trace.c

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/* trace.c 147 2007-04-09 00:44:09Z selinger */

#include <stdio.h>
#include <math.h>
#include <stdlib.h>
#include <string.h>

#include "global.h"
#include "potracelib.h"
#include "curve.h"
#include "lists.h"
#include "auxiliary.h"
#include "trace.h"
#include "pcb-printf.h"
//#include "progress.h"

#define INFTY 10000000

#define COS179 -0.999847695156

/* ---------------------------------------------------------------------- */
#define SAFE_MALLOC(var, n, typ) \
  if ((var = (typ *)malloc((n)*sizeof(typ))) == NULL) goto malloc_error

/* ---------------------------------------------------------------------- */
/* auxiliary functions */

static inline point_t
dorth_infty (dpoint_t p0, dpoint_t p2)
{
  point_t r;

  r.y = sign (p2.x - p0.x);
  r.x = -sign (p2.y - p0.y);

  return r;
}

static inline double
dpara (dpoint_t p0, dpoint_t p1, dpoint_t p2)
{
  double x1, y1, x2, y2;

  x1 = p1.x - p0.x;
  y1 = p1.y - p0.y;
  x2 = p2.x - p0.x;
  y2 = p2.y - p0.y;

  return x1 * y2 - x2 * y1;
}

static inline double
ddenom (dpoint_t p0, dpoint_t p2)
{
  point_t r = dorth_infty (p0, p2);

  return r.y * (p2.x - p0.x) - r.x * (p2.y - p0.y);
}

static inline int
cyclic (int a, int b, int c)
{
  if (a <= c)
    {
      return (a <= b && b < c);
    }
  else
    {
      return (a <= b || b < c);
    }
}

static void
pointslope (privpath_t * pp, int i, int j, dpoint_t * ctr, dpoint_t * dir)
{
  /* assume i<j */

  int n = pp->len;
  sums_t *sums = pp->sums;

  double x, y, x2, xy, y2;
  double k;
  double a, b, c, lambda2, l;
  int r = 0;                    /* rotations from i to j */

  while (j >= n)
    {
      j -= n;
      r += 1;
    }
  while (i >= n)
    {
      i -= n;
      r -= 1;
    }
  while (j < 0)
    {
      j += n;
      r -= 1;
    }
  while (i < 0)
    {
      i += n;
      r += 1;
    }

  x = sums[j + 1].x - sums[i].x + r * sums[n].x;
  y = sums[j + 1].y - sums[i].y + r * sums[n].y;
  x2 = sums[j + 1].x2 - sums[i].x2 + r * sums[n].x2;
  xy = sums[j + 1].xy - sums[i].xy + r * sums[n].xy;
  y2 = sums[j + 1].y2 - sums[i].y2 + r * sums[n].y2;
  k = j + 1 - i + r * n;

  ctr->x = x / k;
  ctr->y = y / k;

  a = (x2 - (double) x * x / k) / k;
  b = (xy - (double) x * y / k) / k;
  c = (y2 - (double) y * y / k) / k;

  lambda2 = (a + c + hypot (a - c, 2 * b)) / 2; /* larger e.value */

  /* now find e.vector for lambda2 */
  a -= lambda2;
  c -= lambda2;

  if (fabs (a) >= fabs (c))
    {
      l = hypot (a, b);
      if (l != 0)
        {
          dir->x = -b / l;
          dir->y = a / l;
        }
    }
  else
    {
      l = hypot (c, b);
      if (l != 0)
        {
          dir->x = -c / l;
          dir->y = b / l;
        }
    }
  if (l == 0)
    {
      dir->x = dir->y = 0;      /* sometimes this can happen when k=4:
                                   the two eigenvalues coincide */
    }
}

typedef double quadform_t[3][3];

static inline double
quadform (quadform_t Q, dpoint_t w)
{
  double v[3];
  int i, j;
  double sum;

  v[0] = w.x;
  v[1] = w.y;
  v[2] = 1;
  sum = 0.0;

  for (i = 0; i < 3; i++)
    {
      for (j = 0; j < 3; j++)
        {
          sum += v[i] * Q[i][j] * v[j];
        }
    }
  return sum;
}

static inline int
xprod (point_t p1, point_t p2)
{
  return p1.x * p2.y - p1.y * p2.x;
}

static inline double
cprod (dpoint_t p0, dpoint_t p1, dpoint_t p2, dpoint_t p3)
{
  double x1, y1, x2, y2;

  x1 = p1.x - p0.x;
  y1 = p1.y - p0.y;
  x2 = p3.x - p2.x;
  y2 = p3.y - p2.y;

  return x1 * y2 - x2 * y1;
}

static inline double
iprod (dpoint_t p0, dpoint_t p1, dpoint_t p2)
{
  double x1, y1, x2, y2;

  x1 = p1.x - p0.x;
  y1 = p1.y - p0.y;
  x2 = p2.x - p0.x;
  y2 = p2.y - p0.y;

  return x1 * x2 + y1 * y2;
}

static inline double
iprod1 (dpoint_t p0, dpoint_t p1, dpoint_t p2, dpoint_t p3)
{
  double x1, y1, x2, y2;

  x1 = p1.x - p0.x;
  y1 = p1.y - p0.y;
  x2 = p3.x - p2.x;
  y2 = p3.y - p2.y;

  return x1 * x2 + y1 * y2;
}

static inline double
ddist (dpoint_t p, dpoint_t q)
{
  return hypot (p.x - q.x, p.y - q.y);
}

static inline dpoint_t
bezier (double t, dpoint_t p0, dpoint_t p1, dpoint_t p2, dpoint_t p3)
{
  double s = 1 - t;
  dpoint_t res;

