chull.c

/*
This code is described in "Computational Geometry in C" (Second Edition),
Chapter 4.  It is not written to be comprehensible without the 
explanation in that book.

Input: 3n integer coordinates for the points.
Output: the 3D convex hull, in postscript with embedded comments
        showing the vertices and faces.

Compile: gcc -o chull chull.c (or simply: make)

Written by Joseph O'Rourke, with contributions by 
  Kristy Anderson, John Kutcher, Catherine Schevon, Susan Weller.
Last modified: May 2000
Questions to orourke@cs.smith.edu.

--------------------------------------------------------------------
This code is Copyright 2000 by Joseph O'Rourke.  It may be freely 
redistributed in its entirety provided that this copyright notice is 
not removed.
--------------------------------------------------------------------
*/

#include "macros.h"

#include <stdio.h>

/*Define Boolean type */
typedef enum { FALSE, TRUE } bool;

/* Define vertex indices. */
#define X 0
#define Y 1
#define Z 2

/* Define structures for vertices, edges and faces */
typedef struct tVertexStructure tsVertex;
typedef tsVertex *tVertex;

typedef struct tEdgeStructure tsEdge;
typedef tsEdge *tEdge;

typedef struct tFaceStructure tsFace;
typedef tsFace *tFace;

struct tVertexStructure
{
    int v[3];
    int vnum;
    tEdge duplicate; /* pointer to incident cone edge (or NULL) */
    bool onhull;     /* T iff point on hull. */
    bool mark;       /* T iff point already processed. */
    tVertex next, prev;
};

struct tEdgeStructure
{
    tFace adjface[2];
    tVertex endpts[2];
    tFace newface; /* pointer to incident cone face. */
    bool delete;   /* T iff edge should be delete. */
    tEdge next, prev;
};

struct tFaceStructure
{
    tEdge edge[3];
    tVertex vertex[3];
    bool visible; /* T iff face visible from new point. */
    tFace next, prev;
};

/* Define flags */
#define ONHULL    TRUE
#define REMOVED   TRUE
#define VISIBLE   TRUE
#define PROCESSED TRUE
#define SAFE      1000000 /* Range of safe coord values. */

/* Global variable definitions */
tVertex vertices = NULL;
tEdge edges      = NULL;
tFace faces      = NULL;
bool debug       = FALSE;
bool check       = FALSE;

/* Function declarations */
tVertex MakeNullVertex(void);
void ReadVertices(void);
void Print(void);
void SubVec(int a[3], int b[3], int c[3]);
void DoubleTriangle(void);
void ConstructHull(void);
bool AddOne(tVertex p);
int VolumeSign(tFace f, tVertex p);
int Volumei(tFace f, tVertex p);
tFace MakeConeFace(tEdge e, tVertex p);
void MakeCcw(tFace f, tEdge e, tVertex p);
tEdge MakeNullEdge(void);
tFace MakeNullFace(void);
tFace MakeFace(tVertex v0, tVertex v1, tVertex v2, tFace f);
void CleanUp(tVertex *pvnext);
void CleanEdges(void);
void CleanFaces(void);
void CleanVertices(tVertex *pvnext);
bool Collinear(tVertex a, tVertex b, tVertex c);
void CheckEuler(int V, int E, int F);
void PrintPoint(tVertex p);
void Checks(void);
void Consistency(void);
void Convexity(void);
void PrintOut(tVertex v);
void PrintVertices(void);
void PrintEdges(void);
void PrintFaces(void);
void CheckEndpts(void);
void EdgeOrderOnFaces(void);

/*-------------------------------------------------------------------*/
/* Geehalel: Use as a library */
/* 
main( int argc, char *argv[] )
{

  if ( argc > 1 && argv[1][0] == '-' ) {
    if( argv[1][1] ==  'd' ) {
      debug = TRUE;
      check = TRUE;
      fprintf( stderr, "Debug and check mode\n");
    }
    if( argv[1][1] == 'c' ) {
      check = TRUE;
      fprintf( stderr, "Check mode\n");
    }
  }
  else if ( argc > 1 && argv[1][0] != '-' ) {
    printf ("Usage:  %s -d[ebug] c[heck]\n", *argv );
    printf ("x y z coords of vertices from stdin\n");
    exit(1);
  }
  
   ReadVertices();
   DoubleTriangle();
   ConstructHull();
   EdgeOrderOnFaces();
   Print();
}
*/
/*---------------------------------------------------------------------
MakeNullVertex: Makes a vertex, nulls out fields.
---------------------------------------------------------------------*/
tVertex MakeNullVertex(void)
{
    tVertex v;

