opennurbs_knot.cpp

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/* $NoKeywords: $ */
/*
//
// Copyright (c) 1993-2012 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>.
//
*/

#include "pcl/surface/3rdparty/opennurbs/opennurbs.h"

//
// Computes tolerance associated with a generic evaluation domain
//

double ON_DomainTolerance( double a, double b )
{
  if ( a == b )
    return 0.0;
  double tol = (fabs(a)+fabs(b)+fabs(a-b))* ON_SQRT_EPSILON;
  if ( tol <  ON_EPSILON )
    tol =  ON_EPSILON;
  return tol;
}

//
// Computes tolerance associated with knot[i]
//

double ON_KnotTolerance( int order, int cv_count, const double* knot, 
                                    int knot_index )
{
  const int knot_count = ON_KnotCount( order, cv_count );
  int i0, i1, j;
  double a, b, tol;
  i0 = knot_index-order+1;
  if ( i0 < 0 )
    i0 = 0;
  i1 = knot_index+order-1;
  if ( i1 >= knot_count )
    i1 = knot_count-1;
  for ( j = knot_index; j > i0; j-- ) {
    if ( knot[j] != knot[knot_index] )
      break;
  }
  a = fabs(knot[knot_index] - knot[j]);
  for ( j = knot_index; j < i1; j++ ) {
    if ( knot[j] != knot[knot_index] )
      break;
  }
  b = fabs(knot[knot_index] - knot[j]);
  tol = (a==0.0 && b==0.0) ? 0.0 : (a + b + fabs(knot[knot_index]))* ON_SQRT_EPSILON;
  return tol;
}

//
// Computes tolerance associated with span of a knot vector
//

double ON_SpanTolerance( int order, int cv_count, const double* knot, int span_index )
{
  const int i0 = span_index+order-2;
  return ON_DomainTolerance( knot[i0], knot[i0+1] );
}

//
// Computes number of knots in knot vector
//

int ON_KnotCount( int order, int cv_count )
{
  return (order+cv_count-2);
}

//
// Computes number of knots in knot vector
//

int ON_KnotMultiplicity(
          int order,          // order (>=2)
          int cv_count,       // cv_count (>=order)
          const double* knot, // knot[]
          int knot_index      // knot_index
          )
{
  int knot_count = order+cv_count-2;
  int km = 0;
  if ( knot && knot_index >= 0 && knot_index < knot_count ) {
    while(knot_index > 0 && knot[knot_index] == knot[knot_index-1])
      knot_index--;
    knot += knot_index;
    knot_count -= knot_index;
    km = 1;
    while ( km < knot_count && knot[0] == knot[km] )
      km++;
  }
  return km;
}

//
// Computes number of non-empty spans
//

int ON_KnotVectorSpanCount(
          int order,         // order (>=2)
          int cv_count,      // cv count
          const double* knot // knot[] array
          )
{
  if ( 0 == knot )
  {
    if ( 0 != order || 0 != cv_count )
    {
      ON_ERROR("NULL knot[] passed to ON_KnotVectorSpanCount.");
    }
    return 0;
  }
  int i, span_count = 0;
  for ( i = order-1; i < cv_count; i++ ) {
    if ( knot[i] > knot[i-1] )
      span_count++;
  }
  return span_count;
}

//
// Gets span vector from knot vector
//

bool ON_GetKnotVectorSpanVector(
          int order,          // order (>=2)
          int cv_count,       // cv count
          const double* knot, // knot[] array
          double* s           // s[] array
          )
{
  if ( 0 == knot || 0 == s )
  {
    if ( 0 != order || 0 != cv_count )
    {
      ON_ERROR("NULL knot[] or s[] passed to ON_KnotVectorSpanCount.");
      return false;
    }
    return true;
  }

  int i, span_count = 0;
  s[span_count++] = knot[order-2];
  for ( i = order-1; i < cv_count; i++ ) {
    if ( knot[i] > knot[i-1] )
      s[span_count++] = knot[i];
  }
  return (span_count>1) ? true : false;
}


//
// Computes span index for evaluation of parameter
//

int ON_NurbsSpanIndex(
          int order,          // (>=2)
          int cv_count,
          const double* knot, // knot[] array or length ON_KnotCount(order,cv_count)
          double t,           // evaluation parameter
          int side,           // side 0 = default, -1 = from below, +1 = from above
          int hint            // hint (or 0 if no hint available)
          )
{
  int j, len;
  
  // shift knot so that domain is knot[0] to knot[len]
  knot += (order-2);
  len = cv_count-order+2;
  
  // see if hint helps 
  if (hint > 0 && hint < len-1) {
    while(hint > 0 && knot[hint-1] == knot[hint]) hint--;
    if (hint > 0) {
      // have knot[hint-1] < knot[hint]
      if (t < knot[hint]) {
        len = hint+1;
        hint = 0;
      }
      else {
        if (side < 0 && t == knot[hint]) 
          hint--;
        knot += hint;
        len -= hint;
      }
    }
  }
  else
    hint = 0;
  
  j = ON_SearchMonotoneArray(knot,len,t);
  if (j < 0)
    j = 0;
  else if (j >= len-1)
    j = len-2;
  else if (side < 0) {
    // if user wants limit from below and t = an internal knot,
    // back up to prevous span
    while(j > 0 && t == knot[j]) 
      j--;
  }
  return (j + hint);
}


int ON_NextNurbsSpanIndex( int order, int cv_count, const double* knot, int span_index )