  /* Note: a good optimizing compiler (such as gcc-3) reduces the
     following to 16 multiplications, using common subexpression
     elimination. */

  res.x =
    s * s * s * p0.x + 3 * (s * s * t) * p1.x + 3 * (t * t * s) * p2.x +
    t * t * t * p3.x;
  res.y =
    s * s * s * p0.y + 3 * (s * s * t) * p1.y + 3 * (t * t * s) * p2.y +
    t * t * t * p3.y;

  return res;
}

static double
tangent (dpoint_t p0, dpoint_t p1, dpoint_t p2, dpoint_t p3, dpoint_t q0,
         dpoint_t q1)
{
  double A, B, C;               /* (1-t)^2 A + 2(1-t)t B + t^2 C = 0 */
  double a, b, c;               /* a t^2 + b t + c = 0 */
  double d, s, r1, r2;

  A = cprod (p0, p1, q0, q1);
  B = cprod (p1, p2, q0, q1);
  C = cprod (p2, p3, q0, q1);

  a = A - 2 * B + C;
  b = -2 * A + 2 * B;
  c = A;

  d = b * b - 4 * a * c;

  if (a == 0 || d < 0)
    {
      return -1.0;
    }

  s = sqrt (d);

  r1 = (-b + s) / (2 * a);
  r2 = (-b - s) / (2 * a);

  if (r1 >= 0 && r1 <= 1)
    {
      return r1;
    }
  else if (r2 >= 0 && r2 <= 1)
    {
      return r2;
    }
  else
    {
      return -1.0;
    }
}

static int
calc_sums (privpath_t * pp)
{
  int i, x, y;
  int n = pp->len;

  SAFE_MALLOC (pp->sums, pp->len + 1, sums_t);

  /* origin */
  pp->x0 = pp->pt[0].x;
  pp->y0 = pp->pt[0].y;

  /* preparatory computation for later fast summing */
  pp->sums[0].x2 = pp->sums[0].xy = pp->sums[0].y2 = pp->sums[0].x =
    pp->sums[0].y = 0;
  for (i = 0; i < n; i++)
    {
      x = pp->pt[i].x - pp->x0;
      y = pp->pt[i].y - pp->y0;
      pp->sums[i + 1].x = pp->sums[i].x + x;
      pp->sums[i + 1].y = pp->sums[i].y + y;
      pp->sums[i + 1].x2 = pp->sums[i].x2 + x * x;
      pp->sums[i + 1].xy = pp->sums[i].xy + x * y;
      pp->sums[i + 1].y2 = pp->sums[i].y2 + y * y;
    }
  return 0;

malloc_error:
  return 1;
}

static int
calc_lon (privpath_t * pp)
{
  point_t *pt = pp->pt;
  int n = pp->len;
  int i, j, k, k1;
  int ct[4], dir;
  point_t constraint[2];
  point_t cur;
  point_t off;
  int *pivk = NULL;             /* pivk[n] */
  int *nc = NULL;               /* nc[n]: next corner */
  point_t dk;                   /* direction of k-k1 */
  int a, b, c, d;

  SAFE_MALLOC (pivk, n, int);
  SAFE_MALLOC (nc, n, int);

  /* initialize the nc data structure. Point from each point to the
     furthest future point to which it is connected by a vertical or
     horizontal segment. We take advantage of the fact that there is
     always a direction change at 0 (due to the path decomposition
     algorithm). But even if this were not so, there is no harm, as
     in practice, correctness does not depend on the word "furthest"
     above.  */
  k = 0;
  for (i = n - 1; i >= 0; i--)
    {
      if (pt[i].x != pt[k].x && pt[i].y != pt[k].y)
        {
          k = i + 1;            /* necessarily i<n-1 in this case */
        }
      nc[i] = k;
    }

  SAFE_MALLOC (pp->lon, n, int);

  /* determine pivot points: for each i, let pivk[i] be the furthest k
     such that all j with i<j<k lie on a line connecting i,k. */

  for (i = n - 1; i >= 0; i--)
    {
      ct[0] = ct[1] = ct[2] = ct[3] = 0;

      /* keep track of "directions" that have occurred */
      dir =
        (3 + 3 * (pt[mod (i + 1, n)].x - pt[i].x) +
         (pt[mod (i + 1, n)].y - pt[i].y)) / 2;
      ct[dir]++;

      constraint[0].x = 0;
      constraint[0].y = 0;
      constraint[1].x = 0;
      constraint[1].y = 0;

      /* find the next k such that no straight line from i to k */
      k = nc[i];
      k1 = i;
      while (1)
        {

          dir =
            (3 + 3 * sign (pt[k].x - pt[k1].x) +
             sign (pt[k].y - pt[k1].y)) / 2;
          ct[dir]++;