    NEW(v, tsVertex);
    v->duplicate = NULL;
    v->onhull    = !ONHULL;
    v->mark      = !PROCESSED;
    ADD(vertices, v);

    return v;
}

/*---------------------------------------------------------------------
ReadVertices: Reads in the vertices, and links them into a circular
list with MakeNullVertex.  There is no need for the # of vertices to be
the first line: the function looks for EOF instead.  Sets the global
variable vertices via the ADD macro.
---------------------------------------------------------------------*/
void ReadVertices(void)
{
    tVertex v;
    int x, y, z;
    int vnum = 0;

    while (scanf("%d %d %d", &x, &y, &z) != EOF)
    {
        v       = MakeNullVertex();
        v->v[X] = x;
        v->v[Y] = y;
        v->v[Z] = z;
        v->vnum = vnum++;
        if ((abs(x) > SAFE) || (abs(y) > SAFE) || (abs(z) > SAFE))
        {
            printf("Coordinate of vertex below might be too large: run with -d flag\n");
            PrintPoint(v);
        }
    }
}

/*---------------------------------------------------------------------
Print: Prints out the vertices and the faces.  Uses the vnum indices 
corresponding to the order in which the vertices were input.
Output is in PostScript format.
---------------------------------------------------------------------*/
void Print(void)
{
    /* Pointers to vertices, edges, faces. */
    tVertex v;
    tEdge e;
    tFace f;
    int xmin, ymin, xmax, ymax;
    int a[3], b[3]; /* used to compute normal vector */
    /* Counters for Euler's formula. */
    int V = 0, E = 0, F = 0;
    /* Note: lowercase==pointer, uppercase==counter. */

    /*-- find X min & max --*/
    v    = vertices;
    xmin = xmax = v->v[X];
    do
    {
        if (v->v[X] > xmax)
            xmax = v->v[X];
        else if (v->v[X] < xmin)
            xmin = v->v[X];
        v = v->next;
    } while (v != vertices);

    /*-- find Y min & max --*/
    v    = vertices;
    ymin = ymax = v->v[Y];
    do
    {
        if (v->v[Y] > ymax)
            ymax = v->v[Y];
        else if (v->v[Y] < ymin)
            ymin = v->v[Y];
        v = v->next;
    } while (v != vertices);

    /* PostScript header */
    printf("%%!PS\n");
    printf("%%%%BoundingBox: %d %d %d %d\n", xmin, ymin, xmax, ymax);
    printf(".00 .00 setlinewidth\n");
    printf("%d %d translate\n", -xmin + 72, -ymin + 72);
    /* The +72 shifts the figure one inch from the lower left corner */

    /* Vertices. */
    v = vertices;
    do
    {
        if (v->mark)
            V++;
        v = v->next;
    } while (v != vertices);
    printf("\n%%%% Vertices:\tV = %d\n", V);
    printf("%%%% index:\tx\ty\tz\n");
    do
    {
        printf("%%%% %5d:\t%d\t%d\t%d\n", v->vnum, v->v[X], v->v[Y], v->v[Z]);
        v = v->next;
    } while (v != vertices);

    /* Faces. */
    /* visible faces are printed as PS output */
    f = faces;
    do
    {
        ++F;
        f = f->next;
    } while (f != faces);
    printf("\n%%%% Faces:\tF = %d\n", F);
    printf("%%%% Visible faces only: \n");
    do
    {
        /* Print face only if it is visible: if normal vector >= 0 */
        SubVec(f->vertex[1]->v, f->vertex[0]->v, a);
        SubVec(f->vertex[2]->v, f->vertex[1]->v, b);
        if ((a[0] * b[1] - a[1] * b[0]) >= 0)
        {
            printf("%%%% vnums:  %d  %d  %d\n", f->vertex[0]->vnum, f->vertex[1]->vnum, f->vertex[2]->vnum);
            printf("newpath\n");
            printf("%d\t%d\tmoveto\n", f->vertex[0]->v[X], f->vertex[0]->v[Y]);
            printf("%d\t%d\tlineto\n", f->vertex[1]->v[X], f->vertex[1]->v[Y]);
            printf("%d\t%d\tlineto\n", f->vertex[2]->v[X], f->vertex[2]->v[Y]);
            printf("closepath stroke\n\n");
        }
        f = f->next;
    } while (f != faces);

    /* prints a list of all faces */
    printf("%%%% List of all faces: \n");
    printf("%%%%\tv0\tv1\tv2\t(vertex indices)\n");
    do
    {
        printf("%%%%\t%d\t%d\t%d\n", f->vertex[0]->vnum, f->vertex[1]->vnum, f->vertex[2]->vnum);
        f = f->next;
    } while (f != faces);