/*
Get index of next non-degenerate NURBS span
 
INPUT:
  order, cv_count, knot
    knot vector
  span_index
    current span index ( >= 0 and <= cv_count-order )
OUTPUT:
  i = ON_NextNurbsSpanIndex()
    i>=0: successful - the index of the next span or
                       cv_count-order if the input value
                       was cv_count-or
                       knot[i+order-2] < knot[i+order-1]
   <0: failure
   
COMMENTS:
  The first span in a NURBS has span_index = 0.  The last span in a NURBS
  has span_index = cv_count-order.
  A span of a degree d NURBS is defined by d+1 CVs and 2*d knots.  For a
  given span_index, the associated knots and CVs are
    {knot[span_index], ..., knot[span_index+2*d-1]}
  and
    {CV[span_index], ..., CV[span_index + d]}
  The domain of the span is 
    [ knot[span_index+order-2], knot[span_index+order-1] ].

EXAMPLE:
  // print the values of all distinct knots in a NURBS's domain
  int span_index = 0;
  int next_span_index = 0;
  for (;;) {
    printf( "knot[%2d] = %g\n", span_index+order-2, knot[span_index+order-2] );
    next_span_index =ON_NextNurbsSpanIndex( order, cv_count, knot, span_index );
    if ( next_span_index < 0 )
      break; // illegal input
    if ( next_span_index == span_index ) {
      // end of the domain
      printf( "knot[%2d] = %g\n", cv_count-1, knot[cv_count-1] );
      break;
    }
    next_span_index = span_index;
  }

  */

{

  if (span_index < 0 || span_index > cv_count-order || !knot)
    return -1;

  if ( span_index < cv_count-order ) {
    do {
      span_index++;
    }
    while ( span_index < cv_count-order && 
            knot[span_index+order-2] == knot[span_index+order-1] );
  }
  return span_index;
}


int ON_GetSpanIndices(int order, 
                            int cv_count, 
                            const double* knot, 
                            int* span_indices)

/* span_indices should have size greater than the number of 
   spans (cv_count is big enough).

  returns span count.
  fills in span_indices with index of last in each bunch of multiple knots at 
  start of span, and first in buch at end of nurb.


  */
{
  int span_index, next_span_index, j;

  span_index = -1;
  next_span_index = 0;
  j = 0;
  while (span_index != next_span_index) {
    span_index = next_span_index;
    span_indices[j] = span_index + order - 2;
    next_span_index = ON_NextNurbsSpanIndex(order, cv_count, knot, span_index);
    if (next_span_index < 0) 
      return next_span_index;
    j++;
  } 
  
  span_indices[j] = span_index + order - 1;

  return j;
}


//
// Computes value for superfluous knot used in systems like OpenGL and 3dsMax
//

double ON_SuperfluousKnot( 
                    int order, int cv_count, const double* knot,
                    int end )
{
  double k;
  const int knot_count = order+cv_count-2;
  // gets superfluous knot for translation to other formats
  k = knot[(end) ? knot_count-1 : 0];
  if (order > 2 && cv_count >= 2*order - 2 && cv_count >= 6 ) {
    // check for non-clamped knots
    if (end) {
      if ( knot[cv_count-1] < knot[knot_count-1] )
        k += (knot[order+1] - knot[order]);
    }
    else {
      if ( knot[0] < knot[order-2] )
        k -= (knot[cv_count-order+1] - knot[cv_count-order]);
    }
  }
  return k;
}

//
// Used to determine when a knot vector is periodic
//

bool ON_IsKnotVectorPeriodic(
       int order,
       int cv_count,
       const double* knot
       )

{
  double tol;
  const double* k1;
  int i;

  if ( order < 2 || cv_count < order || !knot ) {
    ON_ERROR("ON_IsKnotVectorPeriodic(): illegal input");
    return false;
  }
  
  if ( order == 2 )
    return false; // convention is that degree 1 curves cannot be periodic.

  if (order <= 4) {
    if (cv_count < order+2) 
      return false;
  }
  else if ( cv_count < 2*order-2 ) {
    return false;
  }

  tol = fabs(knot[order-1] - knot[order-3])* ON_SQRT_EPSILON;
  if (tol < fabs(knot[cv_count-1] - knot[order-2])* ON_SQRT_EPSILON) 
    tol = fabs(knot[cv_count-1] - knot[order-2])* ON_SQRT_EPSILON;
  k1 = knot+cv_count-order+1;
  i = 2*(order-2);
  while(i--) {
    if (fabs(knot[1] - knot[0] + k1[0] - k1[1]) > tol)
      return false;
    knot++; k1++;
  }
  return true;
}

//
// Used to determine when a knot vector is clamped
//

bool ON_IsKnotVectorClamped(
       int order,
       int cv_count,
       const double* knot,
       int end // (default = 2) 0 = left end, 1 = right end, 2 = both
       )