          /* if all four "directions" have occurred, cut this path */
          if (ct[0] && ct[1] && ct[2] && ct[3])
            {
              pivk[i] = k1;
              goto foundk;
            }

          cur.x = pt[k].x - pt[i].x;
          cur.y = pt[k].y - pt[i].y;

          /* see if current constraint is violated */
          if (xprod (constraint[0], cur) < 0
              || xprod (constraint[1], cur) > 0)
            {
              goto constraint_viol;
            }

          /* else, update constraint */
          if (abs (cur.x) <= 1 && abs (cur.y) <= 1)
            {
              /* no constraint */
            }
          else
            {
              off.x =
                cur.x + ((cur.y >= 0 && (cur.y > 0 || cur.x < 0)) ? 1 : -1);
              off.y =
                cur.y + ((cur.x <= 0 && (cur.x < 0 || cur.y < 0)) ? 1 : -1);
              if (xprod (constraint[0], off) >= 0)
                {
                  constraint[0] = off;
                }
              off.x =
                cur.x + ((cur.y <= 0 && (cur.y < 0 || cur.x < 0)) ? 1 : -1);
              off.y =
                cur.y + ((cur.x >= 0 && (cur.x > 0 || cur.y < 0)) ? 1 : -1);
              if (xprod (constraint[1], off) <= 0)
                {
                  constraint[1] = off;
                }
            }
          k1 = k;
          k = nc[k1];
          if (!cyclic (k, i, k1))
            {
              break;
            }
        }
    constraint_viol:
      /* k1 was the last "corner" satisfying the current constraint, and
         k is the first one violating it. We now need to find the last
         point along k1..k which satisfied the constraint. */
      dk.x = sign (pt[k].x - pt[k1].x);
      dk.y = sign (pt[k].y - pt[k1].y);
      cur.x = pt[k1].x - pt[i].x;
      cur.y = pt[k1].y - pt[i].y;
      /* find largest integer j such that xprod(constraint[0], cur+j*dk)
         >= 0 and xprod(constraint[1], cur+j*dk) <= 0. Use bilinearity
         of xprod. */
      a = xprod (constraint[0], cur);
      b = xprod (constraint[0], dk);
      c = xprod (constraint[1], cur);
      d = xprod (constraint[1], dk);
      /* find largest integer j such that a+j*b>=0 and c+j*d<=0. This
         can be solved with integer arithmetic. */
      j = INFTY;
      if (b < 0)
        {
          j = floordiv (a, -b);
        }
      if (d > 0)
        {
          j = min (j, floordiv (-c, d));
        }
      pivk[i] = mod (k1 + j, n);
    foundk:
      ;
    }                           /* for i */

  /* clean up: for each i, let lon[i] be the largest k such that for
     all i' with i<=i'<k, i'<k<=pivk[i']. */

  j = pivk[n - 1];
  pp->lon[n - 1] = j;
  for (i = n - 2; i >= 0; i--)
    {
      if (cyclic (i + 1, pivk[i], j))
        {
          j = pivk[i];
        }
      pp->lon[i] = j;
    }

  for (i = n - 1; cyclic (mod (i + 1, n), j, pp->lon[i]); i--)
    {
      pp->lon[i] = j;
    }

  free (pivk);
  free (nc);
  return 0;

malloc_error:
  free (pivk);
  free (nc);
  return 1;
}


static double
penalty3 (privpath_t * pp, int i, int j)
{
  int n = pp->len;
  point_t *pt = pp->pt;
  sums_t *sums = pp->sums;

  /* assume 0<=i<j<=n  */
  double x, y, x2, xy, y2;
  double k;
  double a, b, c, s;
  double px, py, ex, ey;

  int r = 0;                    /* rotations from i to j */

  if (j >= n)
    {
      j -= n;
      r += 1;
    }

  x = sums[j + 1].x - sums[i].x + r * sums[n].x;
  y = sums[j + 1].y - sums[i].y + r * sums[n].y;
  x2 = sums[j + 1].x2 - sums[i].x2 + r * sums[n].x2;
  xy = sums[j + 1].xy - sums[i].xy + r * sums[n].xy;
  y2 = sums[j + 1].y2 - sums[i].y2 + r * sums[n].y2;
  k = j + 1 - i + r * n;

  px = (pt[i].x + pt[j].x) / 2.0 - pt[0].x;
  py = (pt[i].y + pt[j].y) / 2.0 - pt[0].y;
  ey = (pt[j].x - pt[i].x);
  ex = -(pt[j].y - pt[i].y);

  a = ((x2 - 2 * x * px) / k + px * px);
  b = ((xy - x * py - y * px) / k + px * py);
  c = ((y2 - 2 * y * py) / k + py * py);

  s = ex * ex * a + 2 * ex * ey * b + ey * ey * c;

  return sqrt (s);
}

static int
bestpolygon (privpath_t * pp)
{
  int i, j, m, k;
  int n = pp->len;
  double *pen = NULL;           /* pen[n+1]: penalty vector */
  int *prev = NULL;             /* prev[n+1]: best path pointer vector */
  int *clip0 = NULL;            /* clip0[n]: longest segment pointer, non-cyclic */
  int *clip1 = NULL;            /* clip1[n+1]: backwards segment pointer, non-cyclic */
  int *seg0 = NULL;             /* seg0[m+1]: forward segment bounds, m<=n */
  int *seg1 = NULL;             /* seg1[m+1]: backward segment bounds, m<=n */
  double thispen;
  double best;
  int c;

