ON_3dVector&, // Ds,
        const ON_3dVector&, // Dt,
        const ON_3dVector&, // Dss,
        const ON_3dVector&, // Dst,
        const ON_3dVector&, // Dtt,
        const ON_3dVector&, // N,   // unit normal to surface (use ON_EvNormal())
        double*, // gauss,  // = Gaussian curvature = kappa1*kappa2
        double*, // mean,   // = mean curvature = (kappa1+kappa2)/2
        double*, // kappa1, // = largest (in absolute value) principal curvature value (may be negative)
        double*, // kappa2, // = smallest (in absolute value) principal curvature value (may be negative)
        ON_3dVector&, // K1,     // kappa1 unit principal curvature direction
        ON_3dVector&  // K2      // kappa2 unit principal curvature direction
                        // output K1,K2,N is right handed frame
        );

ON_DECL
bool ON_EvPrincipalCurvatures( 
        const ON_3dVector&, // Ds,
        const ON_3dVector&, // Dt,
        double l, // Dss*N Second fundamental form coefficients
        double m, // Dst*N,
        double n, // Dtt*N,
        const ON_3dVector&, // N,   // unit normal to surface (use ON_EvNormal())
        double*, // gauss,  // = Gaussian curvature = kappa1*kappa2
        double*, // mean,   // = mean curvature = (kappa1+kappa2)/2
        double*, // kappa1, // = largest (in absolute value) principal curvature value (may be negative)
        double*, // kappa2, // = smallest (in absolute value) principal curvature value (may be negative)
        ON_3dVector&, // K1,     // kappa1 unit principal curvature direction
        ON_3dVector&  // K2      // kappa2 unit principal curvature direction
                        // output K1,K2,N is right handed frame
        );

/*
Description:
  Evaluate sectional curvature from surface derivatives and 
  section plane normal.
Parameters:
  S10, S01 - [in]
    surface 1st partial derivatives
  S20, S11, S02 - [in]
    surface 2nd partial derivatives
  planeNormal - [in]
    unit normal to section plane
  K - [out] Sectional curvature
    Curvature of the intersection curve of the surface
    and plane through the surface point where the partial
    derivatives were evaluated.
Returns:
  True if successful.
  False if first partials are not linearly independent, in
  which case the K is set to zero.
*/
ON_DECL
bool ON_EvSectionalCurvature( 
    const ON_3dVector& S10, 
    const ON_3dVector& S01,
    const ON_3dVector& S20, 
    const ON_3dVector& S11, 
    const ON_3dVector& S02,
    const ON_3dVector& planeNormal,
    ON_3dVector& K 
    );


ON_DECL
ON_3dVector ON_NormalCurvature( 
        const ON_3dVector&, // surface 1rst partial (Ds)
        const ON_3dVector&, // surface 1rst partial (Dt)
        const ON_3dVector&, // surface 1rst partial (Dss)
        const ON_3dVector&, // surface 1rst partial (Dst)
        const ON_3dVector&, // surface 1rst partial (Dtt)
        const ON_3dVector&, // surface unit normal
        const ON_3dVector&  // unit tangent direction
        );

/*
Description:
  Determine if two curvatures are different enough
  to qualify as a curvature discontinuity.
Parameters:
  Km - [in]
  Kp - [in]
    Km and Kp should be curvatures evaluated at the same
    parameters using limits from below (minus) and above (plus).
    The assumption is that you have already compared the
    points and tangents and consider to curve to be G1 at the
    point in question.
  cos_angle_tolerance - [in]
    If the input value of cos_angle_tolerance >= -1.0
    and cos_angle_tolerance <= 1.0 and
    Km o Kp < cos_angle_tolerance*|Km|*|Kp|, then
    true is returned.  Otherwise it is assumed Km and Kp
    are parallel. If the curve being tested is nonplanar,
    then use something like cos(2*tangent angle tolerance)
    for this parameter. If the curve being tested is planar,
    then 0.0 will work fine.
  curvature_tolerance - [in]
    If |Kp-Km| <= curvature_tolerance,
    then false is returned, otherwise other tests are used
    to determine continuity.
  zero_curvature - [in] (ignored if < 2^-110 = 7.7037197787136e-34)
    If |K| <= zero_curvature, then K is treated as zero.
    When in doubt, use ON_ZERO_CURVATURE_TOLERANCE.
  radius_tolerance - [in]
    If radius_tolerance >= 0.0 and the difference between the
    radii of curvature is >= radius_tolerance, then true 
    is returned.
  relative_tolerance - [in]
    If relative_tolerance > 0 and
    |(|Km| - |Kp|)|/max(|Km|,|Kp|) > relative_tolerance,
    then true is returned.  Note that if the curvatures are
    nonzero and rm and rp are the radii of curvature, then
    |(|Km| - |Kp|)|/max(|Km|,|Kp|) = |rm-rp|/max(rm,rp).
    This means the relative_tolerance insures both the scalar
    curvature and the radii of curvature agree to the specified
    number of decimal places.
    When in doubt, use ON_RELATIVE_CURVATURE_TOLERANCE, which
    is currently 0.05.
Returns:
  False if the curvatures should be considered G2.
  True if the curvatures are different enough that the curve should be
  considered not G2.  
  In addition to the tests described under the curvature_tolerance and 
  radius_tolerance checks, other heuristic tests are used.
*/
ON_DECL
bool ON_IsCurvatureDiscontinuity( 
  const ON_3dVector Km, 
  const ON_3dVector Kp,
  double cos_angle_tolerance,
  double curvature_tolerance,
  double zero_curvature,
  double radius_tolerance,
  double relative_tolerance
  );