{
  if ( order <= 1 || cv_count < order || !knot || end < 0 || end > 2 )
    return false;
  bool rc = true;
  if ( (end == 0 || end == 2) && knot[0] != knot[order-2] )
    rc = false;
  if ( (end == 1 || end == 2) && knot[cv_count-1] != knot[order+cv_count-3] )
    rc = false;
  return rc;
}

bool ON_IsKnotVectorUniform(
          int order,
          int cv_count,
          const double* knot 
          )
{
  bool rc = (order >= 2 && cv_count >= order && 0 != knot);
  if (rc)
  {
    const double delta = knot[order-1] - knot[order-2];
    const double delta_tol = ON_SQRT_EPSILON*delta;
    int i0, i1;
    double d;
    if ( ON_IsKnotVectorClamped(order,cv_count,knot) )
    {
      i0 = order;
      i1 = cv_count;
    }
    else
    {
      i0 = 1;
      i1 = ON_KnotCount(order,cv_count);
    }
    for (/*empty*/; i0 < i1 && rc; i0++ )
    {
      d = knot[i0] - knot[i0-1];
      if ( fabs(d - delta) > delta_tol )
        rc = false;
    }
  }
  return rc;
}

//
// Used to determine properties of knot vector
//

bool ON_KnotVectorHasBezierSpans(
          int order,          // order (>=2)
          int cv_count,       // cv count
          const double* knot  // knot[] array
          )
{
  int knot_count = ON_KnotCount( order, cv_count );
  if ( knot_count < 2 )
    return false;
  int span_count = ON_KnotVectorSpanCount( order, cv_count, knot );
  if ( span_count < 1 )
    return false;
  if ( order >= 2 && 
       cv_count >= order && 
       knot_count == (span_count+1)*(order-1) && 
       knot[0] == knot[order-2] && knot[cv_count-1] == knot[knot_count-1])
    return true;
  return false;
}

//
// Used to determine properties of knot vector
//

ON::knot_style ON_KnotVectorStyle( 
       int order,
       int cv_count,
       const double* knot
       )
{
  ON::knot_style s = ON::unknown_knot_style;
  if ( order >= 2 && cv_count >= order && knot && knot[order-2] < knot[cv_count-1] ) {
    const int knot_count = order+cv_count-2;
    const double delta = 0.5*((knot[order-1] - knot[order-2]) + (knot[cv_count-1] - knot[cv_count-2]));
    const double ktol = delta*1.0e-6;
    int i;
    if ( ON_IsKnotVectorClamped( order, cv_count, knot ) ) {
      if ( order == cv_count ) {
        s = ON::piecewise_bezier_knots;
      }
      else {
        s = ON::clamped_end_knots;
        for ( i = order-1; i <= cv_count-1; i++ ) {
          if ( fabs(knot[i] - knot[i-1] - delta) > ktol ) {
            break;
          }
        }
        if ( i >= cv_count ) {
          s = ON::quasi_uniform_knots;
        }
        else {
          const int degree = order-1;
          for ( i = order-1; i < cv_count-1; i += degree ) {
            if ( knot[i] != knot[i+degree-1] )
              break; 
          }
          if ( i >= cv_count-1 )
            s = ON::piecewise_bezier_knots;
        }
      }
    }
    else {
      // check for uniform knots
      s = ON::non_uniform_knots;
      for ( i = 1; i < knot_count; i++ ) {
        if ( fabs(knot[i] - knot[i-1] - delta) > ktol ) {
          break;
        }
      }
      if ( i >= knot_count )
        s = ON::uniform_knots; 
    }
  }
  return s;
}


//
// Used to set the domain of a knot vector
//

bool ON_SetKnotVectorDomain( int order, int cv_count, double* knot, double t0, double t1 )
{
  bool rc = false;
  if ( order < 2 || cv_count < order || 0 == knot || t0 >= t1 || !ON_IsValid(t0) || !ON_IsValid(t1) )
  {
    ON_ERROR("ON_SetKnotVectorDomain - invalid input");
  }
  else if (    knot[order-2] >= knot[cv_count-1] 
            || !ON_IsValid(knot[order-2]) 
            || !ON_IsValid(knot[cv_count-2]) )
  {
    ON_ERROR("ON_SetKnotVectorDomain - invalid input knot vector");
  }
  else
  {
    const ON_Interval oldd(knot[order-2],knot[cv_count-1]);
    const ON_Interval newd(t0,t1);
    if ( oldd != newd )
    {
      int i, knot_count = ON_KnotCount(order,cv_count);
      for ( i = 0; i < knot_count; i++ )
      {
        knot[i] = newd.ParameterAt(oldd.NormalizedParameterAt(knot[i]));
      }
    }
    rc = true;
  }
  return rc;
}


//
// Used to get the domain of a knot vector
//

bool ON_GetKnotVectorDomain(
       int order,
       int cv_count,
       const double* knot,
       double* k0, double* k1
       )
{
  if ( order < 2 || cv_count < order || knot == NULL )
    return false;
  if ( k0 )
    *k0 = knot[order-2];
  if ( k1 )
    *k1 = knot[cv_count-1];
  return true;
}

