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Diffstat (limited to 'generic/tkbltGrElemLineSpline.C')
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diff --git a/generic/tkbltGrElemLineSpline.C b/generic/tkbltGrElemLineSpline.C new file mode 100644 index 0000000..9224d53 --- /dev/null +++ b/generic/tkbltGrElemLineSpline.C @@ -0,0 +1,1086 @@ +/* + * Smithsonian Astrophysical Observatory, Cambridge, MA, USA + * This code has been modified under the terms listed below and is made + * available under the same terms. + */ + +/* + * Copyright 2009 George A Howlett. + * + * Permission is hereby granted, free of charge, to any person obtaining + * a copy of this software and associated documentation files (the + * "Software"), to deal in the Software without restriction, including + * without limitation the rights to use, copy, modify, merge, publish, + * distribute, sublicense, and/or sell copies of the Software, and to + * permit persons to whom the Software is furnished to do so, subject to + * the following conditions: + * + * The above copyright notice and this permission notice shall be + * included in all copies or substantial portions of the Software. + * + * THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, + * EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF + * MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND + * NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE + * LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION + * OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION + * WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE. + * + */ + +#include <float.h> +#include <stdlib.h> +#include <string.h> + +#include <cmath> + +#include "tkbltGrElemLine.h" + +using namespace Blt; + +typedef double TriDiagonalMatrix[3]; +typedef struct { + double b, c, d; +} Cubic2D; + +typedef struct { + double b, c, d, e, f; +} Quint2D; + +// Quadratic spline parameters +#define E1 param[0] +#define E2 param[1] +#define V1 param[2] +#define V2 param[3] +#define W1 param[4] +#define W2 param[5] +#define Z1 param[6] +#define Z2 param[7] +#define Y1 param[8] +#define Y2 param[9] + +/* + *--------------------------------------------------------------------------- + * + * Search -- + * + * Conducts a binary search for a value. This routine is called + * only if key is between x(0) and x(len - 1). + * + * Results: + * Returns the index of the largest value in xtab for which + * x[i] < key. + * + *--------------------------------------------------------------------------- + */ +static int Search(Point2d points[], int nPoints, double key, int *foundPtr) +{ + int low = 0; + int high = nPoints - 1; + + while (high >= low) { + int mid = (high + low) / 2; + if (key > points[mid].x) + low = mid + 1; + else if (key < points[mid].x) + high = mid - 1; + else { + *foundPtr = 1; + return mid; + } + } + *foundPtr = 0; + return low; +} + +/* + *--------------------------------------------------------------------------- + * + * QuadChoose -- + * + * Determines the case needed for the computation of the parame- + * ters of the quadratic spline. + * + * Results: + * Returns a case number (1-4) which controls how the parameters + * of the quadratic spline are evaluated. + * + *--------------------------------------------------------------------------- + */ +static int QuadChoose(Point2d* p, Point2d* q, double m1, double m2, + double epsilon) +{ + // Calculate the slope of the line joining P and Q + double slope = (q->y - p->y) / (q->x - p->x); + + if (slope != 0.0) { + double prod1 = slope * m1; + double prod2 = slope * m2; + + // Find the absolute values of the slopes slope, m1, and m2 + double mref = fabs(slope); + double mref1 = fabs(m1); + double mref2 = fabs(m2); + + // If the relative deviation of m1 or m2 from slope is less than + // epsilon, then choose case 2 or case 3. + double relerr = epsilon * mref; + if ((fabs(slope - m1) > relerr) && (fabs(slope - m2) > relerr) && + (prod1 >= 0.0) && (prod2 >= 0.0)) { + double prod = (mref - mref1) * (mref - mref2); + if (prod < 0.0) { + // l1, the line through (x1,y1) with slope m1, and l2, + // the line through (x2,y2) with slope m2, intersect + // at a point whose abscissa is between x1 and x2. + // The abscissa becomes a knot of the spline. + return 1; + } + if (mref1 > (mref * 2.0)) { + if (mref2 <= ((2.0 - epsilon) * mref)) + return 3; + } + else if (mref2 <= (mref * 2.0)) { + // Both l1 and l2 cross the line through + // (x1+x2)/2.0,y1 and (x1+x2)/2.0,y2, which is the + // midline of the rectangle formed by P and Q or both + // m1 and m2 have signs different than the sign of + // slope, or one of m1 and m2 has opposite sign from + // slope and l1 and l2 intersect to the left of x1 or + // to the right of x2. The point (x1+x2)/2. is a knot + // of the spline. + return 2; + } + else if (mref1 <= ((2.0 - epsilon) * mref)) { + // In cases 3 and 4, sign(m1)=sign(m2)=sign(slope). + // Either l1 or l2 crosses the midline, but not both. + // Choose case 4 if mref1 is greater than + // (2.-epsilon)*mref; otherwise, choose case 3. + return 3; + } + // If neither l1 nor l2 crosses the midline, the spline + // requires two knots between x1 and x2. + return 4; + } + else { + // The sign of at least one of the slopes m1 or m2 does not + // agree with the sign of *slope*. + if ((prod1 < 0.0) && (prod2 < 0.0)) { + return 2; + } + else if (prod1 < 0.0) { + if (mref2 > ((epsilon + 1.0) * mref)) + return 1; + else + return 2; + } + else if (mref1 > ((epsilon + 1.0) * mref)) + return 1; + else + return 2; + } + } + else if ((m1 * m2) >= 0.0) + return 2; + else + return 1; +} + +/* + *--------------------------------------------------------------------------- + * Computes the knots and other parameters of the spline on the + * interval PQ. + * On input-- + * P and Q are the coordinates of the points of interpolation. + * m1 is the slope at P. + * m2 is the slope at Q. + * ncase controls the number and location of the knots. + * On output-- + * + * (v1,v2),(w1,w2),(z1,z2), and (e1,e2) are the coordinates of + * the knots and other parameters of the spline on P. + * (e1,e2) and Q are used only if ncase=4. + *--------------------------------------------------------------------------- + */ +static void QuadCases(Point2d* p, Point2d* q, double m1, double m2, + double param[], int which) +{ + if ((which == 3) || (which == 4)) { + double c1 = p->x + (q->y - p->y) / m1; + double d1 = q->x + (p->y - q->y) / m2; + double h1 = c1 * 2.0 - p->x; + double j1 = d1 * 2.0 - q->x; + double mbar1 = (q->y - p->y) / (h1 - p->x); + double mbar2 = (p->y - q->y) / (j1 - q->x); + + if (which == 4) { + // Case 4 + Y1 = (p->x + c1) / 2.0; + V1 = (p->x + Y1) / 2.0; + V2 = m1 * (V1 - p->x) + p->y; + Z1 = (d1 + q->x) / 2.0; + W1 = (q->x + Z1) / 2.0; + W2 = m2 * (W1 - q->x) + q->y; + double mbar3 = (W2 - V2) / (W1 - V1); + Y2 = mbar3 * (Y1 - V1) + V2; + Z2 = mbar3 * (Z1 - V1) + V2; + E1 = (Y1 + Z1) / 2.0; + E2 = mbar3 * (E1 - V1) + V2; + } + else { + // Case 3 + double k1 = (p->y - q->y + q->x * mbar2 - p->x * mbar1) / (mbar2 - mbar1); + if (fabs(m1) > fabs(m2)) { + Z1 = (k1 + p->x) / 2.0; + } else { + Z1 = (k1 + q->x) / 2.0; + } + V1 = (p->x + Z1) / 2.0; + V2 = p->y + m1 * (V1 - p->x); + W1 = (q->x + Z1) / 2.0; + W2 = q->y + m2 * (W1 - q->x); + Z2 = V2 + (W2 - V2) / (W1 - V1) * (Z1 - V1); + } + } + else