ON_DECL
bool ON_IsCurvatureDiscontinuity( 
  const ON_3dVector Km, 
  const ON_3dVector Kp,
  double cos_angle_tolerance,
  double curvature_tolerance,
  double zero_curvature,
  double radius_tolerance
  );


/*
Description:
  This function is used to test curvature continuity
  in IsContinuous and GetNextDiscontinuity functions
  when the continuity parameter is ON::continuity::G2_continuous.
Parameters:
  Km - [in]
    Curve's vector curvature evaluated from below
  Kp - [in]
    Curve's vector curvature evaluated from below
Returns:
  True if the change from Km to Kp should be considered
  G2 continuous.
*/
ON_DECL
bool ON_IsG2CurvatureContinuous(
  const ON_3dVector Km, 
  const ON_3dVector Kp,
  double cos_angle_tolerance,
  double curvature_tolerance
  );

/*
Description:
  This function is used to test curvature continuity
  in IsContinuous and GetNextDiscontinuity functions
  when the continuity parameter is ON::continuity::Gsmooth_continuous.
Parameters:
  Km - [in]
    Curve's vector curvature evaluated from below
  Kp - [in]
    Curve's vector curvature evaluated from below
Returns:
  True if the change from Km to Kp should be considered
  Gsmooth continuous.
*/
ON_DECL
bool ON_IsGsmoothCurvatureContinuous(
  const ON_3dVector Km, 
  const ON_3dVector Kp,
  double cos_angle_tolerance,
  double curvature_tolerance
  );

/*
Description:
  Test curve continuity from derivative values.
Parameters:
  c - [in] type of continuity to test for. Read ON::continuity
           comments for details.
  Pa - [in] point on curve A.
  D1a - [in] first derivative of curve A.
  D2a - [in] second derivative of curve A.
  Pb - [in] point on curve B.
  D1b - [in] first derivative of curve B.
  D3b - [in] second derivative of curve B.
  point_tolerance - [in] if the distance between two points is
      greater than point_tolerance, then the curve is not C0.
  d1_tolerance - [in] if the difference between two first derivatives is
      greater than d1_tolerance, then the curve is not C1.
  d2_tolerance - [in] if the difference between two second derivatives is
      greater than d2_tolerance, then the curve is not C2.
  cos_angle_tolerance - [in] default = cos(1 degree) Used only when
      c is ON::continuity::G1_continuous or ON::continuity::G2_continuous.  If the cosine
      of the angle between two tangent vectors 
      is <= cos_angle_tolerance, then a G1 discontinuity is reported.
  curvature_tolerance - [in] (default = ON_SQRT_EPSILON) Used only when
      c is ON::continuity::G2_continuous.  If K0 and K1 are curvatures evaluated
      from above and below and |K0 - K1| > curvature_tolerance,
      then a curvature discontinuity is reported.
Returns:
  true if the curve has at least the c type continuity at 
  the parameter t.
*/
ON_DECL
bool ON_IsContinuous(
  ON::continuity c,
  ON_3dPoint Pa,
  ON_3dVector D1a,
  ON_3dVector D2a,
  ON_3dPoint Pb,
  ON_3dVector D1b,
  ON_3dVector D2b,
  double point_tolerance=ON_ZERO_TOLERANCE,
  double d1_tolerance=ON_ZERO_TOLERANCE,
  double d2_tolerance=ON_ZERO_TOLERANCE,
  double cos_angle_tolerance=ON_DEFAULT_ANGLE_TOLERANCE_COSINE,
  double curvature_tolerance=ON_SQRT_EPSILON
  );


