/*
Solving the Nearest Point-on-Curve Problem
and
A Bezier Curve-Based Root-Finder
by Philip J. Schneider
from "Graphics Gems", Academic Press, 1990
*/
/* point_on_curve.c */
#include <stdio.h>
#include <malloc.h>
#include <math.h>
#include "GraphicsGems.h"
#define TESTMODE
/*
* Forward declarations
*/
Point2 NearestPointOnCurve();
static int FindRoots();
static Point2 *ConvertToBezierForm();
static double ComputeXIntercept();
static int ControlPolygonFlatEnough();
static int CrossingCount();
static Point2 Bezier();
static Vector2 V2ScaleII();
int MAXDEPTH = 64; /* Maximum depth for recursion */
#define EPSILON (ldexp(1.0,-MAXDEPTH-1)) /*Flatness control value */
#define DEGREE 3 /* Cubic Bezier curve */
#define W_DEGREE 5 /* Degree of eqn to find roots of */
#ifdef TESTMODE
/*
* main :
* Given a cubic Bezier curve (i.e., its control points), and some
* arbitrary point in the plane, find the point on the curve
* closest to that arbitrary point.
*/
main()
{
static Point2 bezCurve[4] = { /* A cubic Bezier curve */
{ 0.0, 0.0 },
{ 1.0, 2.0 },
{ 3.0, 3.0 },
{ 4.0, 2.0 },
};
static Point2 arbPoint = { 3.5, 2.0 }; /*Some arbitrary point*/
Point2 pointOnCurve; /* Nearest point on the curve */
/* Find the closest point */
pointOnCurve = NearestPointOnCurve(arbPoint, bezCurve);
printf("pointOnCurve : (%4.4f, %4.4f)\n", pointOnCurve.x,
pointOnCurve.y);
}
#endif /* TESTMODE */
/*
* NearestPointOnCurve :
* Compute the parameter value of the point on a Bezier
* curve segment closest to some arbtitrary, user-input point.
* Return the point on the curve at that parameter value.
*
*/
Point2 NearestPointOnCurve(P, V)
Point2 P; /* The user-supplied point */
Point2 *V; /* Control points of cubic Bezier */
{
Point2 *w; /* Ctl pts for 5th-degree eqn */
double t_candidate[W_DEGREE]; /* Possible roots */
int n_solutions; /* Number of roots found */
double t; /* Parameter value of closest pt*/
/* Convert problem to 5th-degree Bezier form */
w = ConvertToBezierForm(P, V);
/* Find all possible roots of 5th-degree equation */
n_solutions = FindRoots(w, W_DEGREE, t_candidate, 0);
free((char *)w);
/* Compare distances of P to all candidates, and to t=0, and t=1 */
{
double dist, new_dist;
Point2 p;
Vector2 v;
int i;
/* Check distance to beginning of curve, where t = 0 */
dist = V2SquaredLength(V2Sub(&P, &V[0], &v));
t = 0.0;
/* Find distances for candidate points */
for (i = 0; i < n_solutions; i++) {
p = Bezier(V, DEGREE, t_candidate[i],
(Point2 *)NULL, (Point2 *)NULL);
new_dist = V2SquaredLength(V2Sub(&P, &p, &v));
if (new_dist < dist) {
dist = new_dist;
t = t_candidate[i];
}
}
/* Finally, look at distance to end point, where t = 1.0 */
new_dist = V2SquaredLength(V2Sub(&P, &V[DEGREE], &v));
if (new_dist < dist) {
dist = new_dist;
t = 1.0;
}
}
/* Return the point on the curve at parameter value t */
printf("t : %4.12f\n", t);
return (Bezier(V, DEGREE, t, (Point2 *)NULL, (Point2 *)NULL));
}
/*
* ConvertToBezierForm :
* Given a point and a Bezier curve, generate a 5th-degree
* Bezier-format equation whose solution finds the point on the
* curve nearest the user-defined point.