if (which == 2) { + // Case 2 + Z1 = (p->x + q->x) / 2.0; + V1 = (p->x + Z1) / 2.0; + V2 = p->y + m1 * (V1 - p->x); + W1 = (Z1 + q->x) / 2.0; + W2 = q->y + m2 * (W1 - q->x); + Z2 = (V2 + W2) / 2.0; + } + else { + // Case 1 + Z1 = (p->y - q->y + m2 * q->x - m1 * p->x) / (m2 - m1); + double ztwo = p->y + m1 * (Z1 - p->x); + V1 = (p->x + Z1) / 2.0; + V2 = (p->y + ztwo) / 2.0; + W1 = (Z1 + q->x) / 2.0; + W2 = (ztwo + q->y) / 2.0; + Z2 = V2 + (W2 - V2) / (W1 - V1) * (Z1 - V1); + } +} + +static int QuadSelect(Point2d* p, Point2d* q, double m1, double m2, + double epsilon, double param[]) +{ + int ncase = QuadChoose(p, q, m1, m2, epsilon); + QuadCases(p, q, m1, m2, param, ncase); + return ncase; +} + +static double QuadGetImage(double p1, double p2, double p3, double x1, + double x2, double x3) +{ + double A = x1 - x2; + double B = x2 - x3; + double C = x1 - x3; + + double y = (p1 * (A * A) + p2 * 2.0 * B * A + p3 * (B * B)) / (C * C); + return y; +} + +/* + *--------------------------------------------------------------------------- + * Finds the image of a point in x. + * On input + * x Contains the value at which the spline is evaluated. + * leftX, leftY + * Coordinates of the left-hand data point used in the + * evaluation of x values. + * rightX, rightY + * Coordinates of the right-hand data point used in the + * evaluation of x values. + * Z1, Z2, Y1, Y2, E2, W2, V2 + * Parameters of the spline. + * ncase Controls the evaluation of the spline by indicating + * whether one or two knots were placed in the interval + * (xtabs,xtabs1). + *--------------------------------------------------------------------------- + */ +static void QuadSpline(Point2d* intp, Point2d* left, Point2d* right, + double param[], int ncase) + +{ + double y; + + if (ncase == 4) { + // Case 4: More than one knot was placed in the interval. + // Determine the location of data point relative to the 1st knot. + if (Y1 > intp->x) + y = QuadGetImage(left->y, V2, Y2, Y1, intp->x, left->x); + else if (Y1 < intp->x) { + // Determine the location of the data point relative to the 2nd knot. + if (Z1 > intp->x) + y = QuadGetImage(Y2, E2, Z2, Z1, intp->x, Y1); + else if (Z1 < intp->x) + y = QuadGetImage(Z2, W2, right->y, right->x, intp->x, Z1); + else + y = Z2; + } + else + y = Y2; + } + else { + // Cases 1, 2, or 3: + // Determine the location of the data point relative to the knot. + if (Z1 < intp->x) + y = QuadGetImage(Z2, W2, right->y, right->x, intp->x, Z1); + else if (Z1 > intp->x) + y = QuadGetImage(left->y, V2, Z2, Z1, intp->x, left->x); + else + y = Z2; + } + + intp->y = y; +} + +/* + *--------------------------------------------------------------------------- + * Calculates the derivative at each of the data points. The + * slopes computed will insure that an osculatory quadratic + * spline will have one additional knot between two adjacent + * points of interpolation. Convexity and monotonicity are + * preserved wherever these conditions are compatible with the + * data. + *--------------------------------------------------------------------------- + */ +static void QuadSlopes(Point2d *points, double *m, int nPoints) +{ + double m1s =0; + double m2s =0; + double m1 =0; + double m2 =0; + int i, n, l; + for (l = 0, i = 1, n = 2; i < (nPoints - 1); l++, i++, n++) { + // Calculate the slopes of the two lines joining three + // consecutive data points. + double ydif1 = points[i].y - points[l].y; + double ydif2 = points[n].y - points[i].y; + m1 = ydif1 / (points[i].x - points[l].x); + m2 = ydif2 / (points[n].x - points[i].x); + if (i == 1) { + // Save slopes of starting point + m1s = m1; + m2s = m2; + } + // If one of the preceding slopes is zero or if