    /* Edges. */
    e = edges;
    do
    {
        E++;
        e = e->next;
    } while (e != edges);
    printf("\n%%%% Edges:\tE = %d\n", E);
    /* Edges not printed out (but easily added). */

    printf("\nshowpage\n\n");

    check = TRUE;
    CheckEuler(V, E, F);
}

/*---------------------------------------------------------------------
SubVec:  Computes a - b and puts it into c.
---------------------------------------------------------------------*/
void SubVec(int a[3], int b[3], int c[3])
{
    int i;

    for (i = 0; i < 2; i++)
        c[i] = a[i] - b[i];
}

/*---------------------------------------------------------------------
 DoubleTriangle builds the initial double triangle.  It first finds 3 
 noncollinear points and makes two faces out of them, in opposite order.
 It then finds a fourth point that is not coplanar with that face.  The  
 vertices are stored in the face structure in counterclockwise order so 
 that the volume between the face and the point is negative. Lastly, the
 3 newfaces to the fourth point are constructed and the data structures
 are cleaned up. 
---------------------------------------------------------------------*/
void DoubleTriangle(void)
{
    tVertex v0, v1, v2, v3;
    tFace f0, f1 = NULL;
    int vol;

    /* Find 3 noncollinear points. */
    v0 = vertices;
    while (Collinear(v0, v0->next, v0->next->next))
        if ((v0 = v0->next) == vertices)
            printf("DoubleTriangle:  All points are Collinear!\n"), exit(0);
    v1 = v0->next;
    v2 = v1->next;

    /* Mark the vertices as processed. */
    v0->mark = PROCESSED;
    v1->mark = PROCESSED;
    v2->mark = PROCESSED;

    /* Create the two "twin" faces. */
    f0 = MakeFace(v0, v1, v2, f1);
    f1 = MakeFace(v2, v1, v0, f0);

    /* Link adjacent face fields. */
    f0->edge[0]->adjface[1] = f1;
    f0->edge[1]->adjface[1] = f1;
    f0->edge[2]->adjface[1] = f1;
    f1->edge[0]->adjface[1] = f0;
    f1->edge[1]->adjface[1] = f0;
    f1->edge[2]->adjface[1] = f0;

    /* Find a fourth, noncoplanar point to form tetrahedron. */
    v3  = v2->next;
    vol = VolumeSign(f0, v3);
    while (!vol)
    {
        if ((v3 = v3->next) == v0)
            printf("DoubleTriangle:  All points are coplanar!\n"), exit(0);
        vol = VolumeSign(f0, v3);
    }

    /* Insure that v3 will be the first added. */
    vertices = v3;
    if (debug)
    {
        fprintf(stderr, "DoubleTriangle: finished. Head repositioned at v3.\n");
        PrintOut(vertices);
    }
}

/*---------------------------------------------------------------------
ConstructHull adds the vertices to the hull one at a time.  The hull
vertices are those in the list marked as onhull.
---------------------------------------------------------------------*/
void ConstructHull(void)
{
    tVertex v, vnext;
    bool changed = FALSE; /* T if addition changes hull; not used. */

    v = vertices;
    do
    {
        vnext = v->next;
        if (!v->mark)
        {
            v->mark = PROCESSED;
            changed = AddOne(v);
            CleanUp(&vnext); /* Pass down vnext in case it gets deleted. */

            if (check)
            {
                fprintf(stderr, "ConstructHull: After Add of %d & Cleanup:\n", v->vnum);
                Checks();
            }
            if (debug)
                PrintOut(v);
        }
        v = vnext;
    } while (v != vertices);
}

/*---------------------------------------------------------------------
AddOne is passed a vertex.  It first determines all faces visible from 
that point.  If none are visible then the point is marked as not 
onhull.  Next is a loop over edges.  If both faces adjacent to an edge
are visible, then the edge is marked for deletion.  If just one of the
adjacent faces is visible then a new face is constructed.
---------------------------------------------------------------------*/
bool AddOne(tVertex p)
{
    tFace f;
    tEdge e, temp;
    int vol;
    bool vis = FALSE;

    if (debug)
    {
        fprintf(stderr, "AddOne: starting to add v%d.\n", p->vnum);
        PrintOut(vertices);
    }

    /* Mark faces visible from p. */
    f = faces;
    do
    {
        vol = VolumeSign(f, p);
        if (debug)
            fprintf(stderr, "faddr: %6p   paddr: %6p   Vol = %d\n", f, p, vol);
        if (vol < 0)
        {
            f->visible = VISIBLE;
            vis        = TRUE;
        }
        f = f->next;
    } while (f != faces);