//
// Used to reverse knot vectors
//

bool ON_ReverseKnotVector(
       int order,
       int cv_count,
       double* knot
       )
{
  if ( order < 2 || cv_count < order || knot == NULL )
    return false;
  const int knot_count = (order+cv_count-2);
  double t;
  int i, j;
  for ( i = 0, j = knot_count-1; i <= j; i++, j-- ) {
    t = knot[i];
    knot[i] = -knot[j];
    knot[j] = -t;
  }
  return true;
}

//
// Used to compare knot vectors
//

int ON_CompareKnotVector( // returns 
                                      // -1: first < second
                                      //  0: first == second
                                      // +1: first > second
          int orderA,
          int cv_countA,
          const double* knotA,
          int orderB,
          int cv_countB,
          const double* knotB
          )
{
  const int knot_count = ON_KnotCount(orderA,cv_countA);
  int i;
  double a, b, atol, btol, ktol, tol;
  if ( orderA < orderB )
    return -1;
  if ( orderA > orderB )
    return 1;
  if ( cv_countA < cv_countB )
    return -1;
  if ( cv_countA > cv_countB )
    return 1;

  if ( !ON_GetKnotVectorDomain( orderA, cv_countA, knotA, &a, &b ) )
    return -1;
  atol = ON_DomainTolerance( a, b );
  if ( !ON_GetKnotVectorDomain( orderA, cv_countA, knotA, &a, &b ) )
    return 1;
  btol = ON_DomainTolerance( a, b );
  tol = (atol <= btol) ? atol : btol;

  for ( i = 0; i < knot_count; i++ ) {
    a = knotA[i];
    b = knotB[i];
    if ( a == b )
      continue;
    if ( a < b-tol )
      return -1;
    if ( b < a-tol )
      return 1;
    atol = ON_KnotTolerance( orderA, cv_countA, knotA, i );
    btol = ON_KnotTolerance( orderB, cv_countB, knotB, i );
    ktol = (atol <= btol) ? atol : btol;
    if ( a < b-ktol )
      return -1;
    if ( b < a-ktol )
      return 1;
  }
  return 0;
}

//
// Used to validate knot vectors
//

static bool ON_KnotVectorIsNotValid(bool bSilentError)
{
  return bSilentError ? false : ON_IsNotValid(); // <-- good place for a breakpoint
}

bool ON_IsValidKnotVector( int order, int cv_count, const double* knot, ON_TextLog* text_logx )
{
  // If low bit of text_log pointer is 1, then ON_Error is not called when the
  // knot vector is invalid.
  const ON__INT_PTR lowbit = 1;
  const ON__INT_PTR hightbits = ~lowbit;
  bool bSilentError = ( 0 != (lowbit & ((ON__INT_PTR)text_logx)) );
  ON_TextLog* text_log = (ON_TextLog*)(((ON__INT_PTR)text_logx) & hightbits);

  const double *k0, *k1;
  int i;
  if ( order < 2 )
  {
    if ( text_log )
    {
      text_log->Print("Knot vector order = %d (should be >= 2 )\n",order);
    }
    return ON_KnotVectorIsNotValid(bSilentError);
  }
  if ( cv_count < order )
  {
    if ( text_log )
    {
      text_log->Print("Knot vector cv_count = %d (should be >= order=%d )\n",cv_count,order);
    }
    return ON_KnotVectorIsNotValid(bSilentError);
  }
  if ( knot == NULL )
  {
    if ( text_log )
    {
      text_log->Print("Knot vector knot array = NULL.\n");
    }
    return ON_KnotVectorIsNotValid(bSilentError);
  }

  for ( i = 0; i < cv_count+order-2; i++ )
  {
    if ( !ON_IsValid(knot[i]) )
    {
      if ( text_log )
      {
        text_log->Print("Knot vector knot[%d]=%g is not valid.\n",i,knot[i]);
      }
      return ON_KnotVectorIsNotValid(bSilentError);
    }
  }

  if ( !(knot[order-2] < knot[order-1]) )
  {
    if ( text_log )
    {
      text_log->Print("Knot vector order=%d and knot[%d]=%g >= knot[%d]=%g (should have knot[order-2] < knot[order-1]).\n",
                       order,order-2,knot[order-2],order-1,knot[order-1]);
    }
    return ON_KnotVectorIsNotValid(bSilentError);
  }
  if ( !(knot[cv_count-2] < knot[cv_count-1]) )
  {
    if ( text_log )
    {
      text_log->Print("Knot vector cv_count=%d and knot[%d]=%g >= knot[%d]=%g (should have knot[cv_count-2] < knot[cv_count-1]).\n",
                       cv_count,cv_count-2,knot[cv_count-2],cv_count-1,knot[cv_count-1]);
    }
    return ON_KnotVectorIsNotValid(bSilentError);
  }

