/*
* GenerateBezier :
* Use least-squares method to find Bezier control points for region.
*
*/
static BezierCurve GenerateBezier(Point2 *d, int first, int last, double *uPrime,
Vector2 tHat1, Vector2 tHat2)
{
int i;
Vector2 A[MAXPOINTS][2]; /* Precomputed rhs for eqn */
int nPts; /* Number of pts in sub-curve */
double C[2][2]; /* Matrix C */
double X[2]; /* Matrix X */
double det_C0_C1, /* Determinants of matrices */
det_C0_X,
det_X_C1;
double alpha_l, /* Alpha values, left and right */
alpha_r;
Vector2 tmp; /* Utility variable */
BezierCurve bezCurve; /* RETURN bezier curve ctl pts */
bezCurve = (Point2 *)malloc(4 * sizeof(Point2));
nPts = last - first + 1;
/* Compute the A's */
for (i = 0; i < nPts; i++) {
Vector2 v1, v2;
v1 = tHat1;
v2 = tHat2;
V2Scale(&v1, B1(uPrime[i]));
V2Scale(&v2, B2(uPrime[i]));
A[i][0] = v1;
A[i][1] = v2;
}
/* Create the C and X matrices */
C[0][0] = 0.0;
C[0][1] = 0.0;
C[1][0] = 0.0;
C[1][1] = 0.0;
X[0] = 0.0;
X[1] = 0.0;
for (i = 0; i < nPts; i++) {
C[0][0] += V2Dot(&A[i][0], &A[i][0]);
C[0][1] += V2Dot(&A[i][0], &A[i][1]);
/* C[1][0] += V2Dot(&A[i][0], &A[i][1]);*/
C[1][0] = C[0][1];
C[1][1] += V2Dot(&A[i][1], &A[i][1]);
tmp = V2SubII(d[first + i],
V2AddII(
V2ScaleIII(d[first], B0(uPrime[i])),
V2AddII(
V2ScaleIII(d[first], B1(uPrime[i])),
V2AddII(
V2ScaleIII(d[last], B2(uPrime[i])),
V2ScaleIII(d[last], B3(uPrime[i]))))));
X[0] += V2Dot(&A[i][0], &tmp);
X[1] += V2Dot(&A[i][1], &tmp);
}
/* Compute the determinants of C and X */
det_C0_C1 = C[0][0] * C[1][1] - C[1][0] * C[0][1];
det_C0_X = C[0][0] * X[1] - C[1][0] * X[0];
det_X_C1 = X[0] * C[1][1] - X[1] * C[0][1];
/* Finally, derive alpha values */
alpha_l = (det_C0_C1 < ZERO_TOLERANCE) ? 0.0 : det_X_C1 / det_C0_C1;
alpha_r = (det_C0_C1 < ZERO_TOLERANCE) ? 0.0 : det_C0_X / det_C0_C1;
/* If alpha negative, use the Wu/Barsky heuristic (see text) */
/* (if alpha is 0, you get coincident control points that lead to
* divide by zero in any subsequent NewtonRaphsonRootFind() call. */
double segLength = V2DistanceBetween2Points(&d[last], &d[first]);
double epsilon = 1.0e-6 * segLength;
if (alpha_l < epsilon || alpha_r < epsilon)
{
/* fall back on standard (probably inaccurate) formula, and subdivide further if needed. */
double dist = segLength / 3.0;
bezCurve[0] = d[first];
bezCurve[3] = d[last];
V2Add(&bezCurve[0], V2Scale(&tHat1, dist), &bezCurve[1]);
V2Add(&bezCurve[3], V2Scale(&tHat2, dist), &bezCurve[2]);
return (bezCurve);
}
/* First and last control points of the Bezier curve are */
/* positioned exactly at the first and last data points */
/* Control points 1 and 2 are positioned an alpha distance out */
/* on the tangent vectors, left and right, respectively */
bezCurve[0] = d[first];
bezCurve[3] = d[last];
V2Add(&bezCurve[0], V2Scale(&tHat1, alpha_l), &bezCurve[1]);
V2Add(&bezCurve[3], V2Scale(&tHat2, alpha_r), &bezCurve[2]);
return (bezCurve);
}
/*
** Compute uv-coordinates of the point
** Return TRUE if the point is in the quadrangle
*/
static boolean point_in_quad (QUAD *Quad, HIT *Hit)
{
char LargestComponent; /* of the normal vector */
Point2 A, B, C, D; /* Projected vertices */
Point2 M; /* Projected intersection point */
Vector2 AB, BC, CD, AD, AM, AE; /* AE = DC - AB = DA - CB */
REAL u, v; /* Parametric coordinates */
REAL a, b, c, SqrtDelta; /* for the quadratic equation */
boolean IsInside = FALSE; /* if u and v are inside */
Vector2 Vector; /* temporary 2D-vector */
/*
** Projection on the plane that is most parallel to the facet
*/
LargestComponent = LARGEST_COMPONENT(Quad->Normal);
if (LargestComponent == 'x') {
A.x = Quad->A.y; B.x = Quad->B.y; C.x = Quad->C.y; D.x = Quad->D.y;
M.x = Hit->Point.y;
}
else {
A.x = Quad->A.x; B.x = Quad->B.x; C.x = Quad->C.x; D.x = Quad->D.x;
M.x = Hit->Point.x;
}
if (LargestComponent == 'z') {
A.y = Quad->A.y; B.y = Quad->B.y; C.y = Quad->C.y; D.y = Quad->D.y;
M.y = Hit->Point.y;
}
else {
A.y = Quad->A.z; B.y = Quad->B.z; C.y = Quad->C.z; D.y = Quad->D.z;