they have opposite + // sign, assign the value zero to the derivative at the middle point. + if ((m1 == 0.0) || (m2 == 0.0) || ((m1 * m2) <= 0.0)) + m[i] = 0.0; + else if (fabs(m1) > fabs(m2)) { + // Calculate the slope by extending the line with slope m1. + double xbar = ydif2 / m1 + points[i].x; + double xhat = (xbar + points[n].x) / 2.0; + m[i] = ydif2 / (xhat - points[i].x); + } + else { + // Calculate the slope by extending the line with slope m2. + double xbar = -ydif1 / m2 + points[i].x; + double xhat = (points[l].x + xbar) / 2.0; + m[i] = ydif1 / (points[i].x - xhat); + } + } + + // Calculate the slope at the last point, x(n). + i = nPoints - 2; + n = nPoints - 1; + if ((m1 * m2) < 0.0) + m[n] = m2 * 2.0; + else { + double xmid = (points[i].x + points[n].x) / 2.0; + double yxmid = m[i] * (xmid - points[i].x) + points[i].y; + m[n] = (points[n].y - yxmid) / (points[n].x - xmid); + if ((m[n] * m2) < 0.0) + m[n] = 0.0; + } + + // Calculate the slope at the first point, x(0). + if ((m1s * m2s) < 0.0) + m[0] = m1s * 2.0; + else { + double xmid = (points[0].x + points[1].x) / 2.0; + double yxmid = m[1] * (xmid - points[1].x) + points[1].y; + m[0] = (yxmid - points[0].y) / (xmid - points[0].x); + if ((m[0] * m1s) < 0.0) + m[0] = 0.0; + } +} + +/* + *--------------------------------------------------------------------------- + * + * QuadEval -- + * + * QuadEval controls the evaluation of an osculatory quadratic + * spline. The user may provide his own slopes at the points of + * interpolation or use the subroutine 'QuadSlopes' to calculate + * slopes which are consistent with the shape of the data. + * + * ON INPUT-- + * intpPts must be a nondecreasing vector of points at which the + * spline will be evaluated. + * origPts contains the abscissas of the data points to be + * interpolated. xtab must be increasing. + * y contains the ordinates of the data points to be + * interpolated. + * m contains the slope of the spline at each point of + * interpolation. + * nPoints number of data points (dimension of xtab and y). + * numEval is the number of points of evaluation (dimension of + * xval and yval). + * epsilon is a relative error tolerance used in subroutine + * 'QuadChoose' to distinguish the situation m(i) or + * m(i+1) is relatively close to the slope or twice + * the slope of the linear segment between xtab(i) and + * xtab(i+1). If this situation occurs, roundoff may + * cause a change in convexity or monotonicity of the + * resulting spline and a change in the case number + * provided by 'QuadChoose'. If epsilon is not equal to zero, + * then epsilon should be greater than or equal to machine + * epsilon. + * ON OUTPUT-- + * yval contains the images of the points in xval. + * err is one of the following error codes: + * 0 - QuadEval ran normally. + * 1 - xval(i) is less than xtab(1) for at least one + * i or xval(i) is greater than xtab(num) for at + * least one i. QuadEval will extrapolate to provide + * function values for these abscissas. + * 2 - xval(i+1) < xval(i) for some i. + * + * + * QuadEval calls the following subroutines or functions: + * Search + * QuadCases + * QuadChoose + * QuadSpline + *--------------------------------------------------------------------------- + */ +static int QuadEval(Point2d origPts[], int nOrigPts, Point2d intpPts[], + int nIntpPts, double *m, double epsilon) +{ + double param[10]; + + // Initialize indices and set error result + int error = 0; + int l = nOrigPts - 1; + int p = l - 1; + int ncase = 1; + + // Determine if abscissas of new vector are non-decreasing. + for (int jj=1; jj<nIntpPts; jj++) { + if (intpPts[jj].x < intpPts[jj - 1].x) + return 2; + } + // Determine if