  // entire array must be monotone increasing
  k0 = knot;
  k1 = knot+1;
  i = order + cv_count - 3;
  while (i--) {
    if ( !(*k1 >= *k0) )
    {
      if ( text_log )
      {
        text_log->Print("Knot vector must be increasing but knot[%d]=%g > knot[%d]=%g\n",
                         order+cv_count-4-i, *k0, order+cv_count-3-i, *k1 );
      }
      return ON_KnotVectorIsNotValid(bSilentError);
    }
    k0++;
    k1++;
  }

  // must have knot[i+order-1] > knot[i]
  k0 = knot;
  k1 = knot + order - 1;
  i = cv_count-1;
  while(i--) {
    if ( !(*k1 > *k0) )
    {
      if ( text_log )
      {
        text_log->Print("Knot vector order = %d but knot[%d]=%g >= knot[%d]=%g\n",
                         order, cv_count-2-i, *k0, cv_count-1-i, *k1 );
      }
      return ON_KnotVectorIsNotValid(bSilentError);
    }
    k0++;
    k1++;
  }

  return true;
}


bool ON_ClampKnotVector(
          int order,          // order (>=2)
          int cv_count,       // cv count
          double* knot,       // knot[] array
          int end             // 0 = clamp start, 1 = clamp end, 2 = clamp both ends
          )
{
  // sets initial/final order-2 knot values to match knot[order-2]/knot[cv_count-1]
  bool rc = false;
  int i, i0;
  if ( knot && order >= 2 && cv_count >= order ) {
    if ( end == 0 || end == 2 ) {
      i0 = order-2;
      for ( i = 0; i < i0; i++ ) {
        knot[i] = knot[i0];
      }
      rc = true;
    }
    if ( end == 1 || end == 2 ) {
      const int knot_count = ON_KnotCount(order,cv_count);
      i0 = cv_count-1;
      for ( i = i0+1; i < knot_count; i++ ) {
        knot[i] = knot[i0];
      }
      rc = true;
    }
  }
  return rc;
}


bool ON_MakeKnotVectorPeriodic(
          int order,          // order (>=2)
          int cv_count,       // cv count (>= (order>=4) ? 2*(order-1) : 5)
          double* knot        // knot[] array
          )
{
  double *k0, *k1;
  int i;
  
  if ( order < 2 || cv_count < order || !knot ) {
    ON_ERROR("ON_MakePeriodicKnotVector(): illegal input");
    return false;
  }

  switch(order) {
  case 2:
    if ( cv_count < 4 ) {
      ON_ERROR("ON_MakePeriodicKnotVector(): illegal input degree=1, cv_count<4");
      return false;
    }
    else {
      return true;
    }
    break;
  case 3:
    if ( cv_count < 4 ) {
      ON_ERROR("ON_MakePeriodicKnotVector(): illegal input degree=2, cv_count<5");
      return false;
    }
    break;
  default:
    if ( cv_count < 2*order-2 ) {
      ON_ERROR("ON_MakePeriodicKnotVector(): illegal input degree>=3, cv_count<2*degree");
      return false;
    }
    break;
  }

  k0 = knot + order-2;
  k1 = knot + cv_count-1;
  i = order-2;
  while (i--) {
    k1[1] = k0[1]-k0[0]+k1[0];
    k0++; k1++;
  }
  k0 = knot + order-2;
  k1 = knot + cv_count-1;
  i = order-2;
  while (i--) {
    k0[-1] = k1[-1] - k1[0] + k0[0];
    k0--; 
    k1--;
  }
  return true;
}

ON_DECL
bool ON_MakeClampedUniformKnotVector(
          int order,
          int cv_count,
          double* knot,
          double delta
          )
{
  bool rc = false;
  if ( order >= 2 && cv_count >= order && knot != NULL && delta > 0.0 )
  {
    double k;
    int i;
    for ( i = order-2, k = 0.0; i < cv_count; i++, k += delta )
      knot[i] = k;
    ON_ClampKnotVector(order,cv_count,knot,2);
    rc = true;
  }
  return rc;
}

// Description:
//   Fill in knot values for a clamped uniform knot
//   vector.
// Parameters:
//   order - [in] (>=2) order (degree+1) of the NURBS
//   cv_count - [in] (>=order) total number of control points
//       in the NURBS.
//   knot - [in/out] Input is an array with room for 
//       ON_KnotCount(order,cv_count) doubles.  Output is
//       a periodic uniform knot vector with domain
//       (0, (1+cv_count-order)*delta).
//   delta - [in] (>0, default=1.0) spacing between knots.
// Returns:
//   true if successful
ON_DECL
bool ON_MakePeriodicUniformKnotVector(
          int order,
          int cv_count,
          double* knot,
          double delta
          )
{
  bool rc = false;
  if ( order >= 2 && cv_count >= order && knot != NULL && delta > 0.0 )
  {
    double k = 0.0;
    int i, knot_count = ON_KnotCount(order,cv_count);
    for ( i = order-2, k = 0.0; i < knot_count; i++, k += delta )
      knot[i] = k;
    for ( i = order-3, k = -delta; i >= 0; i--, k -= delta )
      knot[i] = k;
    rc = true;
  }
  return rc;
}