M.y = Hit->Point.z;
}
V2Sub (&B, &A, &AB); V2Sub (&C, &B, &BC);
V2Sub (&D, &C, &CD); V2Sub (&D, &A, &AD);
V2Add (&CD, &AB, &AE); V2Negate (&AE); V2Sub (&M, &A, &AM);
if (fabs(DETERMINANT(AB, CD)) < EPSILON) /* case AB // CD */
{
V2Sub (&AB, &CD, &Vector);
v = DETERMINANT(AM, Vector) / DETERMINANT(AD, Vector);
if ((v >= 0.0) && (v <= 1.0)) {
b = DETERMINANT(AB, AD) - DETERMINANT(AM, AE);
c = DETERMINANT (AM, AD);
u = ABS(b) < EPSILON ? -1 : c/b;
IsInside = ((u >= 0.0) && (u <= 1.0));
}
}
else if (fabs(DETERMINANT(BC, AD)) < EPSILON) /* case AD // BC */
{
V2Add (&AD, &BC, &Vector);
u = DETERMINANT(AM, Vector) / DETERMINANT(AB, Vector);
if ((u >= 0.0) && (u <= 1.0)) {
b = DETERMINANT(AD, AB) - DETERMINANT(AM, AE);
c = DETERMINANT (AM, AB);
v = ABS(b) < EPSILON ? -1 : c/b;
IsInside = ((v >= 0.0) && (v <= 1.0));
}
}
else /* general case */
{
a = DETERMINANT(AB, AE); c = - DETERMINANT (AM,AD);
b = DETERMINANT(AB, AD) - DETERMINANT(AM, AE);
a = -0.5/a; b *= a; c *= (a + a); SqrtDelta = b*b + c;
if (SqrtDelta >= 0.0) {
SqrtDelta = sqrt(SqrtDelta);
u = b - SqrtDelta;
if ((u < 0.0) || (u > 1.0)) /* to choose u between 0 and 1 */
u = b + SqrtDelta;
if ((u >= 0.0) && (u <= 1.0)) {
v = AD.x + u * AE.x;
if (ABS(v) < EPSILON)
v = (AM.y - u * AB.y) / (AD.y + u * AE.y);
else
v = (AM.x - u * AB.x) / v;
IsInside = ((v >= 0.0) && (v <= 1.0));
}
}
}
if (IsInside) {
Hit->u = u;
Hit->v = v;
}
return (IsInside);
}
/*
* FitCubic :
* Fit a Bezier curve to a (sub)set of digitized points
*/
static void FitCubic(Point2 *d, int first, int last, Vector2 tHat1, Vector2 tHat2,
double error, BezierContour &bezContour)
{
BezierCurve bezCurve; /*Control points of fitted Bezier curve*/
double *u; /* Parameter values for point */
double *uPrime; /* Improved parameter values */
double maxError; /* Maximum fitting error */
int splitPoint; /* Point to split point set at */
int nPts; /* Number of points in subset */
double iterationError; /*Error below which you try iterating */
int maxIterations = 4; /* Max times to try iterating */
Vector2 tHatCenter; /* Unit tangent vector at splitPoint */
int i;
iterationError = error * error;
nPts = last - first + 1;
/* Use heuristic if region only has two points in it */
if (nPts == 2) {
double dist = V2DistanceBetween2Points(&d[last], &d[first]) / 3.0;
bezCurve = (Point2 *)malloc(4 * sizeof(Point2));
bezCurve[0] = d[first];
bezCurve[3] = d[last];
V2Add(&bezCurve[0], V2Scale(&tHat1, dist), &bezCurve[1]);
V2Add(&bezCurve[3], V2Scale(&tHat2, dist), &bezCurve[2]);
DrawBezierCurve(3, bezCurve, bezContour);
free((void *)bezCurve);
return;
}
/* Parameterize points, and attempt to fit curve */
u = ChordLengthParameterize(d, first, last);
bezCurve = GenerateBezier(d, first, last, u, tHat1, tHat2);
/* Find max deviation of points to fitted curve */
maxError = ComputeMaxError(d, first, last, bezCurve, u, &splitPoint);
if (maxError < error) {
DrawBezierCurve(3, bezCurve, bezContour);
free((void *)u);
free((void *)bezCurve);
return;
}
/* If error not too large, try some reparameterization */
/* and iteration */
if (maxError < iterationError) {
for (i = 0; i < maxIterations; i++) {
uPrime = Reparameterize(d, first, last, u, bezCurve);
free((void *)bezCurve);
bezCurve = GenerateBezier(d, first, last, uPrime, tHat1, tHat2);
maxError = ComputeMaxError(d, first, last,
bezCurve, uPrime, &splitPoint);
if (maxError < error) {
DrawBezierCurve(3, bezCurve, bezContour);
free((void *)u);
free((void *)bezCurve);
free((void *)uPrime);
return;
}
free((void *)u);
u = uPrime;
}
}
/* Fitting failed -- split at max error point and fit recursively */
free((void *)u);
free((void *)bezCurve);
tHatCenter = ComputeCenterTangent(d, splitPoint);
FitCubic(d, first, splitPoint, tHat1, tHatCenter, error, bezContour);
V2Negate(&tHatCenter);
FitCubic(d, splitPoint, last, tHatCenter, tHat2, error, bezContour);
}
/*
* GenerateBezier :
* Use least-squares method to find Bezier control points for region.