any of the points in xval are LESS than the + // abscissa of the first data point. + int start; + for (start = 0; start < nIntpPts; start++) { + if (intpPts[start].x >= origPts[0].x) + break; + } + // Determine if any of the points in xval are GREATER than the + // abscissa of the l data point. + int end; + for (end = nIntpPts - 1; end >= 0; end--) { + if (intpPts[end].x <= origPts[l].x) + break; + } + + if (start > 0) { + // Set error value to indicate that extrapolation has occurred + error = 1; + + // Calculate the images of points of evaluation whose abscissas + // are less than the abscissa of the first data point. + ncase = QuadSelect(origPts, origPts + 1, m[0], m[1], epsilon, param); + for (int jj=0; jj<(start - 1); jj++) + QuadSpline(intpPts + jj, origPts, origPts + 1, param, ncase); + if (nIntpPts == 1) + return error; + } + int ii; + int nn; + if ((nIntpPts == 1) && (end != (nIntpPts - 1))) + goto noExtrapolation; + + // Search locates the interval in which the first in-range + // point of evaluation lies. + int found; + ii = Search(origPts, nOrigPts, intpPts[start].x, &found); + + nn = ii + 1; + if (nn >= nOrigPts) { + nn = nOrigPts - 1; + ii = nOrigPts - 2; + } + /* + * If the first in-range point of evaluation is equal to one + * of the data points, assign the appropriate value from y. + * Continue until a point of evaluation is found which is not + * equal to a data point. + */ + if (found) { + do { + intpPts[start].y = origPts[ii].y; + start++; + if (start >= nIntpPts) { + return error; + } + } while (intpPts[start - 1].x == intpPts[start].x); + + for (;;) { + if (intpPts[start].x < origPts[nn].x) { + break; /* Break out of for-loop */ + } + if (intpPts[start].x == origPts[nn].x) { + do { + intpPts[start].y = origPts[nn].y; + start++; + if (start >= nIntpPts) { + return error; + } + } while (intpPts[start].x == intpPts[start - 1].x); + } + ii++; + nn++; + } + } + /* + * Calculate the images of all the points which lie within + * range of the data. + */ + if ((ii > 0) || (error != 1)) + ncase = QuadSelect(origPts+ii, origPts+nn, m[ii], m[nn], epsilon, param); + + for (int jj=start; jj<=end; jj++) { + // If xx(j) - x(n) is negative, do not recalculate + // the parameters for this section of the spline since + // they are already known. + if (intpPts[jj].x == origPts[nn].x) { + intpPts[jj].y = origPts[nn].y; + continue; + } + else if (intpPts[jj].x > origPts[nn].x) { + double delta; + + // Determine that the routine is in the correct part of the spline + do { + ii++; + nn++; + delta = intpPts[jj].x - origPts[nn].x; + } while (delta > 0.0); + + if (delta < 0.0) + ncase = QuadSelect(origPts+ii, origPts+nn, m[ii], m[nn], + epsilon, param); + else if (delta == 0.0) { + intpPts[jj].y = origPts[nn].y; + continue; + } + } + QuadSpline(intpPts+jj, origPts+ii, origPts+nn, param, ncase); + } + + if (end == (nIntpPts - 1)) + return error; + + if ((nn == l) && (intpPts[end].x != origPts[l].x)) + goto noExtrapolation; + + // Set error value to indicate that extrapolation has occurred + error = 1; + ncase = QuadSelect(origPts + p, origPts + l, m[p], m[l], epsilon, param); + + noExtrapolation: + // Calculate the images of the points of evaluation whose + // abscissas are greater than the abscissa of the last data point. + for (int jj=(end + 1); jj<nIntpPts; jj++) + QuadSpline(intpPts + jj, origPts + p, origPts + l, param, ncase); + + return error; +} + +/* + *--------------------------------------------------------------------------- + * + * Shape preserving quadratic splines + * by D.F.Mcallister & J.A.Roulier + * Coded by S.L.Dodd & M.Roulier + * N.C.State University + * + *--------------------------------------------------------------------------- + */ +/* + * Driver routine for quadratic spline package + * On input-- + * X,Y Contain n-long arrays of data (x is increasing) + * XM Contains m-long array of x values (increasing) + * eps Relative error tolerance + * n Number of input data points + * m Number of output data points + * On output-- + * work Contains the value of the first derivative at each data point + * ym Contains the interpolated spline value at each data point + */ +int LineElement::quadraticSpline(Point2d *origPts, int nOrigPts, + Point2d *intpPts, int nIntpPts) +{ + double* work = new double[nOrigPts]; + double epsilon = 0.0; + /* allocate space for vectors used in calculation */ + QuadSlopes(origPts, work, nOrigPts); + int result = QuadEval(origPts, nOrigPts, intpPts, nIntpPts, work, epsilon); + delete [] work; + if (result > 1) { + return 0; + } + return 1; +} + +/* + *--------------------------------------------------------------------------- + * Reference: + * Numerical Analysis, R. Burden, J. Faires and A. Reynolds. + * Prindle, Weber & Schmidt 1981 pp 112 + *--------------------------------------------------------------------------- + */ +int LineElement::naturalSpline(Point2d *origPts, int nOrigPts, + Point2d *intpPts, int nIntpPts) +{ + Point2d *ip, *iend; + double x, dy, alpha; + int isKnot; + int i, j, n; + + double* dx = new double[nOrigPts]; + /* Calculate vector of differences */ + for (i = 0, j = 1; j < nOrigPts; i++, j++) { + dx[i] = origPts[j].x - origPts[i].x; + if (dx[i] < 0.0) { + return 0; + } + } + n = nOrigPts - 1; /* Number of intervals. */ + TriDiagonalMatrix* A = new TriDiagonalMatrix[nOrigPts]; + if (!A) { + delete [] dx; + return 0; + } + /* Vectors to solve the tridiagonal matrix */ + A[0][0] = A[n][0] = 1.0; + A[0][1] = A[n][1] = 0.0; + A[0][2] = A[n][2] = 0.0; + + /* Calculate the intermediate results */ + for (i = 0, j = 1; j < n; j++, i++) { + alpha = 3.0 * ((origPts[j + 1].y / dx[j]) - (origPts[j].y / dx[i]) - + (origPts[j].y / dx[j]) + (origPts[i].y / dx[i])); + A[j][0] = 2 * (dx[j] + dx[i]) - dx[i] * A[i][1]; + A[j][1] = dx[j] / A[j][0]; + A[j][2] = (alpha - dx[i] * A[i][2]) / A[j][0]; + } + + Cubic2D* eq = new Cubic2D[nOrigPts]; + if (!eq) { + delete [] A; + delete [] dx; + return 0; + } + eq[0].c = eq[n].c = 0.0; + for (j = n, i = n - 1; i >= 0; i--, j--) { + eq[i].c = A[i][2] - A[i][1] * eq[j].c; + dy = origPts[i+1].y - origPts[i].y; + eq[i].b = (dy) / dx[i] - dx[i] * (eq[j].c + 2.0 * eq[i].c) / 3.0; + eq[i].d = (eq[j].c - eq[i].c) / (3.0 * dx[i]); + } + delete [] A; + delete [] dx; + + /* Now calculate the new values */ + for (ip = intpPts, iend = ip + nIntpPts; ip < iend; ip++) { + ip->y = 0.0; + x = ip->x; + + /* Is it outside the interval? */ + if ((x < origPts[0].x) || (x > origPts[n].x)) { + continue; + } + /* Search for the interval containing x in the point array */ + i = Search(origPts, nOrigPts, x, &isKnot); + if (isKnot) { + ip->y = origPts[i].y; + } else { + i--; + x -= origPts[i].x; + ip->y = origPts[i].y + x * (eq[i].b + x * (eq[i].c + x * eq[i].d)); + } + } + delete [] eq; + return 1; +} + +typedef struct { + double t; /* Arc length of interval. */ + double x; /* 2nd derivative of X with respect to T */ + double y; /* 2nd derivative of Y with respect to T */ +} CubicSpline; + +/* + * The following two procedures solve the special linear system which arise + * in cubic spline interpolation. If x is assumed cyclic ( x[i]=x[n+i] ) the + * equations can be written as (i=0,1,...,n-1): + * m[i][0] * x[i-1] + m[i][1] * x[i] + m[i][2] * x[i+1] = b[i] . + * In matrix notation one gets A * x = b, where the matrix A is tridiagonal + * with additional elements in the