  SAFE_MALLOC (pen, n + 1, double);
  SAFE_MALLOC (prev, n + 1, int);
  SAFE_MALLOC (clip0, n, int);
  SAFE_MALLOC (clip1, n + 1, int);
  SAFE_MALLOC (seg0, n + 1, int);
  SAFE_MALLOC (seg1, n + 1, int);

  /* calculate clipped paths */
  for (i = 0; i < n; i++)
    {
      c = mod (pp->lon[mod (i - 1, n)] - 1, n);
      if (c == i)
        {
          c = mod (i + 1, n);
        }
      if (c < i)
        {
          clip0[i] = n;
        }
      else
        {
          clip0[i] = c;
        }
    }

  /* calculate backwards path clipping, non-cyclic. j <= clip0[i] iff
     clip1[j] <= i, for i,j=0..n. */
  j = 1;
  for (i = 0; i < n; i++)
    {
      while (j <= clip0[i])
        {
          clip1[j] = i;
          j++;
        }
    }

  /* calculate seg0[j] = longest path from 0 with j segments */
  i = 0;
  for (j = 0; i < n; j++)
    {
      seg0[j] = i;
      i = clip0[i];
    }
  seg0[j] = n;
  m = j;

  /* calculate seg1[j] = longest path to n with m-j segments */
  i = n;
  for (j = m; j > 0; j--)
    {
      seg1[j] = i;
      i = clip1[i];
    }
  seg1[0] = 0;

  /* now find the shortest path with m segments, based on penalty3 */
  /* note: the outer 2 loops jointly have at most n interations, thus
     the worst-case behavior here is quadratic. In practice, it is
     close to linear since the inner loop tends to be short. */
  pen[0] = 0;
  for (j = 1; j <= m; j++)
    {
      for (i = seg1[j]; i <= seg0[j]; i++)
        {
          best = -1;
          for (k = seg0[j - 1]; k >= clip1[i]; k--)
            {
              thispen = penalty3 (pp, k, i) + pen[k];
              if (best < 0 || thispen < best)
                {
                  prev[i] = k;
                  best = thispen;
                }
            }
          pen[i] = best;
        }
    }

  pp->m = m;
  SAFE_MALLOC (pp->po, m, int);

  /* read off shortest path */
  for (i = n, j = m - 1; i > 0; j--)
    {
      i = prev[i];
      pp->po[j] = i;
    }

  free (pen);
  free (prev);
  free (clip0);
  free (clip1);
  free (seg0);
  free (seg1);
  return 0;

malloc_error:
  free (pen);
  free (prev);
  free (clip0);
  free (clip1);
  free (seg0);
  free (seg1);
  return 1;
}

static int
adjust_vertices (privpath_t * pp)
{
  int m = pp->m;
  int *po = pp->po;
  int n = pp->len;
  point_t *pt = pp->pt;
  int x0 = pp->x0;
  int y0 = pp->y0;

  dpoint_t *ctr = NULL;         /* ctr[m] */
  dpoint_t *dir = NULL;         /* dir[m] */
  quadform_t *q = NULL;         /* q[m] */
  double v[3];
  double d;
  int i, j, k, l;
  dpoint_t s;
  int r;

  SAFE_MALLOC (ctr, m, dpoint_t);
  SAFE_MALLOC (dir, m, dpoint_t);
  SAFE_MALLOC (q, m, quadform_t);

  r = privcurve_init (&pp->curve, m);
  if (r)
    {
      goto malloc_error;
    }

  /* calculate "optimal" point-slope representation for each line
     segment */
  for (i = 0; i < m; i++)
    {
      j = po[mod (i + 1, m)];
      j = mod (j - po[i], n) + po[i];
      pointslope (pp, po[i], j, &ctr[i], &dir[i]);
    }

  /* represent each line segment as a singular quadratic form; the
     distance of a point (x,y) from the line segment will be
     (x,y,1)Q(x,y,1)^t, where Q=q[i]. */
  for (i = 0; i < m; i++)
    {
      d = sq (dir[i].x) + sq (dir[i].y);
      if (d == 0.0)
        {
          for (j = 0; j < 3; j++)
            {
              for (k = 0; k < 3; k++)
                {
                  q[i][j][k] = 0;
                }
            }
        }
      else
        {
          v[0] = dir[i].y;
          v[1] = -dir[i].x;
          v[2] = -v[1] * ctr[i].y - v[0] * ctr[i].x;
          for (l = 0; l < 3; l++)
            {
              for (k = 0; k < 3; k++)
                {
                  q[i][l][k] = v[l] * v[k] / d;
                }
            }
        }
    }