    /* If no faces are visible from p, then p is inside the hull. */
    if (!vis)
    {
        p->onhull = !ONHULL;
        return FALSE;
    }

    /* Mark edges in interior of visible region for deletion.
      Erect a newface based on each border edge. */
    e = edges;
    do
    {
        temp = e->next;
        if (e->adjface[0]->visible && e->adjface[1]->visible)
            /* e interior: mark for deletion. */
            e->delete = REMOVED;
        else if (e->adjface[0]->visible || e->adjface[1]->visible)
            /* e border: make a new face. */
            e->newface = MakeConeFace(e, p);
        e = temp;
    } while (e != edges);
    return TRUE;
}

/*---------------------------------------------------------------------
VolumeSign returns the sign of the volume of the tetrahedron determined by f
and p.  VolumeSign is +1 iff p is on the negative side of f,
where the positive side is determined by the rh-rule.  So the volume 
is positive if the ccw normal to f points outside the tetrahedron.
The final fewer-multiplications form is due to Bob Williamson.
---------------------------------------------------------------------*/
int VolumeSign(tFace f, tVertex p)
{
    double vol;
    int voli;
    double ax, ay, az, bx, by, bz, cx, cy, cz;

    ax = f->vertex[0]->v[X] - p->v[X];
    ay = f->vertex[0]->v[Y] - p->v[Y];
    az = f->vertex[0]->v[Z] - p->v[Z];
    bx = f->vertex[1]->v[X] - p->v[X];
    by = f->vertex[1]->v[Y] - p->v[Y];
    bz = f->vertex[1]->v[Z] - p->v[Z];
    cx = f->vertex[2]->v[X] - p->v[X];
    cy = f->vertex[2]->v[Y] - p->v[Y];
    cz = f->vertex[2]->v[Z] - p->v[Z];

    vol = ax * (by * cz - bz * cy) + ay * (bz * cx - bx * cz) + az * (bx * cy - by * cx);

    if (debug)
    {
        /* Compute the volume using integers for comparison. */
        voli = Volumei(f, p);
        fprintf(stderr, "Face=%6p; Vertex=%d: vol(int) = %d, vol(double) = %lf\n", f, p->vnum, voli, vol);
    }

    /* The volume should be an integer. */
    if (vol > 0.5)
        return 1;
    else if (vol < -0.5)
        return -1;
    else
        return 0;
}
/*---------------------------------------------------------------------
Same computation, but computes using ints, and returns the actual volume.
---------------------------------------------------------------------*/
int Volumei(tFace f, tVertex p)
{
    int vol = 0;
    int ax = 0, ay = 0, az = 0, bx = 0, by = 0, bz = 0, cx = 0, cy = 0, cz = 0;

    ax = f->vertex[0]->v[X] - p->v[X];
    ay = f->vertex[0]->v[Y] - p->v[Y];
    az = f->vertex[0]->v[Z] - p->v[Z];
    bx = f->vertex[1]->v[X] - p->v[X];
    by = f->vertex[1]->v[Y] - p->v[Y];
    bz = f->vertex[1]->v[Z] - p->v[Z];
    cx = f->vertex[2]->v[X] - p->v[X];
    cy = f->vertex[2]->v[Y] - p->v[Y];
    cz = f->vertex[2]->v[Z] - p->v[Z];

    vol = (ax * (by * cz - bz * cy) + ay * (bz * cx - bx * cz) + az * (bx * cy - by * cx));

    return vol;
}

/*-------------------------------------------------------------------*/
void PrintPoint(tVertex p)
{
    int i;

    for (i = 0; i < 3; i++)
        printf("\t%d", p->v[i]);
    putchar('\n');
}

/*---------------------------------------------------------------------
MakeConeFace makes a new face and two new edges between the 
edge and the point that are passed to it. It returns a pointer to
the new face.
---------------------------------------------------------------------*/
tFace MakeConeFace(tEdge e, tVertex p)
{
    tEdge new_edge[2];
    tFace new_face;
    int i, j;

    /* Make two new edges (if don't already exist). */
    for (i = 0; i < 2; ++i)
        /* If the edge exists, copy it into new_edge. */
        if (!(new_edge[i] = e->endpts[i]->duplicate))
        {
            /* Otherwise (duplicate is NULL), MakeNullEdge. */
            new_edge[i]             = MakeNullEdge();
            new_edge[i]->endpts[0]  = e->endpts[i];
            new_edge[i]->endpts[1]  = p;
            e->endpts[i]->duplicate = new_edge[i];
        }

