/* Triangulate
Efficient Triangulation Algorithm Suitable for Terrain Modelling
or
An Algorithm for Interpolating Irregularly-Spaced Data
with Applications in Terrain Modelling
Written by Paul Bourke
Presented at Pan Pacific Computer Conference, Beijing, China.
January 1989
Abstract
A discussion of a method that has been used with success in terrain
modelling to estimate the height at any point on the land surface
from irregularly distributed samples. The special requirements of
terrain modelling are discussed as well as a detailed description
of the algorithm and an example of its application.
http://paulbourke.net/papers/triangulate/
http://paulbourke.net/papers/triangulate/triangulate.c
*/
#include <stdlib.h>
#include <math.h>
#include "delaunay.h"
typedef struct {
int p1,p2;
} IEDGE;
#define TRUE 1
#define FALSE 0
#define EPSILON 0.000001
/*
Return TRUE if a point (xp,yp) is inside the circumcircle made up
of the points (x1,y1), (x2,y2), (x3,y3)
The circumcircle centre is returned in (xc,yc) and the radius r
NOTE: A point on the edge is inside the circumcircle
*/
static int
circumcircle (double xp,double yp,
double x1,double y1,double x2,double y2,double x3,double y3,
double *xc,double *yc,double *rsqr)
{
double m1,m2,mx1,mx2,my1,my2;
double dx,dy,drsqr;
double fabsy1y2 = fabs(y1-y2);
double fabsy2y3 = fabs(y2-y3);
/* Check for coincident points */
if (fabsy1y2 < EPSILON && fabsy2y3 < EPSILON)
return(FALSE);
if (fabsy1y2 < EPSILON) {
m2 = - (x3-x2) / (y3-y2);
mx2 = (x2 + x3) / 2.0;
my2 = (y2 + y3) / 2.0;
*xc = (x2 + x1) / 2.0;
*yc = m2 * (*xc - mx2) + my2;
} else if (fabsy2y3 < EPSILON) {
m1 = - (x2-x1) / (y2-y1);
mx1 = (x1 + x2) / 2.0;
my1 = (y1 + y2) / 2.0;
*xc = (x3 + x2) / 2.0;
*yc = m1 * (*xc - mx1) + my1;
} else {
m1 = - (x2-x1) / (y2-y1);
m2 = - (x3-x2) / (y3-y2);
mx1 = (x1 + x2) / 2.0;
mx2 = (x2 + x3) / 2.0;
my1 = (y1 + y2) / 2.0;
my2 = (y2 + y3) / 2.0;
*xc = (m1 * mx1 - m2 * mx2 + my2 - my1) / (m1 - m2);
if (fabsy1y2 > fabsy2y3) {
*yc = m1 * (*xc - mx1) + my1;
} else {
*yc = m2 * (*xc - mx2) + my2;
}
}
dx = x2 - *xc;
dy = y2 - *yc;
*rsqr = dx*dx + dy*dy;
dx = xp - *xc;
dy = yp - *yc;
drsqr = dx*dx + dy*dy;
/* Original
return((drsqr <= *rsqr) ? TRUE : FALSE);
Proposed by Chuck Morris */
return((drsqr - *rsqr) <= EPSILON ? TRUE : FALSE);
}
/*
Triangulation subroutine
Takes as input NV vertices in array pxyz
Returned is a list of ntri triangular faces in the array v
These triangles are arranged in a consistent clockwise order.
The triangle array 'v' should be malloced to 3 * nv
The vertex array pxyz must be big enough to hold 3 more points
The vertex array must be sorted in increasing x values say
qsort(p,nv,sizeof(XYZ),XYZCompare);
*/
int
delaunay (int nv,XYZ *pxyz,ITRIANGLE *v,int *ntri)
{
int *complete = NULL;
IEDGE *edges = NULL;
int nedge = 0;
int trimax,emax = 200;
int status = 0;
int inside;
int i,j,k;
double xp,yp,x1,y1,x2,y2,x3,y3,xc=0,yc=0,r=0;
double xmin,xmax,ymin,ymax,xmid,ymid;
double dx,dy,dmax;
/* Allocate memory for the completeness list, flag for each triangle */
trimax = 4 * nv;
if ((complete = malloc(trimax*sizeof(int))) == NULL) {
status = 1;
goto skip;
}
/* Allocate memory for the edge list */
if ((edges = malloc(emax*(long)sizeof(IEDGE))) == NULL) {
status = 2;
goto skip;
}
/*
Find the maximum and minimum vertex bounds.