  /* now calculate the "intersections" of consecutive segments.
     Instead of using the actual intersection, we find the point
     within a given unit square which minimizes the square distance to
     the two lines. */
  for (i = 0; i < m; i++)
    {
      quadform_t Q;
      dpoint_t w;
      double dx, dy;
      double det;
      double min, cand;         /* minimum and candidate for minimum of quad. form */
      double xmin, ymin;        /* coordinates of minimum */
      int z;

      /* let s be the vertex, in coordinates relative to x0/y0 */
      s.x = pt[po[i]].x - x0;
      s.y = pt[po[i]].y - y0;

      /* intersect segments i-1 and i */

      j = mod (i - 1, m);

      /* add quadratic forms */
      for (l = 0; l < 3; l++)
        {
          for (k = 0; k < 3; k++)
            {
              Q[l][k] = q[j][l][k] + q[i][l][k];
            }
        }

      while (1)
        {
          /* minimize the quadratic form Q on the unit square */
          /* find intersection */

#ifdef HAVE_GCC_LOOP_BUG
          /* work around gcc bug #12243 */
          free (NULL);
#endif

          det = Q[0][0] * Q[1][1] - Q[0][1] * Q[1][0];
          if (det != 0.0)
            {
              w.x = (-Q[0][2] * Q[1][1] + Q[1][2] * Q[0][1]) / det;
              w.y = (Q[0][2] * Q[1][0] - Q[1][2] * Q[0][0]) / det;
              break;
            }

          /* matrix is singular - lines are parallel. Add another,
             orthogonal axis, through the center of the unit square */
          if (Q[0][0] > Q[1][1])
            {
              v[0] = -Q[0][1];
              v[1] = Q[0][0];
            }
          else if (Q[1][1])
            {
              v[0] = -Q[1][1];
              v[1] = Q[1][0];
            }
          else
            {
              v[0] = 1;
              v[1] = 0;
            }
          d = sq (v[0]) + sq (v[1]);
          v[2] = -v[1] * s.y - v[0] * s.x;
          for (l = 0; l < 3; l++)
            {
              for (k = 0; k < 3; k++)
                {
                  Q[l][k] += v[l] * v[k] / d;
                }
            }
        }
      dx = fabs (w.x - s.x);
      dy = fabs (w.y - s.y);
      if (dx <= .5 && dy <= .5)
        {
          pp->curve.vertex[i].x = w.x + x0;
          pp->curve.vertex[i].y = w.y + y0;
          continue;
        }

      /* the minimum was not in the unit square; now minimize quadratic
         on boundary of square */
      min = quadform (Q, s);
      xmin = s.x;
      ymin = s.y;

      if (Q[0][0] == 0.0)
        {
          goto fixx;
        }
      for (z = 0; z < 2; z++)
        {                       /* value of the y-coordinate */
          w.y = s.y - 0.5 + z;
          w.x = -(Q[0][1] * w.y + Q[0][2]) / Q[0][0];
          dx = fabs (w.x - s.x);
          cand = quadform (Q, w);
          if (dx <= .5 && cand < min)
            {
              min = cand;
              xmin = w.x;
              ymin = w.y;
            }
        }
    fixx:
      if (Q[1][1] == 0.0)
        {
          goto corners;
        }
      for (z = 0; z < 2; z++)
        {                       /* value of the x-coordinate */
          w.x = s.x - 0.5 + z;
          w.y = -(Q[1][0] * w.x + Q[1][2]) / Q[1][1];
          dy = fabs (w.y - s.y);
          cand = quadform (Q, w);
          if (dy <= .5 && cand < min)
            {
              min = cand;
              xmin = w.x;
              ymin = w.y;
            }
        }
    corners:
      /* check four corners */
      for (l = 0; l < 2; l++)
        {
          for (k = 0; k < 2; k++)
            {
              w.x = s.x - 0.5 + l;
              w.y = s.y - 0.5 + k;
              cand = quadform (Q, w);
              if (cand < min)
                {
                  min = cand;
                  xmin = w.x;
                  ymin = w.y;
                }
            }
        }

      pp->curve.vertex[i].x = xmin + x0;
      pp->curve.vertex[i].y = ymin + y0;
      continue;
    }

  free (ctr);
  free (dir);
  free (q);
  return 0;

malloc_error:
  free (ctr);
  free (dir);
  free (q);
  return 1;
}

static int
ATTRIBUTE_UNUSED smooth (privcurve_t * curve, int sign, double alphamax)
{
  int m = curve->n;

  int i, j, k;
  double dd, denom, alpha;
  dpoint_t p2, p3, p4;

  if (sign == '-')
    {
      /* reverse orientation of negative paths */
      for (i = 0, j = m - 1; i < j; i++, j--)
        {
          dpoint_t tmp;
          tmp = curve->vertex[i];
          curve->vertex[i] = curve->vertex[j];
          curve->vertex[j] = tmp;
        }
    }