*/
static Point2 *ConvertToBezierForm(P, V)
Point2 P; /* The point to find t for */
Point2 *V; /* The control points */
{
int i, j, k, m, n, ub, lb;
int row, column; /* Table indices */
Vector2 c[DEGREE+1]; /* V(i)'s - P */
Vector2 d[DEGREE]; /* V(i+1) - V(i) */
Point2 *w; /* Ctl pts of 5th-degree curve */
double cdTable[3][4]; /* Dot product of c, d */
static double z[3][4] = { /* 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 <= DEGREE; i++) {
V2Sub(&V[i], &P, &c[i]);
}
/* Determine the d's -- these are vectors created by subtracting*/
/* each control point from the next */
for (i = 0; i <= DEGREE - 1; i++) {
d[i] = V2ScaleII(V2Sub(&V[i+1], &V[i], &d[i]), 3.0);
}
/* Create the c,d table -- this is a table of dot products of the */
/* c's and d's */
for (row = 0; row <= DEGREE - 1; row++) {
for (column = 0; column <= DEGREE; column++) {
cdTable[row][column] = V2Dot(&d[row], &c[column]);
}
}
/* Now, apply the z's to the dot products, on the skew diagonal*/
/* Also, set up the x-values, making these "points" */
w = (Point2 *)malloc((unsigned)(W_DEGREE+1) * sizeof(Point2));
for (i = 0; i <= W_DEGREE; i++) {
w[i].y = 0.0;
w[i].x = (double)(i) / W_DEGREE;
}
n = DEGREE;
m = DEGREE-1;
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].y += cdTable[j][i] * z[j][i];
}
}
return (w);
}
/*
* FindRoots :
* Given a 5th-degree equation in Bernstein-Bezier form, find
* all of the roots in the interval [0, 1]. Return the number
* of roots found.
*/
static int FindRoots(w, degree, t, depth)
Point2 *w; /* The control points */
int degree; /* The degree of the polynomial */
double *t; /* RETURN candidate t-values */
int depth; /* The depth of the recursion */
{
int i;
Point2 Left[W_DEGREE+1], /* New left and right */
Right[W_DEGREE+1]; /* control polygons */
int left_count, /* Solution count from */
right_count; /* children */
double left_t[W_DEGREE+1], /* Solutions from kids */
right_t[W_DEGREE+1];
switch (CrossingCount(w, degree)) {
case 0 : { /* No solutions here */
return 0;
}
case 1 : { /* Unique solution */
/* Stop recursion when the tree is deep enough */
/* if deep enough, return 1 solution at midpoint */
if (depth >= MAXDEPTH) {
t[0] = (w[0].x + w[W_DEGREE].x) / 2.0;
return 1;
}
if (ControlPolygonFlatEnough(w, degree)) {
t[0] = ComputeXIntercept(w, degree);
return 1;
}
break;
}
}
/* Otherwise, solve recursively after */
/* subdividing control polygon */
Bezier(w, degree, 0.5, Left, Right);
left_count = FindRoots(Left, degree, left_t, depth+1);
right_count = FindRoots(Right, degree, right_t, depth+1);
/* Gather solutions together */
for (i = 0; i < left_count; i++) {
t[i] = left_t[i];
}
for (i = 0; i < right_count; i++) {
t[i+left_count] = right_t[i];
}
/* Send back total number of solutions */
return (left_count+right_count);
}
/*
* CrossingCount :
* Count the number of times a Bezier control polygon
* crosses the 0-axis. This number is >= the number of roots.
*
*/
static int CrossingCount(V, degree)
Point2 *V; /* Control pts of Bezier curve */
int degree; /* Degreee of Bezier curve */
{
int i;
int n_crossings = 0; /* Number of zero-crossings */
int sign, old_sign; /* Sign of coefficients */
sign = old_sign = SGN(V[0].y);
for (i = 1; i <= degree; i++) {
sign = SGN(V[i].y);
if (sign != old_sign) n_crossings++;
old_sign = sign;
}
return n_crossings;
}
/*
* ControlPolygonFlatEnough :
* Check if the control polygon of a Bezier curve is flat enough
* for recursive subdivision to bottom out.