    /* Make the new face. */
    new_face          = MakeNullFace();
    new_face->edge[0] = e;
    new_face->edge[1] = new_edge[0];
    new_face->edge[2] = new_edge[1];
    MakeCcw(new_face, e, p);

    /* Set the adjacent face pointers. */
    for (i = 0; i < 2; ++i)
        for (j = 0; j < 2; ++j)
            /* Only one NULL link should be set to new_face. */
            if (!new_edge[i]->adjface[j])
            {
                new_edge[i]->adjface[j] = new_face;
                break;
            }

    return new_face;
}

/*---------------------------------------------------------------------
MakeCcw puts the vertices in the face structure in counterclock wise 
order.  We want to store the vertices in the same 
order as in the visible face.  The third vertex is always p.

Although no specific ordering of the edges of a face are used
by the code, the following condition is maintained for each face f:
one of the two endpoints of f->edge[i] matches f->vertex[i]. 
But note that this does not imply that f->edge[i] is between
f->vertex[i] and f->vertex[(i+1)%3].  (Thanks to Bob Williamson.)
---------------------------------------------------------------------*/
void MakeCcw(tFace f, tEdge e, tVertex p)
{
    tFace fv; /* The visible face adjacent to e */
    int i;    /* Index of e->endpoint[0] in fv. */
    tEdge s;  /* Temporary, for swapping */

    if (e->adjface[0]->visible)
        fv = e->adjface[0];
    else
        fv = e->adjface[1];

    /* Set vertex[0] & [1] of f to have the same orientation
      as do the corresponding vertices of fv. */
    for (i = 0; fv->vertex[i] != e->endpts[0]; ++i)
        ;
    /* Orient f the same as fv. */
    if (fv->vertex[(i + 1) % 3] != e->endpts[1])
    {
        f->vertex[0] = e->endpts[1];
        f->vertex[1] = e->endpts[0];
    }
    else
    {
        f->vertex[0] = e->endpts[0];
        f->vertex[1] = e->endpts[1];
        SWAP(s, f->edge[1], f->edge[2]);
    }
    /* This swap is tricky. e is edge[0]. edge[1] is based on endpt[0],
      edge[2] on endpt[1].  So if e is oriented "forwards," we
      need to move edge[1] to follow [0], because it precedes. */

    f->vertex[2] = p;
}

/*---------------------------------------------------------------------
MakeNullEdge creates a new cell and initializes all pointers to NULL
and sets all flags to off.  It returns a pointer to the empty cell.
---------------------------------------------------------------------*/
tEdge MakeNullEdge(void)
{
    tEdge e;

    NEW(e, tsEdge);
    e->adjface[0] = e->adjface[1] = e->newface = NULL;
    e->endpts[0] = e->endpts[1] = NULL;
    e->delete                   = !REMOVED;
    ADD(edges, e);
    return e;
}

/*--------------------------------------------------------------------
MakeNullFace creates a new face structure and initializes all of its
flags to NULL and sets all the flags to off.  It returns a pointer
to the empty cell.
---------------------------------------------------------------------*/
tFace MakeNullFace(void)
{
    tFace f;
    int i;

    NEW(f, tsFace);
    for (i = 0; i < 3; ++i)
    {
        f->edge[i]   = NULL;
        f->vertex[i] = NULL;
    }
    f->visible = !VISIBLE;
    ADD(faces, f);
    return f;
}

/*---------------------------------------------------------------------
MakeFace creates a new face structure from three vertices (in ccw
order).  It returns a pointer to the face.
---------------------------------------------------------------------*/
tFace MakeFace(tVertex v0, tVertex v1, tVertex v2, tFace fold)
{
    tFace f;
    tEdge e0, e1, e2;

    /* Create edges of the initial triangle. */
    if (!fold)
    {
        e0 = MakeNullEdge();
        e1 = MakeNullEdge();
        e2 = MakeNullEdge();
    }
    else
    { /* Copy from fold, in reverse order. */
        e0 = fold->edge[2];
        e1 = fold->edge[1];
        e2 = fold->edge[0];
    }
    e0->endpts[0] = v0;
    e0->endpts[1] = v1;
    e1->endpts[0] = v1;
    e1->endpts[1] = v2;
    e2->endpts[0] = v2;
    e2->endpts[1] = v0;

    /* Create face for triangle. */
    f            = MakeNullFace();
    f->edge[0]   = e0;
    f->edge[1]   = e1;
    f->edge[2]   = e2;
    f->vertex[0] = v0;
    f->vertex[1] = v1;
    f->vertex[2] = v2;

