cubicspline5th.osl

#define MAXDEPTH 20
// stacksize is a function of maxdepth but array sizes are integers only, not expressions :-(
#define STACKSIZE 10000
// e=ldexp(1,-maxdepth)
#define EPSILON exp2(-MAXDEPTH)

// undefine 2D macro to make all calculations 3D
#define CALC2D

void ConvertToBezierForm(point p, point V[4], point w[6])
{
    int 	i, j, k, m, n, ub, lb;	
    int 	row, column;		// Table indices		
    vector 	c[4];		// V(i)'s - P			
    vector 	d[3];		// V(i+1) - V(i)		
    float 	cdTable[12];		// Dot product of c, d		
    float z[12] = {	// Precomputed "z" for cubics	
	 1.0, 0.6, 0.3, 0.1,
	 0.4, 0.6, 0.6, 0.4,
	 0.1, 0.3, 0.6, 1.0
    };

    //Determine the c's -- these are vectors created by subtracting
    // point P from each of the control points				
    for (i = 0; i <= 3; i++) {
#ifdef CALC2D
      c[i][0]=V[i][0]-p[0];
      c[i][1]=V[i][1]-p[1];
#else
      c[i]=V[i]-p;
#endif
    }
    // Determine the d's -- these are vectors created by subtracting
    // each control point from the next					
    for (i = 0; i <= 3 - 1; i++) { 
#ifdef CALC2D
      d[i][0]=3*(V[i+1][0]-V[i][0]);
      d[i][1]=3*(V[i+1][1]-V[i][1]);
#else
      d[i]=3*(V[i+1]-V[i]);
#endif
    }

    // Create the c,d table -- this is a table of dot products of the 
    // c's and d's							
    for (row = 0; row <= 3 - 1; row++) {
		  for (column = 0; column <= 3; column++) {
#ifdef CALC2D
	    	cdTable[row*4+column] = d[row][0]*c[column][0]+d[row][1]*c[column][1];
#else
	    	cdTable[row*4+column] = dot(d[row], c[column]);
#endif
		  }
    }

    // Now, apply the z's to the dot products, on the skew diagonal
    // Also, set up the x-values, making these "points"		
    for (i = 0; i <= 5; i++) {
		  w[i][1] = 0.0;
		  w[i][0] = (float)(i) / 5;
    }

    n = 3;
    m = 2;
    for (k = 0; k <= n + m; k++) {
		  lb = max(0, k - m);
		  ub = min(k, n);
		  for (i = lb; i <= ub; i++) {
	    	j = k - i;
	    	w[i+j][1] += cdTable[j*4+i] * z[j*4+i];
		  }
    }
}

int CrossingCount(point w[], int wtop)
{
    int 	i;	
    int 	n_crossings = 0;	//  Number of zero-crossings	
    int		sign, old_sign;		//  Sign of coefficients	

    sign = old_sign = (w[wtop][1] >= 0);
    for (i = 1; i <= 5; i++) {
	  	sign = (w[wtop+i][1] >= 0);
		  if (sign != old_sign) n_crossings++;
		    old_sign = sign;
    }
    return n_crossings;
}

float ComputeXIntercept(point w[], int wtop)
{
    float XLK, YLK, XNM, YNM, XMK, YMK;
    float det, detInv;
    float	 S,	X;

    XLK = 1.0;
    //YLK = 0.0;
    XNM = w[wtop+5][0] - w[wtop][0];
    YNM = w[wtop+5][1] - w[wtop][1];
    XMK = w[wtop][0];
    YMK = w[wtop][1];

    //det = XNM*YLK - YNM*XLK;
    //detInv = 1.0/det;

    //S = (XNM*YMK - YNM*XMK) * detInv;

    //X = XLK * S;

    return (XNM*YMK - YNM*XMK) / - YNM;
}

int ControlPolygonFlatEnough(point w[], int wtop)
{
    int     i;        // Index variable        
    float  value;
    float  max_distance_above;
    float  max_distance_below;
    float  error;        // Precision of root        
    float  intercept_1,
            intercept_2,
            left_intercept,
            right_intercept;
    float  a, b, c;    // Coefficients of implicit    
            // eqn for line from V[0]-V[deg]
    float  det, dInv;
    float  a1, b1, c1, a2, b2, c2;