ON_DECL
bool ON_TuneupEvaluationParameter( 
   int side,
   double s0, double s1, // segment domain
   double *s             // segment parameter
   );


ON_DECL
int ON_Compare2dex( const ON_2dex* a, const ON_2dex* b);

ON_DECL
int ON_Compare2udex(const ON_2udex* a, const ON_2udex* b);

ON_DECL
int ON_Compare3dex( const ON_3dex* a, const ON_3dex* b);

ON_DECL
int ON_Compare4dex( const ON_4dex* a, const ON_4dex* b);

ON_DECL
const ON_2dex* ON_BinarySearch2dexArray( 
          int key_i, 
          const ON_2dex* base, 
          size_t nel
          );

ON_DECL
const ON_2udex* ON_BinarySearch2udexArray(
  unsigned int key_i,
  const ON_2udex* base,
  size_t nel
);

// These simple intersectors are fast and detect transverse intersections.
// If the intersection is not a simple transverse case, then they
// return false and you will have to use one of the slower but fancier
// models.

// Description:
//   Find the unique closest-points pair between two infinite lines, if it exists.
// Parameters:
//   lineA - [in]
//   lineB - [in]
//   double* a - [out]
//   double* b - [out] The shortest distance between the lines is the
//                    chord from lineA.PointAt(*a) to lineB.PointAt(*b).
// Returns:
//   True if points are found and false if the lines are numerically parallel.
//   Numerically parallel means the 2x2 matrix
// 
//             AoA  -AoB
//            -AoB   BoB
// 
//  is numerically singular, where A = lineA.to-lineA.from
//  and B = lineB.to-lineB.from.
// See Also:
//   ON_IntersectLineLine
ON_DECL
bool ON_Intersect( 
          const ON_Line&, 
          const ON_Line&, 
          double*, // parameter on first line
          double*  // parameter on second line
          );

// Description:
//   Finds the intersection point of a line and plane, if it exists.
// Parameters:
//   line - [in]
//   plane - [in]
//   line_parameter - [out]
//     If the returned parameter is < 0 or > 1, then the
//     line segment between line.from and line.to
//     does not intersect the plane.
// Returns:
//   true if the intersection is a single point,
//   and false otherwise.
ON_DECL
bool ON_Intersect( 
          const ON_Line&, 
          const ON_Plane&, 
          double* // parameter on line
          );

// Description:
//   Finds the intersection point of a line and plane, if it exists.
// Parameters:
//   line - [in]
//   plane_equation - [in]
//   line_parameter - [out]
//     If the returned parameter is < 0 or > 1, then the
//     line segment between line.from and line.to
//     does not intersect the plane.
// Returns:
//   true if the intersection is a single point,
//   and false otherwise.
ON_DECL
bool ON_Intersect(
          const ON_Line& line, 
          const ON_PlaneEquation& plane_equation, 
          double* line_parameter
          );

// Description:
//   Finds the intersection line between two planes, if it exists.
// Parameters:
//   planeA - [in]
//   planeB - [in]
//   line_out - [out]
//     Intersection line is returned here.
// Returns:
//   true if the intersection is a single line,
//   and false otherwise.
ON_DECL
bool ON_Intersect( 
        const ON_Plane&, 
        const ON_Plane&, 
        ON_Line& // intersection line is returned here
        );

// Description:
//   Finds the intersection point of three planes, if it exists.
// Parameters:
//   planeA - [in]
//   planeB - [in]
//   planeC - [in]
//   point_out - [out]
//     Intersection point is returned here.
// Returns:
//   true if the intersection is a single point,
//   and false otherwise.
ON_DECL
bool ON_Intersect( 
        const ON_Plane&, 
        const ON_Plane&, 
        const ON_Plane&,
        ON_3dPoint& // intersection point is returned here
        );

// Description:
//   Intersect a plane and a sphere.
// Parameters:
//   plane - [in]
//   sphere - [in]
//   circle - [out]
//     Intersection circle is returned here.
// Returns:
//   0: no intersection
//     circle radius = 0,
//     and circle origin = point on the plane closest to the sphere.
//   1: intersection is a single point
//     circle radius = 0,
//     and circle origin = tangent point on the plane.
//   2: intersection is a circle
//     circle radius > 0.
ON_DECL
int ON_Intersect(
        const ON_Plane&,
        const ON_Sphere&,
        ON_Circle&  // intersection circle is returned here
        );