    /* Link edges to face. */
    e0->adjface[0] = e1->adjface[0] = e2->adjface[0] = f;

    return f;
}

/*---------------------------------------------------------------------
CleanUp goes through each data structure list and clears all
flags and NULLs out some pointers.  The order of processing
(edges, faces, vertices) is important.
---------------------------------------------------------------------*/
void CleanUp(tVertex *pvnext)
{
    CleanEdges();
    CleanFaces();
    CleanVertices(pvnext);
}

/*---------------------------------------------------------------------
CleanEdges runs through the edge list and cleans up the structure.
If there is a newface then it will put that face in place of the 
visible face and NULL out newface. It also deletes so marked edges.
---------------------------------------------------------------------*/
void CleanEdges(void)
{
    tEdge e; /* Primary index into edge list. */
    tEdge t; /* Temporary edge pointer. */

    /* Integrate the newface's into the data structure. */
    /* Check every edge. */
    e = edges;
    do
    {
        if (e->newface)
        {
            if (e->adjface[0]->visible)
                e->adjface[0] = e->newface;
            else
                e->adjface[1] = e->newface;
            e->newface = NULL;
        }
        e = e->next;
    } while (e != edges);

    /* Delete any edges marked for deletion. */
    while (edges && edges->delete)
    {
        e = edges;
        DELETE(edges, e);
    }
    e = edges->next;
    do
    {
        if (e->delete)
        {
            t = e;
            e = e->next;
            DELETE(edges, t);
        }
        else
            e = e->next;
    } while (e != edges);
}

/*---------------------------------------------------------------------
CleanFaces runs through the face list and deletes any face marked visible.
---------------------------------------------------------------------*/
void CleanFaces(void)
{
    tFace f; /* Primary pointer into face list. */
    tFace t; /* Temporary pointer, for deleting. */

    while (faces && faces->visible)
    {
        f = faces;
        DELETE(faces, f);
    }
    f = faces->next;
    do
    {
        if (f->visible)
        {
            t = f;
            f = f->next;
            DELETE(faces, t);
        }
        else
            f = f->next;
    } while (f != faces);
}

/*---------------------------------------------------------------------
CleanVertices runs through the vertex list and deletes the 
vertices that are marked as processed but are not incident to any 
undeleted edges. 
The pointer to vnext, pvnext, is used to alter vnext in
ConstructHull() if we are about to delete vnext.
---------------------------------------------------------------------*/
void CleanVertices(tVertex *pvnext)
{
    tEdge e;
    tVertex v, t;

    /* Mark all vertices incident to some undeleted edge as on the hull. */
    e = edges;
    do
    {
        e->endpts[0]->onhull = e->endpts[1]->onhull = ONHULL;
        e                                           = e->next;
    } while (e != edges);

    /* Delete all vertices that have been processed but
      are not on the hull. */
    while (vertices && vertices->mark && !vertices->onhull)
    {
        /* If about to delete vnext, advance it first. */
        v = vertices;
        if (v == *pvnext)
            *pvnext = v->next;
        DELETE(vertices, v);
    }
    v = vertices->next;
    do
    {
        if (v->mark && !v->onhull)
        {
            t = v;
            v = v->next;
            if (t == *pvnext)
                *pvnext = t->next;
            DELETE(vertices, t);
        }
        else
            v = v->next;
    } while (v != vertices);

    /* Reset flags. */
    v = vertices;
    do
    {
        v->duplicate = NULL;
        v->onhull    = !ONHULL;
        v            = v->next;
    } while (v != vertices);
}

/*---------------------------------------------------------------------
Collinear checks to see if the three points given are collinear,
by checking to see if each element of the cross product is zero.
---------------------------------------------------------------------*/
bool Collinear(tVertex a, tVertex b, tVertex c)
{
    return (c->v[Z] - a->v[Z]) * (b->v[Y] - a->v[Y]) - (b->v[Z] - a->v[Z]) * (c->v[Y] - a->v[Y]) == 0 &&
           (b->v[Z] - a->v[Z]) * (c->v[X] - a->v[X]) - (b->v[X] - a->v[X]) * (c->v[Z] - a->v[Z]) == 0 &&
           (b->v[X] - a->v[X]) * (c->v[Y] - a->v[Y]) - (b->v[Y] - a->v[Y]) * (c->v[X] - a->v[X]) == 0;
}