double ON_GrevilleAbcissa( // get Greville abcissa
          int order,          // order (>=2)
          const double* knot  // knot[order-1] array
          )
{
  double g=0.0;
  if ( order <= 2 || knot[0] == knot[order-2]) {
    g = knot[0]; // degree = 1 or fully multiple knot
  }
  else {
    // g = (knot[i]+...+knot[i+degree-1])/degree
    order--;
    const double k = knot[order/2];
    const double tol = (knot[order-1]-knot[0]);
    const double d = 1.0/((double)order);
    while ( order--) {
      g += *knot++;
    }
    g *= d;
    if ( fabs(g-k) <= (fabs(g)+tol)* ON_SQRT_EPSILON )
      g = k; // sets g to exact value when knot vector is uniform
  }
  return g;
}


bool ON_GetGrevilleAbcissae( // get Greville abcissa from knots
          int order,          // order (>=2)
          int cv_count,       // cv count (>=order)
          const double* knot, // knot[] array
          bool bPeriodic,
          double* g           // has length cv_count in non-periodic case
                              // and length cv_count-order+1 in periodic case
          )
{
  // Grevielle abscissae for a given knot vector
  double x, t0; 
  int gi, periodic_check;

  if ( order < 2 || cv_count < order || !knot || !g )
    return false;
  
  const int g_count = (bPeriodic) ? cv_count-order+1 : cv_count;
  
  if (order == 2) {
    // g[i] = knot[i] in degree 1 case
    memcpy( g, knot, g_count*sizeof(*g) );
  }    
  else {
    // g = (knot[i]+...+knot[i+degree-1])/degree
    t0 = knot[order-2];
    gi = 0;
    periodic_check = (bPeriodic) ? order-2 : 0;
    while (gi < g_count) {
      x = ON_GrevilleAbcissa( order, knot++ );
      if ( periodic_check ) {
        periodic_check--;
        if ( x < t0 )
          continue;
      }
      g[gi++] = x;
    }
  }
  
  return true;
}


bool ON_GetGrevilleKnotVector( // get knots from Greville abcissa
          int g_stride,     
          const double *g,  // if not periodic, g[cv_count], 
                            // if periodic, g[cv_count-order+2]
                            // usually, g[0] = 0, g[i] = |P[i]-P[i-1]|^q
          bool bPeriodic,
          int order, 
          int cv_count, 
          double* knot
          )
{
  bool rc = false;
  double* p = NULL;
  int ki, knot_count, g_count, gi, j, degree;
  double k, dd;

  if ( g_stride < 1 || !g || !knot || order < 2 || cv_count < order )
    return false;
  if ( bPeriodic && order == 2 )
    return false;
  if ( bPeriodic && cv_count - order + 2 < 3 )
    return false;

  degree = order-1;
  if ( degree == 1 ) {
    for ( j = 0; j < cv_count; j++ ) {
      knot[j] = g[j*g_stride];
    }
    return true;
  }
  dd = 1.0/degree;
  knot_count = ON_KnotCount( order, cv_count );
  g_count = (bPeriodic) ? cv_count-order+2 : cv_count;

  if ( bPeriodic ) {
    int half_degree = (degree%2) ? degree/2 : 0;
    // step 1: set p[] = fully periodic list of abcissa
    p = (double*)onmalloc((g_count + 2*degree)*sizeof(*p));
    for ( j = 0, gi = g_count-order; j < degree; j++, gi++ ) {
      p[j] = g[0] - g[g_count-1] + g[gi];
    }
    for ( gi = 0, j = degree; gi < g_count; gi++, j++ ) {
      p[j] = g[gi];
    }
    for ( j = g_count+degree, gi = 1; j < g_count+2*degree; j++, gi++ ) {
      p[j] = g[g_count-1] - g[0] + g[gi];
    }

    // step 2: set new p[i] = old (p[i] + ... + p[i+degree-1]) / degree
    for ( j = 0; j < g_count+order; j++ ) {
      k = p[j];
      for ( ki = 1; ki < degree; ki++ ) 
        k += p[j+ki];
      k *= dd;
      if ( half_degree ) {
        // if g[]'s are uniform and degree is odd, then knots = g[]'s
        if ( fabs(k-p[j+half_degree]) <=  ON_SQRT_EPSILON*(p[j+degree-1]-p[j]) )
          k = p[j+half_degree];
      }
      p[j] = k;
    }

    // step 3: determine where g[0] maximizes NURBS basis functions
    {
      double* B = (double*)alloca(order*order*sizeof(B[0]));
      double maxB  = 0.0;
      int maxBj = 0;
      for ( j = 0; j < 2*degree; j++ ) {
        if ( g[0] > p[j+degree] )
          continue;
        if ( g[0] < p[j+degree-1] )
          break;
        ON_EvaluateNurbsBasis( order, p+j, g[0], B );
        if ( B[0] > maxB ) {
          maxB  = B[0];
          maxBj = j;
        }
      }
      memcpy( knot, &p[maxBj], knot_count*sizeof(*knot) );
    }