// Description:
//   Intersect an infinite line and a sphere.
// Parameters:
//   line - [in]
//   sphere - [in]
//   pointA - [out]
//     First intersection point is returned here.
//   pointB - [out]
//     Second intersection point is returned here.
// Returns:
//   0: no intersection
//     pointA is the point on line closest to sphere,
//     pointB is the point on sphere closest to line.
//   1: intersection is a single point
//     pointA is obtained by evaluating the line
//     pointB is obtained by evaluating the sphere
//   2: 2 distinct intersection points
ON_DECL
int ON_Intersect(                  
        const ON_Line&, 
        const ON_Sphere&,
        ON_3dPoint&, // [out] first intersection point
        ON_3dPoint&  // [out] second intersection point
        );

// Description:
//   Intersect an infinite line and a cylinder.
// Parameters:
//   line - [in]
//   cylinder - [in]
//   pointA - [out]
//   pointB - [out]
// Returns:
//   0: no intersection
//     pointA is the point on line closest to cylinder,
//     pointB is the point on cylinder closest to line.
//   1: intersection is a single point
//     pointA is obtained by evaluating the line
//     pointB is obtained by evaluating the cylinder
//   2: 2 distinct intersection points
//   3: line lies on cylinder
// Remarks:
//   The value of cylinder.IsFinite() determines if the
//   intersection is performed on the finite or infinite cylinder.
ON_DECL
int ON_Intersect( 
      const ON_Line&,
      const ON_Cylinder&,
      ON_3dPoint&, // [out] first intersection point
      ON_3dPoint&  // [out] second intersection point
      );

// Description:
//   Intersect an infinite line and circle.
// Parameters:
//   line - [in]
//   circle - [in]
//   line_t0 - [out] line parameter of first intersection point
//   circle_point0 - [out] first intersection point on circle
//   line_t1 - [out] line parameter of second intersection point
//   circle_point1 - [out] second intersection point on circle
// Returns:
//   0     No intersection
//   1     One intersection at line.PointAt(*line_t0)
//   2     Two intersections at line.PointAt(*line_t0)
//         and line.PointAt(*line_t1).
ON_DECL
int ON_Intersect( 
                  const ON_Line& line, 
                  const ON_Circle& circle,
                  double* line_t0,
                  ON_3dPoint& circle_point0,
                  double* line_t1,
                  ON_3dPoint& circle_point1
                  );



// Description:
//   Intersect a infinite line and arc.
// Parameters:
//   line - [in]
//   arc - [in]
//   line_t0 - [out] line parameter of first intersection point
//   arc_point0 - [out] first intersection point on arc
//   line_t1 - [out] line parameter of second intersection point
//   arc_point1 - [out] second intersection point on arc
// Returns:
//   0     No intersection
//   1     One intersection at line.PointAt(*line_t0)
//   2     Two intersections at line.PointAt(*line_t0)
//         and line.PointAt(*line_t1).
ON_DECL
int ON_Intersect( 
                  const ON_Line& line, 
                  const ON_Arc& arc,
                  double* line_t0,
                  ON_3dPoint& arc_point0,
                  double* line_t1,
                  ON_3dPoint& arc_point1
                  );

// Description:
//   Intersect a plane and a circle.
// Parameters:
//   plane - [in]
//   circle - [in]
//   point0 - [out] first intersection point 
//   point1 - [out] second intersection point
// Returns:
//   0     No intersection
//   1     One intersection at point0
//   2     Two intersections at point0
//         and point1.
//	   3		  Circle lies on plane
ON_DECL
int ON_Intersect( 
                  const ON_Plane& plane, 
                  const ON_Circle& circle,
                  ON_3dPoint& point0,
                  ON_3dPoint& point1
                  );

// Description:
//   Intersect a plane and an arc.
// Parameters:
//   plane - [in]
//   arc - [in]
//   point0 - [out] first intersection point 
//   point1 - [out] second intersection point
// Returns:
//   0     No intersection
//   1     One intersection at point0
//   2     Two intersections at point0
//         and point1.
//   3	    Arc lies on plane
ON_DECL
int ON_Intersect( 
                  const ON_Plane& plane, 
                  const ON_Arc& arc,
                  ON_3dPoint& point0,
                  ON_3dPoint& point1
                  );