  /* examine each vertex and find its best fit */
  for (i = 0; i < m; i++)
    {
      j = mod (i + 1, m);
      k = mod (i + 2, m);
      p4 = interval (1 / 2.0, curve->vertex[k], curve->vertex[j]);

      denom = ddenom (curve->vertex[i], curve->vertex[k]);
      if (denom != 0.0)
        {
          dd =
            dpara (curve->vertex[i], curve->vertex[j],
                   curve->vertex[k]) / denom;
          dd = fabs (dd);
          alpha = dd > 1 ? (1 - 1.0 / dd) : 0;
          alpha = alpha / 0.75;
        }
      else
        {
          alpha = 4 / 3.0;
        }
      curve->alpha0[j] = alpha; /* remember "original" value of alpha */

      if (alpha > alphamax)
        {                       /* pointed corner */
          curve->tag[j] = POTRACE_CORNER;
          curve->c[j][1] = curve->vertex[j];
          curve->c[j][2] = p4;
        }
      else
        {
          if (alpha < 0.55)
            {
              alpha = 0.55;
            }
          else if (alpha > 1)
            {
              alpha = 1;
            }
          p2 = interval (.5 + .5 * alpha, curve->vertex[i], curve->vertex[j]);
          p3 = interval (.5 + .5 * alpha, curve->vertex[k], curve->vertex[j]);
          curve->tag[j] = POTRACE_CURVETO;
          curve->c[j][0] = p2;
          curve->c[j][1] = p3;
          curve->c[j][2] = p4;
        }
      curve->alpha[j] = alpha;  /* store the "cropped" value of alpha */
      curve->beta[j] = 0.5;
    }
  curve->alphacurve = 1;

  return 0;
}

/* ---------------------------------------------------------------------- */
/* Stage 5: Curve optimization (Sec. 2.4) */

struct opti_s
{
  double pen; 
  dpoint_t c[2]; 
  double t; 
  double s; 
  double alpha; 
};
typedef struct opti_s opti_t;

static int
opti_penalty (privpath_t * pp, int i, int j, opti_t * res,
              double opttolerance, int *convc, double *areac)
{
  int m = pp->curve.n;
  int k, k1, k2, conv, i1;
  double area, alpha, d, d1, d2;
  dpoint_t p0, p1, p2, p3, pt;
  double A, R, A1, A2, A3, A4;
  double s, t;

  /* check convexity, corner-freeness, and maximum bend < 179 degrees */

  if (i == j)
    {                           /* sanity - a full loop can never be an opticurve */
      return 1;
    }

  k = i;
  i1 = mod (i + 1, m);
  k1 = mod (k + 1, m);
  conv = convc[k1];
  if (conv == 0)
    {
      return 1;
    }
  d = ddist (pp->curve.vertex[i], pp->curve.vertex[i1]);
  for (k = k1; k != j; k = k1)
    {
      k1 = mod (k + 1, m);
      k2 = mod (k + 2, m);
      if (convc[k1] != conv)
        {
          return 1;
        }
      if (sign
          (cprod
           (pp->curve.vertex[i], pp->curve.vertex[i1], pp->curve.vertex[k1],
            pp->curve.vertex[k2])) != conv)
        {
          return 1;
        }
      if (iprod1
          (pp->curve.vertex[i], pp->curve.vertex[i1], pp->curve.vertex[k1],
           pp->curve.vertex[k2]) < d * ddist (pp->curve.vertex[k1],
                                              pp->curve.vertex[k2]) * COS179)
        {
          return 1;
        }
    }

  /* the curve we're working in: */
  p0 = pp->curve.c[mod (i, m)][2];
  p1 = pp->curve.vertex[mod (i + 1, m)];
  p2 = pp->curve.vertex[mod (j, m)];
  p3 = pp->curve.c[mod (j, m)][2];

  /* determine its area */
  area = areac[j] - areac[i];
  area -=
    dpara (pp->curve.vertex[0], pp->curve.c[i][2], pp->curve.c[j][2]) / 2;
  if (i >= j)
    {
      area += areac[m];
    }

  /* find intersection o of p0p1 and p2p3. Let t,s such that o =
     interval(t,p0,p1) = interval(s,p3,p2). Let A be the area of the
     triangle (p0,o,p3). */

  A1 = dpara (p0, p1, p2);
  A2 = dpara (p0, p1, p3);
  A3 = dpara (p0, p2, p3);
  /* A4 = dpara(p1, p2, p3); */
  A4 = A1 + A3 - A2;

  if (A2 == A1)
    {                           /* this should never happen */
      return 1;
    }

  t = A3 / (A3 - A4);
  s = A2 / (A2 - A1);
  A = A2 * t / 2.0;

  if (A == 0.0)
    {                           /* this should never happen */
      return 1;
    }

  R = area / A;                 /* relative area */
  alpha = 2 - sqrt (4 - R / 0.3);       /* overall alpha for p0-o-p3 curve */

  res->c[0] = interval (t * alpha, p0, p1);
  res->c[1] = interval (s * alpha, p3, p2);
  res->alpha = alpha;
  res->t = t;
  res->s = s;

  p1 = res->c[0];
  p2 = res->c[1];               /* the proposed curve is now (p0,p1,p2,p3) */

  res->pen = 0;