    rc = ON_MakeKnotVectorPeriodic( order, cv_count, knot );
  }
  else {
    // clamped knots
    rc = true;
    if ( g > knot && g < knot+knot_count ) {
      p = (double*)onmalloc(cv_count*sizeof(*p));
      for( j = 0; j < cv_count; j++ ) {
        p[j] = g[j*g_stride];
      }
      g = p;
      g_stride = 1;
    }
    for ( ki = 0; ki < degree; ki++ ) {
      knot[ki] = g[0];
    }
    for ( ki = degree, gi = 1; ki < cv_count; ki++, gi++ ) {
      k = 0.0;
      for ( j = 0; j < degree; j++ ) {
        k += g[(gi+j)*g_stride];
      }
      knot[ki] = k*dd;
      if ( knot[ki] < knot[ki-1] || knot[ki] <= knot[ki-degree] ) {
        rc = false;
      }
    }
    for ( ki = cv_count-1; ki < knot_count; ki++ ) {
      knot[ki] = g[(cv_count-1)*g_stride];
    }
  }

  if (p)
    onfree(p);
  return rc;
}

bool ON_ClampKnotVector(
        int cv_dim,   // dimension of cv's = ( = dim+1 for rational cvs )
        int order, 
        int cv_count,
        int cv_stride, 
        double* cv,   // NULL or cv array with room for at least knot_multiplicity new cvs
        double* knot, // knot array with room for at least knot_multiplicity new knots
        int end       // 0 = clamp start, 1 = clamp end, 2 = clamp both ends
        )
{
  // sets initial/final order-2 knot values to match knot[order-2]/knot[cv_count-1]
  bool rc = false;
  int i, i0;

  if ( knot && order >= 2 && cv_count >= order ) {
    if ( end == 0 || end == 2 ) {
      if ( cv ) {
        ON_EvaluateNurbsDeBoor(cv_dim,order,cv_stride,cv,knot,1,0.0,knot[order-2]);
      }
      i0 = order-2;
      for (i = 0; i < i0; i++)
        knot[i] = knot[i0];
      rc = true;
    }
    if ( end == 1 || end == 2 ) {
      i0 = cv_count-order;
      knot += i0;
      if ( cv ) {
        cv += i0*cv_stride;
        ON_EvaluateNurbsDeBoor(cv_dim,order,cv_stride,cv,knot,-1,0.0,knot[order-1]);
      }
      i0 = order-1;
      for (i = 2*order-3; i > i0; i--)
        knot[i] = knot[i0];
      rc = true;
    }
  }
  return rc;
}


static bool ON_InsertSingleKnot( int cv_dim, int order, 
                             int cv_stride, 
                             double *cv,   // NULL or array of length at least order*cv_stride+cv_dim
                             double *knot, // array of length at least 2*order-1 and existing knot values in
                                           // knot[0], ..., knot[2*order-3]
                             double knot_value // knot[order-2] <= knot_value < knot[order-1]
                                               // and knot[0] < knot_vale
                             )
{
  double alpha0, alpha1;
  double *k0, *k1, *prev_cv;
  int    i, d, cv_inc, degree;

  if ( order < 2 || !knot || knot_value < knot[order-2] || knot[order-1] <= knot_value ) {
    ON_ERROR( "ON_InsertSingleKnot() - illegal knot input" );
    return false;
  }

  if ( cv ) {
    if ( cv_dim < 1 || cv_stride < cv_dim  ) {
      ON_ERROR( "ON_InsertSingleKnot() - illegal cv input" );
      return false;
    }
  }

  degree = order-1;

  // move last degree many knots over one spot
  k1 = knot + 2*degree;
  k0 = k1-1;
  i = degree;
  while (i--) 
    *k1-- = *k0--;

  // insert new knot value 
  *k1 = knot_value;

  if ( cv ) {
    // move last cv over one spot
    memcpy( cv+cv_dim*order, cv+cv_dim*degree, cv_dim*sizeof(*cv) );

    // compute new cv values
    k0 = knot + degree-1;
    k1 = k0 + order;
    cv += order*cv_stride;
    prev_cv = cv - cv_stride;
    cv_inc = cv_stride - cv_dim;
    i = degree;
    if (knot_value - *k0 <= *k1 - knot_value) {
      while (i--) {
        alpha1 = (knot_value - *k0)/(*k1 - *k0);
        alpha0 = 1.0 - alpha1;
        k0--; k1--;
        cv -= cv_inc;
        prev_cv -= cv_inc;
        d = cv_dim;
        while (d--) {
          --cv;
          --prev_cv;
          *cv = *cv * alpha1 + *prev_cv * alpha0;
        }
      }
    }
    else {
      while (i--) {
        alpha0 = (*k1 - knot_value)/(*k1 - *k0);
        alpha1 = 1.0 - alpha0;
        k0--; k1--;
        cv -= cv_inc;
        prev_cv -= cv_inc;
        d = cv_dim;
        while (d--) {
          --cv;
          --prev_cv;
          *cv = *cv * alpha1 + *prev_cv * alpha0;
        }
      }
    }
  }
    