// Description:
//   Intersect a pair of circles
// Parameters:
//   circle0 - [in]
//   circle11 - [in]
//   point0 - [out] first intersection point 
//   point1 - [out] second intersection point
// Returns:
//   0     No intersection
//   1     One intersection at point0
//   2     Two intersections at point0
//         and point1.
//	 3		 Circles are identical
ON_DECL
int ON_Intersect(
  const ON_Circle& c0,
  const ON_Circle& c1,
  ON_3dPoint& point0,
  ON_3dPoint& point1
);
 

// Description:
//   Intersect a pair of arcs, note either arc could be a full circle
// Parameters:
//   arc0 - [in]
//   arc1 - [in]
//   point0 - [out] first intersection point 
//   point1 - [out] second intersection point
// Returns:
//   0     No intersection
//   1     One intersection at point0
//   2     Two intersections at point0
//         and point1.
//	 3		 arcs are cocircular and overlap
ON_DECL
int ON_Intersect(
  const ON_Arc& arc0,
  const ON_Arc& arc1,
  ON_3dPoint& point0,
  ON_3dPoint& point1
);

// returns 0 = no, 1 = yes, 2 = points are coincident and on line
ON_DECL
int ON_ArePointsOnLine(
        int, // dimension of points
        bool, // is_rat = true if homogeneous rational
        int, // count = number of points
        int, // stride ( >= is_rat?(dim+1) :dim)
        const double*, // point array
        const ON_BoundingBox&, // if needed, use ON_GetBoundingBox(dim,is_rat,count,stride,point)
        const ON_Line&,
        double         // tolerance (if 0.0, a tolerance based on bounding box size is used)
        );

// returns 0 = no, 1 = yes, 2 = points are coincident and on line
ON_DECL
int ON_ArePointsOnPlane(
        int, // dimension of points
        bool, // is_rat = true if homogeneous rational
        int, // count = number of points
        int, // stride ( >= is_rat?(dim+1) :dim)
        const double*, // point array
        const ON_BoundingBox&, // if needed, use ON_GetBoundingBox(dim,is_rat,count,stride,point)
        const ON_Plane&,
        double         // tolerance (if 0.0, a tolerance based on bounding box size is used)
        );

/*
Description:
  Use the quotient rule to compute derivatives of a one parameter
  rational function F(t) = X(t)/W(t), where W is a scalar
  and F and X are vectors of dimension dim.
Parameters:
  dim - [in]
  der_count - [in] number of derivative (>=0)
  v_stride - [in] (>= dim+1)
  v - [in/out]
    v[] is an array of length (der_count+1)*v_stride.
    The input v[] array contains  derivatives of the numerator and
    denominator	functions in the order (X, W), (Xt, Wt), (Xtt, Wtt), ...
    In general, the (dim+1) coordinates of the d-th derivative 
    are in (v[n],...,v[n+dim]) where n = d*v_stride.
    In the output v[] array the derivatives of X are replaced with
    the derivatives of F and the derivatives of W are divided by
    w = v[dim].
Returns:
  True if input is valid; i.e., v[dim] != 0.
See Also:
  ON_EvaluateQuotientRule2
  ON_EvaluateQuotientRule3
*/
ON_DECL
bool ON_EvaluateQuotientRule( 
          int dim, 
          int der_count,
          int v_stride, 
          double *v 
          );

/*
Description:
  Use the quotient rule to compute partial derivatives of a two parameter
  rational function F(s,t) = X(s,t)/W(s,t), where W is a scalar
  and F and X are vectors of dimension dim.
Parameters:
  dim - [in]
  der_count - [in] number of derivative (>=0)
  v_stride - [in] (>= dim+1)
  v - [in/out]
    v[] is an array of length (der_count+2)*(der_count+1)*v_stride.
    The input array contains derivatives of the numerator and denominator
		functions in the order X, W, Xs, Ws, Xt, Wt, Xss, Wss, Xst, Wst, Xtt, Wtt, ...
    In general, the (i,j)-th derivatives are in the (dim+1) entries of v[]
		v[k], ..., answer[k+dim], where	k = ((i+j)*(i+j+1)/2 + j)*v_stride.
    In the output v[] array the derivatives of X are replaced with
    the derivatives of F and the derivatives of W are divided by
    w = v[dim].
Returns:
  True if input is valid; i.e., v[dim] != 0.
See Also:
  ON_EvaluateQuotientRule
  ON_EvaluateQuotientRule3
*/
ON_DECL
bool ON_EvaluateQuotientRule2( 
          int dim, 
          int der_count, 
          int v_stride, 
          double *v 
          );