    // Derive the implicit equation for line connecting first *'
    //  and last control points 
    a = w[wtop][1] - w[wtop+5][1];
    b = w[wtop+5][0] - w[wtop][0];
    c = w[wtop][0] * w[wtop+5][1] - w[wtop+5][0] * w[wtop][1];

    max_distance_above = max_distance_below = 0.0;
    
    for (i = 1; i < 5; i++)
    {
        value = a * w[wtop+i][0] + b * w[wtop+i][1] + c;
       
        if (value > max_distance_above)
        {
            max_distance_above = value;
        }
        else if (value < max_distance_below)
        {
            max_distance_below = value;
        }
    }

    //  Implicit equation for zero line 
    //  a1 = 0.0;
    //b1 = 1.0;
    //c1 = 0.0;

    //  Implicit equation for "above" line 
    a2 = a;
    b2 = b;
    c2 = c - max_distance_above;

    //det = a1 * b2 - a2 * b1;
    //dInv = 1.0/det;

    //intercept_1 = (b1 * c2 - b2 * c1) * dInv;
    intercept_1 = ( c2 ) / (-a2);

    //  Implicit equation for "below" line 
    a2 = a;
    b2 = b;
    c2 = c - max_distance_below;

    //det = a1 * b2 - a2 * b1;
    //dInv = 1.0/det;

    //intercept_2 = (b1 * c2 - b2 * c1) * dInv;
    intercept_2 = ( c2 ) / -a2;

    // Compute intercepts of bounding box    
    left_intercept = min(intercept_1, intercept_2);
    right_intercept = max(intercept_1, intercept_2);

    error = right_intercept - left_intercept;

    return (error < EPSILON)? 1 : 0;
}

point Bezier(point V[], int degree, float t)
{
    int 	i, j;		// Index variables	
    point 	Vtemp[36];


    // Copy control points	
    for (j =0; j <= degree; j++) {
#ifdef CALC2D
		  Vtemp[j][0] = V[j][0];
		  Vtemp[j][1] = V[j][1];
#else
		  Vtemp[j] = V[j];
#endif
    }

    // Triangle computation	
    for (i = 1; i <= degree; i++) {	
		  for (j =0 ; j <= degree - i; j++) {
	    	Vtemp[i*6+j][0] =
	      		(1.0 - t) * Vtemp[6*(i-1)+j][0] + t * Vtemp[6*(i-1)+j+1][0];
	    	Vtemp[i*6+j][1] =
	      		(1.0 - t) * Vtemp[6*(i-1)+j][1] + t * Vtemp[6*(i-1)+j+1][1];
#ifndef CALC2D
	    	Vtemp[i*6+j][2] =
	      		(1.0 - t) * Vtemp[6*(i-1)+j][2] + t * Vtemp[6*(i-1)+j+1][2];
#endif
		  }
    }
    
    return (Vtemp[6*degree]);
}

// this one is used for the 5th order interpolation only so should always be just 2d
void DeCasteljau(point w[], float t, int wtop, int wend)
{
    int 	i, j;		// Index variables	
    point 	Vtemp[36];


    // Copy control points	
    for (j =0; j <= 5; j++) {
#ifdef CALC2D
		  Vtemp[j][0] = w[wtop+j][0];
		  Vtemp[j][1] = w[wtop+j][1];
#else
		  Vtemp[j] = w[wtop+j];
#endif
    }

    // Triangle computation
    int d=0;	
    for (i = 1; i <= 5; i++) {	
		  for (j =0 ; j <= 5 - i; j++) {
	    	Vtemp[i*6+j][0] =
	      		(1.0 - t) * Vtemp[d+j][0] + t * Vtemp[d+j+1][0];
	    	Vtemp[i*6+j][1] =
	      		(1.0 - t) * Vtemp[d+j][1] + t * Vtemp[d+j+1][1];
		  }
          d+=6;
    }
    
    for (j = 0; j <= 5; j++) {
	    	w[wend+j]  = Vtemp[j*6];
		  }
 	for (j = 0; j <= 5; j++) {
	    	w[wend+6+j] = Vtemp[6*(5-j)+j];
		  }
}