  return true;
}

int ON_InsertKnot( 
        double knot_value, 
        int knot_multiplicity, 
        int cv_dim,   // dimension of cv's = ( = dim+1 for rational cvs )
        int order, 
        int cv_count,
        int cv_stride, 
        double* cv,   // NULL or cv array with room for at least knot_multiplicity new cvs
        double* knot, // knot array with room for at least knot_multiplicity new knots
        int* hint     // optional hint about where to search for span to add knots to
                      // pass NULL if no hint is available
        )
{
  int rc = 0; // return code = number of knots added

  if ( order < 2 || cv_count < order || !knot ) 
  {
    ON_ERROR("ON_InsertKnot(): illegal input" );
    return 0;
  }

  if ( cv ) 
  {
    if ( cv_dim < 1 || cv_stride < cv_dim ) 
    {
      ON_ERROR("ON_InsertKnot(): illegal input" );
      return 0;
    }
  }

  if ( knot_multiplicity >= order ) 
  {
    ON_ERROR("ON_InsertKnot(): requested knot_multiplicity > degree" );
    return 0;
  }

  // shift knot vector and cv array so that knot_value lies in first span
  int span_index = ON_NurbsSpanIndex( order, cv_count, knot, knot_value, 1, hint?*hint:0 );
  knot += span_index;
  if ( cv )
    cv += (span_index*cv_stride);
  cv_count -= span_index;

  const double knot_tolerance = ON_SpanTolerance( order, cv_count, knot, 0 );

  // check that knot_value is interior to NURBS domain
  if ( span_index == 0 ) 
  {
    if ( knot_value < knot[order-1] ) 
    {
      if ( knot_value <= knot[order-2] + knot_tolerance ) 
      {
        ON_ERROR("ON_InsertKnot(): requested knot_value at start of NURBS domain" );
        return 0;
      }
    }
  }
  if ( span_index == cv_count-order ) 
  {
    if ( knot_value > knot[order-2] && knot_value >= knot[order-1] - knot_tolerance ) 
    {
      ON_ERROR("ON_InsertKnot(): requested knot_value at end of NURBS domain" );
      return 0;
    }
  }

  // if knot_value is nearly equal to an existing knot, make it exactly equal
  if ( knot_value <= 0.5*(knot[order-2]+knot[order-1]) && fabs( knot_value - knot[order-2] ) <= knot_tolerance ) {
    knot_value = knot[order-2];
  }
  else if ( fabs( knot_value - knot[order-1] ) <= knot_tolerance ) {
    knot_value = knot[order-1];
  }

  const int degree = order-1;

  // set m = number of knots to add
  int m = 0; 
  int j;
  if ( knot_value == knot[order-2] ) {
    for ( j = order-2; m < knot_multiplicity && knot[j-m] == knot_value; m++ )
      ; // empty for
  }
  else if ( knot_value == knot[order-1] ) {
    for ( j = order-1; m < knot_multiplicity && knot[j+m] == knot_value; m++ )
      ; // empty for
  }
  m = knot_multiplicity - m;
  if ( hint )
    *hint = span_index+m;

  if ( m <= 0 )
    return 0; // no knots need to be added

  double* new_knot = (double*)onmalloc( ((2*degree+m) + (order+m)*cv_dim)*sizeof(*new_knot) );
  if ( !new_knot ) {
    ON_ERROR("ON_InsertKnot(): out of memory");
    return 0;
  }
  double* new_cv = 0;
  memcpy( new_knot, knot, 2*degree*sizeof(*new_knot) );
  if ( cv ) {
    new_cv = new_knot + (2*degree+m);
    for ( j = 0; j < order; j++ ) {
      memcpy( new_cv + j*cv_dim, cv + j*cv_stride, cv_dim*sizeof(*new_cv) );
    }
  }

  // add m more knots at knot_value
  rc = 0;
  while (m>0) {
    if ( !ON_InsertSingleKnot(cv_dim,order,cv_dim,new_cv,new_knot,knot_value) )
      break;
    m--;
    if ( new_cv )
      new_cv += cv_stride; 
    new_knot++; 
    rc++;
  }
  new_knot -= rc;
  new_cv -= rc*cv_stride;

  if ( rc > 0 ) {
    // make room for rc many new knots
    int i0 = ON_KnotCount( order, cv_count ) - 1; // knot[i0] = last input knot
    int i1 = i0 + rc;
    int j  = (cv_count-order);
    while (j--)
      knot[i1--] = knot[i0--];
    
    // update knot vector
    memcpy ( knot+degree, new_knot+degree, (degree+rc)*sizeof(*new_knot) );

    if ( cv ) {
      // make room for rc many new CVs
      i0 = (cv_count-1)*cv_stride;             // cv[i0] = last coord of last input cv */
      i1 = i0 + rc*cv_stride;
      j = cv_count-order;
      while (j--) {
        memcpy( cv+i1, cv+i0, cv_dim*sizeof(*cv) );
        i1 -= cv_stride;
        i0 -= cv_stride;
      }

      // update cv values
      for ( j = 0; j < order+rc; j++ ) {
        memcpy( cv, new_cv, cv_dim*sizeof(*new_cv) );
        cv += cv_stride;
        new_cv += cv_dim;
      }
    }

  }

  onfree(new_knot);

  return rc;
}