/*
Description:
  Use the quotient rule to compute partial derivatives of a 3 parameter
  rational function F(r,s,t) = X(r,s,t)/W(r,s,t), where W is a scalar
  and F and X are vectors of dimension dim.
Parameters:
  dim - [in]
  der_count - [in] number of derivative (>=0)
  v_stride - [in] (>= dim+1)
  v - [in/out]
    v[] is an array of length 
    v_stride*(der_count+1)*(der_count+2)*(der_count+3)/6.
    The input v[] array contains  derivatives of the numerator and
    denominator	functions in the order (X, W), (Xr, Wr), (Xs, Ws),
    (Xt, Wt), (Xrr, Wrr), (Xrs, Wrs), (Xrt, Wrt), (Xss, Wss), 
    (Xst, Wst), (Xtt, Wtt), ...
    In general, the (dim+1) coordinates of the derivative 
    (Dr^i Ds^j Dt^k, i+j+k=d) are at v[n], ..., v[n+dim] where 
    n = v_stride*( d*(d+1)*(d+2)/6  +  (d-i)*(d-i+1)/2  +  k ).
    In the output v[] array the derivatives of X are replaced with
    the derivatives of F and the derivatives of W are divided by
    w = v[dim].
Returns:
  True if input is valid; i.e., v[dim] != 0.
See Also:
  ON_EvaluateQuotientRule
  ON_EvaluateQuotientRule2
*/
ON_DECL
bool ON_EvaluateQuotientRule3( 
          int dim, 
          int der_count, 
          int v_stride,
          double *v 
          );

ON_DECL
bool ON_GetPolylineLength(
        int,           // dimension of points
        bool,          // bIsRational true if points are homogeneous rational
        int,           // number of points
        int,           // stride between points
        const double*, // points
        double*        // length returned here
        );


/*
Description:
  Find the index of the point in the point_list that is closest to P.
Parameters:
  point_count - [in]
  point_list - [in]
  P - [in]
  closest_point_index - [out]
Returns:
  True if successful and *closest_point_index is set.
  False if input is not valid, in which case *closest_point_index
  is undefined.
*/
ON_DECL
bool ON_GetClosestPointInPointList( 
          int point_count,
          const ON_3dPoint* point_list,
          ON_3dPoint P,
          int* closest_point_index
          );

/*
Description:
  Test math library functions.
Parameters:
  function_index - [in]  Determines which math library function is called.

           1:    z = x+y
           2:    z = x-y
           3:    z = x*y
           4:    z = x/y
           5:    z = fabs(x)
           6:    z = exp(x)
           7:    z = log(x)
           8:    z = logb(x)
           9:    z = log10(x)
          10:    z = pow(x,y)
          11:    z = sqrt(x)
          12:    z = sin(x)
          13:    z = cos(x)
          14:    z = tan(x)
          15:    z = sinh(x)
          16:    z = cosh(x)
          17:    z = tanh(x)
          18:    z = asin(x)
          19:    z = acos(x)
          20:    z = atan(x)
          21:    z = atan2(y,x)
          22:    z = fmod(x,y)
          23:    z = modf(x,&y)
          24:    z = frexp(x,&y)

  double x - [in]
  double y - [in]
Returns:
  Returns the "z" value listed in the function_index parameter
  description.
Remarks:
  This function is used to test the results of class floating
  point functions.  It is primarily used to see what happens
  when opennurbs is used as a DLL and illegal operations are
  performed.
*/
ON_DECL
double ON_TestMathFunction( 
        int function_index, 
        double x, 
        double y 
        );

// If performance is important, then
// you are better off using ((b<a)?a:b)
ON_DECL double ON_Max(double a, double b);

// If performance is important, then
// you are better off using ((b<a)?a:b)
ON_DECL float ON_Max(float a, float b);

// If performance is important, then
// you are better off using ((b<a)?a:b)
ON_DECL int ON_Max(int a, int b);

// If performance is important, then
// you are better off using ((a<b)?a:b)
ON_DECL double ON_Min(double a, double b);

// If performance is important, then
// you are better off using ((a<b)?a:b)
ON_DECL float ON_Min(float a, float b);

// If performance is important, then
// you are better off using ((a<b)?a:b)
ON_DECL int ON_Min(int a, int b);

// Do not call ON_Round() in any opennurbs code, tl code
// or any other code that does critical calculations or
// when there is any possibility that x is invalid or
// fabs(x)>2147483647. Use floor(x+0.5) instead.
ON_DECL int ON_Round(double x);