/*---------------------------------------------------------------------
Consistency runs through the edge list and checks that all
adjacent faces have their endpoints in opposite order.  This verifies
that the vertices are in counterclockwise order.
---------------------------------------------------------------------*/
void Consistency(void)
{
    register tEdge e;
    register int i, j;

    e = edges;

    do
    {
        /* find index of endpoint[0] in adjacent face[0] */
        for (i = 0; e->adjface[0]->vertex[i] != e->endpts[0]; ++i)
            ;

        /* find index of endpoint[0] in adjacent face[1] */
        for (j = 0; e->adjface[1]->vertex[j] != e->endpts[0]; ++j)
            ;

        /* check if the endpoints occur in opposite order */
        if (!(e->adjface[0]->vertex[(i + 1) % 3] == e->adjface[1]->vertex[(j + 2) % 3] ||
              e->adjface[0]->vertex[(i + 2) % 3] == e->adjface[1]->vertex[(j + 1) % 3]))
            break;
        e = e->next;

    } while (e != edges);

    if (e != edges)
        fprintf(stderr, "Checks: edges are NOT consistent.\n");
    else
        fprintf(stderr, "Checks: edges consistent.\n");
}

/*---------------------------------------------------------------------
Convexity checks that the volume between every face and every
point is negative.  This shows that each point is inside every face
and therefore the hull is convex.
---------------------------------------------------------------------*/
void Convexity(void)
{
    register tFace f;
    register tVertex v;
    int vol;

    f = faces;

    do
    {
        v = vertices;
        do
        {
            if (v->mark)
            {
                vol = VolumeSign(f, v);
                if (vol < 0)
                    break;
            }
            v = v->next;
        } while (v != vertices);

        f = f->next;

    } while (f != faces);

    if (f != faces)
        fprintf(stderr, "Checks: NOT convex.\n");
    else if (check)
        fprintf(stderr, "Checks: convex.\n");
}

/*---------------------------------------------------------------------
CheckEuler checks Euler's relation, as well as its implications when
all faces are known to be triangles.  Only prints positive information
when debug is true, but always prints negative information.
---------------------------------------------------------------------*/
void CheckEuler(int V, int E, int F)
{
    if (check)
        fprintf(stderr, "Checks: V, E, F = %d %d %d:\t", V, E, F);

    if ((V - E + F) != 2)
        fprintf(stderr, "Checks: V-E+F != 2\n");
    else if (check)
        fprintf(stderr, "V-E+F = 2\t");

    if (F != (2 * V - 4))
        fprintf(stderr, "Checks: F=%d != 2V-4=%d; V=%d\n", F, 2 * V - 4, V);
    else if (check)
        fprintf(stderr, "F = 2V-4\t");

    if ((2 * E) != (3 * F))
        fprintf(stderr, "Checks: 2E=%d != 3F=%d; E=%d, F=%d\n", 2 * E, 3 * F, E, F);
    else if (check)
        fprintf(stderr, "2E = 3F\n");
}

/*-------------------------------------------------------------------*/
void Checks(void)
{
    tVertex v;
    tEdge e;
    tFace f;
    int V = 0, E = 0, F = 0;

    Consistency();
    Convexity();
    if ((v = vertices))
        do
        {
            if (v->mark)
                V++;
            v = v->next;
        } while (v != vertices);
    if ((e = edges))
        do
        {
            E++;
            e = e->next;
        } while (e != edges);
    if ((f = faces))
        do
        {
            F++;
            f = f->next;
        } while (f != faces);
    CheckEuler(V, E, F);
    CheckEndpts();
}

/*===================================================================
These functions are used whenever the debug flag is set.
They print out the entire contents of each data structure.  
Printing is to standard error.  To grab the output in a file in the csh, 
use this:
	chull < i.file >&! o.file
=====================================================================*/
/*-------------------------------------------------------------------*/
void PrintOut(tVertex v)
{
    fprintf(stderr, "\nHead vertex %d = %6p :\n", v->vnum, v);
    PrintVertices();
    PrintEdges();
    PrintFaces();
}

/*-------------------------------------------------------------------*/
void PrintVertices(void)
{
    tVertex temp;

    temp = vertices;
    fprintf(stderr, "Vertex List\n");
    if (vertices)
        do
        {
            fprintf(stderr, "  addr %6p\t", vertices);
            fprintf(stderr, "  vnum %4d", vertices->vnum);
            fprintf(stderr, "   (%6d,%6d,%6d)", vertices->v[X], vertices->v[Y], vertices->v[Z]);
            fprintf(stderr, "   active:%3d", vertices->onhull);
            fprintf(stderr, "   dup:%5p", vertices->duplicate);
            fprintf(stderr, "   mark:%2d\n", vertices->mark);
            vertices = vertices->next;
        } while (vertices != temp);
}