  /* calculate penalty */
  /* check tangency with edges */
  for (k = mod (i + 1, m); k != j; k = k1)
    {
      k1 = mod (k + 1, m);
      t = tangent (p0, p1, p2, p3, pp->curve.vertex[k], pp->curve.vertex[k1]);
      if (t < -.5)
        {
          return 1;
        }
      pt = bezier (t, p0, p1, p2, p3);
      d = ddist (pp->curve.vertex[k], pp->curve.vertex[k1]);
      if (d == 0.0)
        {                       /* this should never happen */
          return 1;
        }
      d1 = dpara (pp->curve.vertex[k], pp->curve.vertex[k1], pt) / d;
      if (fabs (d1) > opttolerance)
        {
          return 1;
        }
      if (iprod (pp->curve.vertex[k], pp->curve.vertex[k1], pt) < 0
          || iprod (pp->curve.vertex[k1], pp->curve.vertex[k], pt) < 0)
        {
          return 1;
        }
      res->pen += sq (d1);
    }

  /* check corners */
  for (k = i; k != j; k = k1)
    {
      k1 = mod (k + 1, m);
      t = tangent (p0, p1, p2, p3, pp->curve.c[k][2], pp->curve.c[k1][2]);
      if (t < -.5)
        {
          return 1;
        }
      pt = bezier (t, p0, p1, p2, p3);
      d = ddist (pp->curve.c[k][2], pp->curve.c[k1][2]);
      if (d == 0.0)
        {                       /* this should never happen */
          return 1;
        }
      d1 = dpara (pp->curve.c[k][2], pp->curve.c[k1][2], pt) / d;
      d2 =
        dpara (pp->curve.c[k][2], pp->curve.c[k1][2],
               pp->curve.vertex[k1]) / d;
      d2 *= 0.75 * pp->curve.alpha[k1];
      if (d2 < 0)
        {
          d1 = -d1;
          d2 = -d2;
        }
      if (d1 < d2 - opttolerance)
        {
          return 1;
        }
      if (d1 < d2)
        {
          res->pen += sq (d1 - d2);
        }
    }

  return 0;
}

static int
ATTRIBUTE_UNUSED opticurve (privpath_t * pp, double opttolerance)
{
  int m = pp->curve.n;
  int *pt = NULL;               /* pt[m+1] */
  double *pen = NULL;           /* pen[m+1] */
  int *len = NULL;              /* len[m+1] */
  opti_t *opt = NULL;           /* opt[m+1] */
  int om;
  int i, j, r;
  opti_t o;
  dpoint_t p0;
  int i1;
  double area;
  double alpha;
  double *s = NULL;
  double *t = NULL;

  int *convc = NULL;            /* conv[m]: pre-computed convexities */
  double *areac = NULL;         /* cumarea[m+1]: cache for fast area computation */

  SAFE_MALLOC (pt, m + 1, int);
  SAFE_MALLOC (pen, m + 1, double);
  SAFE_MALLOC (len, m + 1, int);
  SAFE_MALLOC (opt, m + 1, opti_t);
  SAFE_MALLOC (convc, m, int);
  SAFE_MALLOC (areac, m + 1, double);

  /* pre-calculate convexity: +1 = right turn, -1 = left turn, 0 = corner */
  for (i = 0; i < m; i++)
    {
      if (pp->curve.tag[i] == POTRACE_CURVETO)
        {
          convc[i] =
            sign (dpara
                  (pp->curve.vertex[mod (i - 1, m)], pp->curve.vertex[i],
                   pp->curve.vertex[mod (i + 1, m)]));
        }
      else
        {
          convc[i] = 0;
        }
    }

  /* pre-calculate areas */
  area = 0.0;
  areac[0] = 0.0;
  p0 = pp->curve.vertex[0];
  for (i = 0; i < m; i++)
    {
      i1 = mod (i + 1, m);
      if (pp->curve.tag[i1] == POTRACE_CURVETO)
        {
          alpha = pp->curve.alpha[i1];
          area +=
            0.3 * alpha * (4 - alpha) * dpara (pp->curve.c[i][2],
                                               pp->curve.vertex[i1],
                                               pp->curve.c[i1][2]) / 2;
          area += dpara (p0, pp->curve.c[i][2], pp->curve.c[i1][2]) / 2;
        }
      areac[i + 1] = area;
    }

  pt[0] = -1;
  pen[0] = 0;
  len[0] = 0;

  /* Fixme: we always start from a fixed point -- should find the best
     curve cyclically ### */

  for (j = 1; j <= m; j++)
    {
      /* calculate best path from 0 to j */
      pt[j] = j - 1;
      pen[j] = pen[j - 1];
      len[j] = len[j - 1] + 1;

      for (i = j - 2; i >= 0; i--)
        {
          r =
            opti_penalty (pp, i, mod (j, m), &o, opttolerance, convc, areac);
          if (r)
            {
              break;
            }
          if (len[j] > len[i] + 1
              || (len[j] == len[i] + 1 && pen[j] > pen[i] + o.pen))
            {
              pt[j] = i;
              pen[j] = pen[i] + o.pen;
              len[j] = len[i] + 1;
              opt[j] = o;
            }
        }
    }
  om = len[m];
  r = privcurve_init (&pp->ocurve, om);
  if (r)
    {
      goto malloc_error;
    }
  SAFE_MALLOC (s, om, double);
  SAFE_MALLOC (t, om, double);