/*
Description:
  Calculate the value of (1.0-t)*x + t*y so that, 
  if 0.0 <= t <= 1.0, then the result is between x and y and
  if x == y and t is a valid double, the result is x.
Returns:
  (1.0-t)*x + t*y
*/
ON_DECL double ON_LinearInterpolation(
  double t, 
  double x, 
  double y
  );

/*
Description:
  Find the equation of the parabola, ellipse or hyperbola 
  (non-degenerate conic) that passes through six distinct points.
Parameters:
  stride - [in] (>=2) 
    points array stride
  points2d - [in] (>=2) 
    i-th point is (points[i*stride],points[i*stride+1])
  conic - [out]
    Coefficients of the conic equation.
    The points on the conic satisfy the equation
      0 = conic[0]*x^2 + conic[1]*xy + conic[2]*y^2 
        + conic[3]*x + conic[4]*y + conic[5]
  max_pivot - [out] (can be null)
  min_pivot - [out] (can be null)
  zero_pivot - [out] (can be null)
    If there are some near duplicates in the input point set,
    the calculation is not stable.  If you want to get an
    estimate of the validity of the solution, then inspect
    the returned values.  max_pivot should around 1, 
    min_pivot should be > 1e-4 or so, and zero_pivot should
    be < 1e-10 or so.  If the returned pivots don't satisfy
    these conditions, then exercise caution when using the
    returned solution.
Returns:
  True if a there is an ellipse, parabola or hyperbola through the  
  six points.
  False if the input is invalid or the conic degenerate (the
  points lie on one or two lines).
  If false is returned, then conic[0]=...=conic[5] = 0 and
  *min_pivot = *max_pivot = *zero_pivot = 0.
*/
ON_DECL bool ON_GetConicEquationThrough6Points( 
        int stride, 
        const double* points2d, 
        double conic[6],
        double* max_pivot,
        double* min_pivot,
        double* zero_pivot
        );

/*
Description:
  Test a conic equation to see if it defines and ellipse. If so,
  return the center and axes of the ellipse.
Parameters:
  conic - [in]
    Coefficients of the conic equation.
    The points on the conic satisfy the equation
      0 = conic[0]*x^2 + conic[1]*xy + conic[2]*y^2 
        + conic[3]*x + conic[4]*y + conic[5]
  center - [out]
  major_axis - [out]
  minor_axis - [out]
  major_radius - [out]
  minor_radius - [out]
Returns:
  True if the conic is an ellipse and the center and axes were found.
  False if the conic is not an ellipse, in which case the input values
  of center, major_axis, minor_axis, major_radius, and minor_radius
  are not changed.
*/
ON_DECL bool ON_IsConicEquationAnEllipse( 
        const double conic[6], 
        ON_2dPoint& center, 
        ON_2dVector& major_axis, 
        ON_2dVector& minor_axis, 
        double* major_radius, 
        double* minor_radius
        );

/*
Description:
  Get the conic equation of an ellipse.
Parameters:
  a - [in] (a>0)
  b - [in] (b>0)
    a and b are the lengths of the axes. Either one
    may be largest and they can be equal.
  x0 - [in]
  y0 - [in]
    (x0,y0) is the enter of the ellipse.
  alpha - [in] (angle in radians)
    When alpha is 0, a corresponds to the x-axis and
    b corresponds to the y axis.  If alpha is non-zero
    it specifies the rotation of these axes.
  conic - [out]
    Coefficients of the conic equation.
    The points on the conic satisfy the equation
      0 = conic[0]*x^2 + conic[1]*xy + conic[2]*y^2 
        + conic[3]*x + conic[4]*y + conic[5]
  center - [out]
  major_axis - [out]
  minor_axis - [out]
  major_radius - [out]
  minor_radius - [out]
Remarks:
  Here is the way to evaluate a point on the ellipse:

          
          double t = ellipse parameter in radians;
          double x = a*cos(t);
          double y = b*sin(t);
          ON_2dPoint ellipse_point;
          ellipse_point.x = x0 + x*cos(alpha) + y*sin(alpha);
          ellipse_point.y = y0 - x*sin(alpha) + y*cos(alpha);

Returns:
  True if the input is valid and conic[] was filled in.
  Falis if the input is not valid.  In this case the values in conic[]
  are not changed.
*/
ON_DECL bool ON_GetEllipseConicEquation( 
      double a, double b, 
      double x0, double y0, 
      double alpha,
      double conic[6]
      );