/*-------------------------------------------------------------------*/
void PrintEdges(void)
{
    tEdge temp;
    int i;

    temp = edges;
    fprintf(stderr, "Edge List\n");
    if (edges)
        do
        {
            fprintf(stderr, "  addr: %6p\t", edges);
            fprintf(stderr, "adj: ");
            for (i = 0; i < 2; ++i)
                fprintf(stderr, "%6p", (edges->adjface[i]));
            fprintf(stderr, "  endpts:");
            for (i = 0; i < 2; ++i)
                fprintf(stderr, "%4d", edges->endpts[i]->vnum);
            fprintf(stderr, "  del:%3d\n", edges->delete);
            edges = edges->next;
        } while (edges != temp);
}

/*-------------------------------------------------------------------*/
void PrintFaces(void)
{
    int i;
    tFace temp;

    temp = faces;
    fprintf(stderr, "Face List\n");
    if (faces)
        do
        {
            fprintf(stderr, "  addr: %10p  ", faces);
            fprintf(stderr, "  edges:");
            for (i = 0; i < 3; ++i)
                fprintf(stderr, "%10p ", faces->edge[i]);
            fprintf(stderr, "  vert:");
            for (i = 0; i < 3; ++i)
                fprintf(stderr, "%4d", faces->vertex[i]->vnum);
            fprintf(stderr, "  vis: %d\n", faces->visible);
            faces = faces->next;
        } while (faces != temp);
}

/*-------------------------------------------------------------------
Checks that, for each face, for each i={0,1,2}, the [i]th vertex of
that face is either the [0]th or [1]st endpoint of the [ith] edge of
the face.
-------------------------------------------------------------------*/
void CheckEndpts(void)
{
    int i;
    tFace fstart;
    tEdge e;
    tVertex v;
    bool error = FALSE;

    fstart = faces;
    if (faces)
        do
        {
            for (i = 0; i < 3; ++i)
            {
                v = faces->vertex[i];
                e = faces->edge[i];
                if (v != e->endpts[0] && v != e->endpts[1])
                {
                    error = TRUE;
                    fprintf(stderr, "CheckEndpts: Error!\n");
                    fprintf(stderr, "  addr: %8p;", faces);
                    fprintf(stderr, "  edges:");
                    fprintf(stderr, "(%3d,%3d)", e->endpts[0]->vnum, e->endpts[1]->vnum);
                    fprintf(stderr, "\n");
                }
            }
            faces = faces->next;
        } while (faces != fstart);

    if (error)
        fprintf(stderr, "Checks: ERROR found and reported above.\n");
    else
        fprintf(stderr, "Checks: All endpts of all edges of all faces check.\n");
}

/*------------------------------------------------------------------
  EdgeOrderOnFaces: puts e0 between v0 and v1, e1 between v1 and v2,
  e2 between v2 and v0 on each face.  This should be unnecessary, alas.
  Not used in code, but useful for other purposes.
------------------------------------------------------------------*/
void EdgeOrderOnFaces(void)
{
    tFace f = faces;
    tEdge new;
    int i, j;

    do
    {
        for (i = 0; i < 3; i++)
        {
            if (!(((f->edge[i]->endpts[0] == f->vertex[i]) && (f->edge[i]->endpts[1] == f->vertex[(i + 1) % 3])) ||
                  ((f->edge[i]->endpts[1] == f->vertex[i]) && (f->edge[i]->endpts[0] == f->vertex[(i + 1) % 3]))))
            {
                /* Change the order of the edges on the face: */
                for (j = 0; j < 3; j++)
                {
                    /* find the edge that should be there */
                    if (((f->edge[j]->endpts[0] == f->vertex[i]) &&
                         (f->edge[j]->endpts[1] == f->vertex[(i + 1) % 3])) ||
                        ((f->edge[j]->endpts[1] == f->vertex[i]) && (f->edge[j]->endpts[0] == f->vertex[(i + 1) % 3])))
                    {
                        /* Swap it with the one erroneously put into its place: */
                        if (debug)
                            fprintf(stderr, "Making a swap in EdgeOrderOnFaces: F(%d,%d,%d): e[%d] and e[%d]\n",
                                    f->vertex[0]->vnum, f->vertex[1]->vnum, f->vertex[2]->vnum, i, j);
                        new        = f->edge[i];
                        f->edge[i] = f->edge[j];
                        f->edge[j] = new;
                    }
                }
            }
        }
        f = f->next;
    } while (f != faces);
}