*
*/
static int ControlPolygonFlatEnough(V, degree)
Point2 *V; /* Control points */
int degree; /* Degree of polynomial */
{
int i; /* Index variable */
double *distance; /* Distances from pts to line */
double max_distance_above; /* maximum of these */
double max_distance_below;
double error; /* Precision of root */
double intercept_1,
intercept_2,
left_intercept,
right_intercept;
double a, b, c; /* Coefficients of implicit */
/* eqn for line from V[0]-V[deg]*/
/* Find the perpendicular distance */
/* from each interior control point to */
/* line connecting V[0] and V[degree] */
distance = (double *)malloc((unsigned)(degree + 1) * sizeof(double));
{
double abSquared;
/* Derive the implicit equation for line connecting first *'
/* and last control points */
a = V[0].y - V[degree].y;
b = V[degree].x - V[0].x;
c = V[0].x * V[degree].y - V[degree].x * V[0].y;
abSquared = (a * a) + (b * b);
for (i = 1; i < degree; i++) {
/* Compute distance from each of the points to that line */
distance[i] = a * V[i].x + b * V[i].y + c;
if (distance[i] > 0.0) {
distance[i] = (distance[i] * distance[i]) / abSquared;
}
if (distance[i] < 0.0) {
distance[i] = -((distance[i] * distance[i]) / abSquared);
}
}
}
/* Find the largest distance */
max_distance_above = 0.0;
max_distance_below = 0.0;
for (i = 1; i < degree; i++) {
if (distance[i] < 0.0) {
max_distance_below = MIN(max_distance_below, distance[i]);
};
if (distance[i] > 0.0) {
max_distance_above = MAX(max_distance_above, distance[i]);
}
}
free((char *)distance);
{
double det, dInv;
double a1, b1, c1, a2, b2, c2;
/* 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;
/* 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;
}
/* Compute intercepts of bounding box */
left_intercept = MIN(intercept_1, intercept_2);
right_intercept = MAX(intercept_1, intercept_2);
error = 0.5 * (right_intercept-left_intercept);
if (error < EPSILON) {
return 1;
}
else {
return 0;
}
}
/*
* ComputeXIntercept :
* Compute intersection of chord from first control point to last
* with 0-axis.
*
*/
/* NOTE: "T" and "Y" do not have to be computed, and there are many useless
* operations in the following (e.g. "0.0 - 0.0").
*/
static double ComputeXIntercept(V, degree)
Point2 *V; /* Control points */
int degree; /* Degree of curve */
{
double XLK, YLK, XNM, YNM, XMK, YMK;
double det, detInv;
double S, T;
double X, Y;
XLK = 1.0 - 0.0;
YLK = 0.0 - 0.0;
XNM = V[degree].x - V[0].x;
YNM = V[degree].y - V[0].y;
XMK = V[0].x - 0.0;
YMK = V[0].y - 0.0;
det = XNM*YLK - YNM*XLK;
detInv = 1.0/det;
S = (XNM*YMK - YNM*XMK) * detInv;
/* T = (XLK*YMK - YLK*XMK) * detInv; */
X = 0.0 + XLK * S;
/* Y = 0.0 + YLK * S; */
return X;
}
/*
* Bezier :
* Evaluate a Bezier curve at a particular parameter value
* Fill in control points for resulting sub-curves if "Left" and
* "Right" are non-null.
*
*/
static Point2 Bezier(V, degree, t, Left, Right)
int degree; /* Degree of bezier curve */
Point2 *V; /* Control pts */
double t; /* Parameter value */
Point2 *Left; /* RETURN left half ctl pts */
Point2 *Right; /* RETURN right half ctl pts */
{
int i, j; /* Index variables */
Point2 Vtemp[W_DEGREE+1][W_DEGREE+1];
/* Copy control points */
for (j =0; j <= degree; j++) {
Vtemp[0][j] = V[j];
}
/* Triangle computation */
for (i = 1; i <= degree; i++) {
for (j =0 ; j <= degree - i; j++) {
Vtemp[i][j].x =
(1.0 - t) * Vtemp[i-1][j].x + t * Vtemp[i-1][j+1].x;
Vtemp[i][j].y =
(1.0 - t) * Vtemp[i-1][j].y + t * Vtemp[i-1][j+1].y;
}
}
if (Left != NULL) {
for (j = 0; j <= degree; j++) {
Left[j] = Vtemp[j][0];
}
}
if (Right != NULL) {
for (j = 0; j <= degree; j++) {
Right[j] = Vtemp[degree-j][j];
}
}
return (Vtemp[degree][0]);
}
static Vector2 V2ScaleII(v, s)
Vector2 *v;
double s;
{
Vector2 result;
result.x = v->x * s; result.y = v->y * s;
return (result);
}