/*
Description:
  Return the length of a 2d vector (x,y)
Returns:
 sqrt(x^2 + y^2) calculated in as precisely and safely as possible.
*/
ON_DECL double ON_Length2d( double x, double y );

/*
Description:
  Return the squared length of a 2d vector (x,y)
Returns:
 (x^2 + y^2)
*/
ON_DECL double ON_Length2dSquared(double x, double y);

/*
Description:
  Return the length of a 3d vector (x,y,z)
Returns:
 sqrt(x^2 + y^2 + z^2) calculated in as precisely and safely as possible.
*/
ON_DECL double ON_Length3d( double x, double y, double z );

/*
Description:
  Return the squared length of a 3d vector (x,y,z)
Returns:
 (x^2 + y^2 + z^2)
*/
ON_DECL double ON_Length3dSquared(double x, double y, double z);

/*
Description:
  Get the area of a 3d triangle.
Parameters:
  A - [in]
  B - [in]
  C - [in]
    Triangle corners
Returns:
  Area of a 3d triangle with corners at A, B, C.
*/
ON_DECL
double ON_TriangleArea3d(
  ON_3dPoint A,
  ON_3dPoint B, 
  ON_3dPoint C
  );

/*
Description:
  Get the area of a 2d triangle.
Parameters:
  A - [in]
  B - [in]
  C - [in]
    Triangle corners
Returns:
  Area of a 2d triangle with corners at A, B, C.
*/
ON_DECL
double ON_TriangleArea2d(
  ON_2dPoint A,
  ON_2dPoint B,
  ON_2dPoint C
  );


/*
Description:
  Convert a double x to the largest float f such that
  the mathematical value of f is at most the value of x.
Parameters:
  x - [in]
Returns
  The largest float f such that the mathematical value
  of f is at most the value of x.
*/
ON_DECL float ON_FloatFloor(double x);

/*
Description:
  Convert a double x to the smallest float f such that
  the mathematical value of f is at least the value of x.
Parameters:
  x - [in]
Returns
  The smallest float f such that the mathematical value
  of f is at least the value of x.
*/
ON_DECL float ON_FloatCeil(double x);

/*
Description:
  Determine if a polyline is convex.
Parameters:
  point_dim - [in]
    2 or 3
  point_count - [in]
  points - [in]
    If point_count >= 4 and the first and last points are equal, 
    then the zero length segment between those points is ignored.
  point_stride - [in]
    number of doubles between points (>=point_dim)
  bStrictlyConvex - [in]
    If false, colinear segments are considered convex.  
Returns
  True if the polyline is convex.
*/
ON_DECL bool ON_IsConvexPolyline(
  size_t point_dim,
  size_t point_count,
  const double* points,
  size_t point_stride,
  bool bStrictlyConvex
);

/*
Description:
  Determine if a polyline is convex.
Parameters:
  points - [in]
    If points.Count() >= 4 and the first and last points are equal, 
    then the zero length segment between those points is ignored.
  bStrictlyConvex - [in]
    If false, colinear segments are considered convex.  
Returns
  True if the polyline is convex.
*/
ON_DECL bool ON_IsConvexPolyline(
  const ON_SimpleArray<ON_3dPoint>& points,
  bool bStrictlyConvex
);

/// <summary>
/// 
/// </summary>
/// <param name="a"></param>
/// <param name="b"></param>
/// <returns>
/// If a &gt; 0 or b &gt; 0, then the greatest common divisor of a and b is returned (nonzero).
/// Note that if n &gt; 0, then gcd(0,n) = gcd(n,0) = n and gcd(1,n) = gcd(n,1) = 1.
/// If a = 0 and b = 0, then the greatest common divisor is not defined and 0 is returned.
/// </returns>
ON_DECL unsigned ON_GreatestCommonDivisor(
  unsigned a,
  unsigned b
);

/// <summary>
/// The least common multiple of a and b is (a/gcd)*(b/gcd)*(gcd), where gcd = greatest common divisor of and b.
/// </summary>
/// <param name="a"></param>
/// <param name="b"></param>
/// <returns>
/// If a $gt; 0 and b &gt; and the least common multiple of a and b &lt;= ON_UINT_MAX, then the
/// least common multiple of a and b is returned.
/// Otherwise 0 is returned.
/// </returns>
ON_DECL unsigned ON_LeastCommonMultiple(
  unsigned a,
  unsigned b
);


#endif