*
*/
static BezierCurve GenerateBezier(
Point2 *d, /* Array of digitized points */
int first, int last, /* Indices defining region */
double *uPrime, /* Parameter values for region */
Vector2 tHat1, Vector2 tHat2) /* Unit tangents at endpoints */
{
int i;
// Vector2 A[MAXPOINTS][2]; /* Precomputed rhs for eqn */
int nPts; /* Number of pts in sub-curve */
double C[2][2]; /* Matrix C */
double X[2]; /* Matrix X */
double det_C0_C1, /* Determinants of matrices */
det_C0_X,
det_X_C1;
double alpha_l, /* Alpha values, left and right */
alpha_r;
Vector2 tmp; /* Utility variable */
BezierCurve bezCurve; /* RETURN bezier curve ctl pts */
bezCurve = (Point2 *)malloc(4 * sizeof(Point2));
nPts = last - first + 1;
Vector2 (*A)[2];
A = new Vector2[nPts][2]; /* Precomputed rhs for eqn */
/* Compute the A's */
for (i = 0; i < nPts; i++) {
Vector2 v1, v2;
v1 = tHat1;
v2 = tHat2;
V2Scale(&v1, B1(uPrime[i]));
V2Scale(&v2, B2(uPrime[i]));
A[i][0] = v1;
A[i][1] = v2;
}
/* Create the C and X matrices */
C[0][0] = 0.0;
C[0][1] = 0.0;
C[1][0] = 0.0;
C[1][1] = 0.0;
X[0] = 0.0;
X[1] = 0.0;
for (i = 0; i < nPts; i++) {
C[0][0] += V2Dot(&A[i][0], &A[i][0]);
C[0][1] += V2Dot(&A[i][0], &A[i][1]);
/* C[1][0] += V2Dot(&A[i][0], &A[i][1]);*/
C[1][0] = C[0][1];
C[1][1] += V2Dot(&A[i][1], &A[i][1]);
tmp = V2SubII(d[first + i],
V2AddII(
V2ScaleIII(d[first], B0(uPrime[i])),
V2AddII(
V2ScaleIII(d[first], B1(uPrime[i])),
V2AddII(
V2ScaleIII(d[last], B2(uPrime[i])),
V2ScaleIII(d[last], B3(uPrime[i]))))));
X[0] += V2Dot(&A[i][0], &tmp);
X[1] += V2Dot(&A[i][1], &tmp);
}
/* Compute the determinants of C and X */
det_C0_C1 = C[0][0] * C[1][1] - C[1][0] * C[0][1];
det_C0_X = C[0][0] * X[1] - C[0][1] * X[0];
det_X_C1 = X[0] * C[1][1] - X[1] * C[0][1];
/* Finally, derive alpha values */
if (det_C0_C1 == 0.0) {
det_C0_C1 = (C[0][0] * C[1][1]) * 10e-12;
}
alpha_l = det_X_C1 / det_C0_C1;
alpha_r = det_C0_X / det_C0_C1;
/* If alpha negative, use the Wu/Barsky heuristic (see text) */
if (alpha_l < 0.0 || alpha_r < 0.0) {
double dist = V2DistanceBetween2Points(&d[last], &d[first]) /
3.0;
bezCurve[0] = d[first];
bezCurve[3] = d[last];
V2Add(&bezCurve[0], V2Scale(&tHat1, dist), &bezCurve[1]);
V2Add(&bezCurve[3], V2Scale(&tHat2, dist), &bezCurve[2]);
delete[] A;
return (bezCurve);
}
/* First and last control points of the Bezier curve are */
/* positioned exactly at the first and last data points */
/* Control points 1 and 2 are positioned an alpha distance out */
/* on the tangent vectors, left and right, respectively */
bezCurve[0] = d[first];
bezCurve[3] = d[last];
V2Add(&bezCurve[0], V2Scale(&tHat1, alpha_l), &bezCurve[1]);
V2Add(&bezCurve[3], V2Scale(&tHat2, alpha_r), &bezCurve[2]);
delete[] A;
return (bezCurve);
}