  j = m;
  for (i = om - 1; i >= 0; i--)
    {
      if (pt[j] == j - 1)
        {
          pp->ocurve.tag[i] = pp->curve.tag[mod (j, m)];
          pp->ocurve.c[i][0] = pp->curve.c[mod (j, m)][0];
          pp->ocurve.c[i][1] = pp->curve.c[mod (j, m)][1];
          pp->ocurve.c[i][2] = pp->curve.c[mod (j, m)][2];
          pp->ocurve.vertex[i] = pp->curve.vertex[mod (j, m)];
          pp->ocurve.alpha[i] = pp->curve.alpha[mod (j, m)];
          pp->ocurve.alpha0[i] = pp->curve.alpha0[mod (j, m)];
          pp->ocurve.beta[i] = pp->curve.beta[mod (j, m)];
          s[i] = t[i] = 1.0;
        }
      else
        {
          pp->ocurve.tag[i] = POTRACE_CURVETO;
          pp->ocurve.c[i][0] = opt[j].c[0];
          pp->ocurve.c[i][1] = opt[j].c[1];
          pp->ocurve.c[i][2] = pp->curve.c[mod (j, m)][2];
          pp->ocurve.vertex[i] =
            interval (opt[j].s, pp->curve.c[mod (j, m)][2],
                      pp->curve.vertex[mod (j, m)]);
          pp->ocurve.alpha[i] = opt[j].alpha;
          pp->ocurve.alpha0[i] = opt[j].alpha;
          s[i] = opt[j].s;
          t[i] = opt[j].t;
        }
      j = pt[j];
    }

  /* calculate beta parameters */
  for (i = 0; i < om; i++)
    {
      i1 = mod (i + 1, om);
      pp->ocurve.beta[i] = s[i] / (s[i] + t[i1]);
    }
  pp->ocurve.alphacurve = 1;

  free (pt);
  free (pen);
  free (len);
  free (opt);
  free (s);
  free (t);
  free (convc);
  free (areac);
  return 0;

malloc_error:
  free (pt);
  free (pen);
  free (len);
  free (opt);
  free (s);
  free (t);
  free (convc);
  free (areac);
  return 1;
}

/* ---------------------------------------------------------------------- */
double
plotpolygon (privpath_t * pp, FILE * f, double scale,
             const char *var_cutdepth, const char *var_safeZ,
             const char *var_plunge, const char *var_feedrate)
{
  int i;
  int m = pp->m;
  int *po = pp->po;
  point_t *pt = pp->pt;
  /* double scale=1.0/dpi; */
  double dm = 0;

  if (!m)
    return 0;

  po = pp->po;
  pt = pp->pt;

  pcb_fprintf (f, "G0 X%`f Y%`f    (start point)\n", pt[po[0]].x * scale,
               pt[po[0]].y * scale);
  pcb_fprintf (f, "G1 Z%s F%s\n", var_cutdepth, var_plunge);
  pcb_fprintf (f, "F%s\n", var_feedrate);
  for (i = 1; i < m; i++)
    {
      pcb_fprintf (f, "G1 X%`f Y%`f\n",
                   pt[po[i]].x * scale, pt[po[i]].y * scale);
      dm += fabs (scale) * hypot (pt[po[i]].x - pt[po[i - 1]].x, pt[po[i]].y - pt[po[i - 1]].y);
    }
  pcb_fprintf (f, "G1 X%`f Y%`f\n", pt[po[0]].x * scale, pt[po[0]].y * scale);
  pcb_fprintf (f, "G0 Z%s\n", var_safeZ);
  dm += fabs (scale) * hypot (pt[po[m - 1]].x - pt[po[0]].x, pt[po[m - 1]].y - pt[po[0]].y);
  pcb_fprintf (f, "(polygon end, distance %`.2f)\n", dm);
  return dm;
}

#define TRY(x) if (x) goto try_error

double
process_path (path_t * plist, const potrace_param_t * param,
              const potrace_bitmap_t * bm, FILE * f, double scale,
              const char *var_cutdepth, const char *var_safeZ,
              const char *var_plunge, const char *var_feedrate)
{
  path_t *p;
  double dm = 0;
  int n = 0;
  /* call downstream function with each path */
  list_forall (p, plist)
  {
    TRY (calc_sums (p->priv));
    TRY (calc_lon (p->priv));
    TRY (bestpolygon (p->priv));
    TRY (adjust_vertices (p->priv));
    fprintf (f, "(polygon %d)\n", ++n);
    dm += plotpolygon (p->priv, f, scale, var_cutdepth, var_safeZ, var_plunge,
                       var_feedrate);
/*  No need to extract curves
        TRY(smooth(&p->priv->curve, p->sign, param->alphamax));
    if (param->opticurve) {
      TRY(opticurve(p->priv, param->opttolerance));
      p->priv->fcurve = &p->priv->ocurve;
    } else {
      p->priv->fcurve = &p->priv->curve;
    }
    privcurve_to_curve(p->priv->fcurve, &p->curve);*/
  }
/*      fprintf(f,"(end, total distance %.2fmm = %.2fin)\n",25.4*dm,dm); */
  return dm;

try_error:
  return -1;
}