This is to allow calculation of the bounding triangle
*/
xmin = pxyz[0].x;
ymin = pxyz[0].y;
xmax = xmin;
ymax = ymin;
for (i=1;i<nv;i++) {
if (pxyz[i].x < xmin) xmin = pxyz[i].x;
if (pxyz[i].x > xmax) xmax = pxyz[i].x;
if (pxyz[i].y < ymin) ymin = pxyz[i].y;
if (pxyz[i].y > ymax) ymax = pxyz[i].y;
}
dx = xmax - xmin;
dy = ymax - ymin;
dmax = (dx > dy) ? dx : dy;
xmid = (xmax + xmin) / 2.0;
ymid = (ymax + ymin) / 2.0;
/*
Set up the supertriangle
This is a triangle which encompasses all the sample points.
The supertriangle coordinates are added to the end of the
vertex list. The supertriangle is the first triangle in
the triangle list.
*/
pxyz[nv+0].x = xmid - 20 * dmax;
pxyz[nv+0].y = ymid - dmax;
pxyz[nv+0].z = 0.0;
pxyz[nv+1].x = xmid;
pxyz[nv+1].y = ymid + 20 * dmax;
pxyz[nv+1].z = 0.0;
pxyz[nv+2].x = xmid + 20 * dmax;
pxyz[nv+2].y = ymid - dmax;
pxyz[nv+2].z = 0.0;
v[0].p1 = nv;
v[0].p2 = nv+1;
v[0].p3 = nv+2;
complete[0] = FALSE;
*ntri = 1;
/*
Include each point one at a time into the existing mesh
*/
for (i=0;i<nv;i++) {
xp = pxyz[i].x;
yp = pxyz[i].y;
nedge = 0;
/*
Set up the edge buffer.
If the point (xp,yp) lies inside the circumcircle then the
three edges of that triangle are added to the edge buffer
and that triangle is removed.
*/
for (j=0;j<(*ntri);j++) {
if (complete[j])
continue;
x1 = pxyz[v[j].p1].x;
y1 = pxyz[v[j].p1].y;
x2 = pxyz[v[j].p2].x;
y2 = pxyz[v[j].p2].y;
x3 = pxyz[v[j].p3].x;
y3 = pxyz[v[j].p3].y;
inside = circumcircle(xp,yp,x1,y1,x2,y2,x3,y3,&xc,&yc,&r);
if (xc < xp && ((xp-xc)*(xp-xc)) > r)
complete[j] = TRUE;
if (inside) {
/* Check that we haven't exceeded the edge list size */
if (nedge+3 >= emax) {
emax += 100;
if ((edges = realloc(edges,emax*(long)sizeof(IEDGE))) == NULL) {
status = 3;
goto skip;
}
}
edges[nedge+0].p1 = v[j].p1;
edges[nedge+0].p2 = v[j].p2;
edges[nedge+1].p1 = v[j].p2;
edges[nedge+1].p2 = v[j].p3;
edges[nedge+2].p1 = v[j].p3;
edges[nedge+2].p2 = v[j].p1;
nedge += 3;
v[j] = v[(*ntri)-1];
complete[j] = complete[(*ntri)-1];
(*ntri)--;
j--;
}
}
/*
Tag multiple edges
Note: if all triangles are specified anticlockwise then all
interior edges are opposite pointing in direction.
*/
for (j=0;j<nedge-1;j++) {
for (k=j+1;k<nedge;k++) {
if ((edges[j].p1 == edges[k].p2) && (edges[j].p2 == edges[k].p1)) {
edges[j].p1 = -1;
edges[j].p2 = -1;
edges[k].p1 = -1;
edges[k].p2 = -1;
}
/* Shouldn't need the following, see note above */
if ((edges[j].p1 == edges[k].p1) && (edges[j].p2 == edges[k].p2)) {
edges[j].p1 = -1;
edges[j].p2 = -1;
edges[k].p1 = -1;
edges[k].p2 = -1;
}
}
}
/*
Form new triangles for the current point
Skipping over any tagged edges.
All edges are arranged in clockwise order.
*/
for (j=0;j<nedge;j++) {
if (edges[j].p1 < 0 || edges[j].p2 < 0)
continue;
if ((*ntri) >= trimax) {
status = 4;
goto skip;
}
v[*ntri].p1 = edges[j].p1;
v[*ntri].p2 = edges[j].p2;
v[*ntri].p3 = i;
complete[*ntri] = FALSE;
(*ntri)++;
}
}
/*
Remove triangles with supertriangle vertices
These are triangles which have a vertex number greater than nv
*/
for (i=0;i<(*ntri);i++) {
if (v[i].p1 >= nv || v[i].p2 >= nv || v[i].p3 >= nv) {
v[i] = v[(*ntri)-1];
(*ntri)--;
i--;
}
}
skip:
free(edges);
free(complete);
return(status);
}
int
delaunay_xyzcompare (const void *v1, const void *v2)
{
const XYZ *p1,*p2;
p1 = v1;
p2 = v2;
if (p1->x < p2->x)
return(-1);
else if (p1->x > p2->x)
return(1);
else
return(0);
}