#define POP wtop+=6;wn--

int FindRoots(point w0[6], float t[5]){
  point w[STACKSIZE];
  int wtop=0;
  int wend=0;
  int wn=0;
  for(int k=0;k<6;k++){w[wend++]=w0[k]  ;}
  wn++;
  int nc=0;
  while(wn>0){
    //printf("[%d]",wn);
    int cc=CrossingCount(w,wtop); // w[:5]
    if(cc==0){
      POP;
      continue;
    }
    if(cc==1){
      if(wn>=MAXDEPTH){
        t[nc++]= (w[wtop][0]+w[wtop+5][0])/2;
        //printf(" MAX ");
        POP;
        continue;
      }
      if(ControlPolygonFlatEnough(w,wtop)){ // w[:5]
        t[nc++]= ComputeXIntercept(w,wtop);
        //printf(" FLAT (%.1f | %.1f)",w[wtop],w[wtop+5]);
        POP;
        continue;
      }
    }
    // not flat or more than one crossing
    DeCasteljau(w,0.5,wtop,wend);  //left=[wend:wend+5], right=[wend+6;wend+11]
    for(int k=0;k<6;k++){w[wtop+k]=w[wend+6+k];};
    wn++; wend+=6;
  }
  return nc;
}

// note that might gain something by directly comparing to width and breaking out instead of trying to find the nearest point so this adaption from NearestPoint does just that

int CloseEnough(point p, point V[4], float d, float Width2){
    // Convert problem to 5th-degree Bezier form
    point w0[6];
    ConvertToBezierForm(p, V, w0);

    //Find all possible roots of 5th-degree equation
    float t[5],dt;
    int n_solutions = FindRoots(w0,t);

    //printf("%d %.2f\n",n_solutions,t[0]);
    
    // check which root is closest, also check end points
#ifdef CALC2D
    vector dv=vector(V[0][0]-p[0],V[0][1]-p[1],0);
    d=dv[0]*dv[0]+dv[1]*dv[1];
    if(d < Width2) return 1;
    for(int n=0; n<n_solutions; n++){
      point pt=Bezier(V,3,t[n]);
      dv[0]=pt[0]-p[0];
      dv[1]=pt[1]-p[1];
      d=dv[0]*dv[0]+dv[1]*dv[1];
      if(d < Width2) return 1;
    }
    dv[0]=V[3][0]-p[0];
    dv[1]=V[3][1]-p[1];
    d=dv[0]*dv[0]+dv[1]*dv[1];
    if(d < Width2) return 1;
#else
    vector dv=V[0]-p;
    d=dot(dv,dv);
    if(d < Width2) return 1;
    for(int n=0; n<n_solutions; n++){
      point pt=Bezier(V,3,t[n]);
      dv=pt-p;
      d=dot(dv,dv);
      if(d < Width2) return 1;
    }
    dv=V[3]-p;
    d=dot(dv,dv);
    if(d < Width2) return 1;
#endif
    return 0;
}

point NearestPoint(point p, point V[4], float d, float tb){
    // Convert problem to 5th-degree Bezier form
    point w0[6];
    ConvertToBezierForm(p, V, w0);

    //Find all possible roots of 5th-degree equation
    float t[5],dt;
    int n_solutions = FindRoots(w0,t);

    //printf("%d %.2f\n",n_solutions,t[0]);
    
    // check which root is closest, also check end points
    tb=0;
#ifdef CALC2D
    point bp=point(V[0][0],V[0][1],0);
    vector dv=vector(V[0][0]-p[0],V[0][1]-p[1],0);
    d=dv[0]*dv[0]+dv[1]*dv[1];
    for(int n=0; n<n_solutions; n++){
      point pt=Bezier(V,3,t[n]);
      dv[0]=pt[0]-p[0];
      dv[1]=pt[1]-p[1];
      dt=dv[0]*dv[0]+dv[1]*dv[1];
      if(dt<d){ d=dt; tb=t[n]; bp=pt; }
    }
    dv[0]=V[3][0]-p[0];
    dv[1]=V[3][1]-p[1];
    dt=dv[0]*dv[0]+dv[1]*dv[1];
#else
    point bp=V[0];
    vector dv=V[0]-p;
    d=dot(dv,dv);
    for(int n=0; n<n_solutions; n++){
      point pt=Bezier(V,3,t[n]);
      dv=pt-p;
      dt=dot(dv,dv);
      if(dt<d){ d=dt; tb=t[n]; bp=pt; }
    }
    dv=V[3]-p;
    dt=dot(dv,dv);
#endif
    if(dt<d){ d=dt; tb=1; bp=V[3]; }
    return bp;
}

shader cubicsplinetest(
  point Pos=P,
  point P0=0,
  point P1=0,
  point P2=0,
  point P3=0,
  float Width=0.02,
  output float Fac=0
){
  float d,t;
  point V[4]={P0,P1,P2,P3};
  if(CloseEnough(Pos,V,d,Width*Width)){
    Fac=1;
  }
}