upper right and lower left position: + * A[i][0] = A_{i,i-1} for i=1,2,...,n-1 and m[0][0] = A_{0,n-1} , + * A[i][1] = A_{i, i } for i=0,1,...,n-1 + * A[i][2] = A_{i,i+1} for i=0,1,...,n-2 and m[n-1][2] = A_{n-1,0}. + * A should be symmetric (A[i+1][0] == A[i][2]) and positive definite. + * The size of the system is given in n (n>=1). + * + * In the first procedure the Cholesky decomposition A = C^T * D * C + * (C is upper triangle with unit diagonal, D is diagonal) is calculated. + * Return TRUE if decomposition exist. + */ +static int SolveCubic1(TriDiagonalMatrix A[], int n) +{ + int i; + double m_ij, m_n, m_nn, d; + + if (n < 1) { + return 0; /* Dimension should be at least 1 */ + } + d = A[0][1]; /* D_{0,0} = A_{0,0} */ + if (d <= 0.0) { + return 0; /* A (or D) should be positive definite */ + } + m_n = A[0][0]; /* A_{0,n-1} */ + m_nn = A[n - 1][1]; /* A_{n-1,n-1} */ + for (i = 0; i < n - 2; i++) { + m_ij = A[i][2]; /* A_{i,1} */ + A[i][2] = m_ij / d; /* C_{i,i+1} */ + A[i][0] = m_n / d; /* C_{i,n-1} */ + m_nn -= A[i][0] * m_n; /* to get C_{n-1,n-1} */ + m_n = -A[i][2] * m_n; /* to get C_{i+1,n-1} */ + d = A[i + 1][1] - A[i][2] * m_ij; /* D_{i+1,i+1} */ + if (d <= 0.0) { + return 0; /* Elements of D should be positive */ + } + A[i + 1][1] = d; + } + if (n >= 2) { /* Complete last column */ + m_n += A[n - 2][2]; /* add A_{n-2,n-1} */ + A[n - 2][0] = m_n / d; /* C_{n-2,n-1} */ + A[n - 1][1] = d = m_nn - A[n - 2][0] * m_n; /* D_{n-1,n-1} */ + if (d <= 0.0) { + return 0; + } + } + return 1; +} + +/* + * The second procedure solves the linear system, with the Cholesky + * decomposition calculated above (in m[][]) and the right side b given + * in x[]. The solution x overwrites the right side in x[]. + */ +static void SolveCubic2(TriDiagonalMatrix A[], CubicSpline spline[], + int nIntervals) +{ + int n = nIntervals - 2; + int m = nIntervals - 1; + + // Division by transpose of C : b = C^{-T} * b + double x = spline[m].x; + double y = spline[m].y; + for (int ii=0; ii<n; ii++) { + spline[ii + 1].x -= A[ii][2] * spline[ii].x; /* C_{i,i+1} * x(i) */ + spline[ii + 1].y -= A[ii][2] * spline[ii].y; /* C_{i,i+1} * x(i) */ + x -= A[ii][0] * spline[ii].x; /* C_{i,n-1} * x(i) */ + y -= A[ii][0] * spline[ii].y; /* C_{i,n-1} * x(i) */ + } + if (n >= 0) { + // C_{n-2,n-1} * x_{n-1} + spline[m].x = x - A[n][0] * spline[n].x; + spline[m].y = y - A[n][0] * spline[n].y; + } + // Division by D: b = D^{-1} * b + for (int ii=0; ii<nIntervals; ii++) { + spline[ii].x /= A[ii][1]; + spline[ii].y /= A[ii][1]; + } + + // Division by C: b = C^{-1} * b + x = spline[m].x; + y = spline[m].y; + if (n >= 0) { + // C_{n-2,n-1} * x_{n-1} + spline[n].x -= A[n][0] * x; + spline[n].y -= A[n][0] * y; + } + for (int ii=(n - 1); ii>=0; ii--) { + // C_{i,i+1} * x_{i+1} + C_{i,n-1} * x_{n-1} + spline[ii].x -= A[ii][2] * spline[ii + 1].x + A[ii][0] * x; + spline[ii].y -= A[ii][2] * spline[ii + 1].y + A[ii][0] * y; + } +} + +/* + * Find second derivatives (x''(t_i),y''(t_i)) of cubic spline interpolation + * through list of points (x_i,y_i). The parameter t is calculated as the + * length of the linear stroke. The number of points must be at least 3. + * Note: For CLOSED_CONTOURs the first and last point must be equal. + */ +static CubicSpline* CubicSlopes(Point2d points[], int nPoints, + int isClosed, double unitX, double unitY) +{ + CubicSpline *s1, *s2; + int n, i; + double norm, dx, dy; + + CubicSpline* spline = new CubicSpline[nPoints]; + if (!spline) + return NULL; + + TriDiagonalMatrix *A = new TriDiagonalMatrix[nPoints]; + if (!A) { + delete [] spline; + return NULL; + } + /* + * Calculate first differences in (dxdt2[i], y[i]) and interval lengths + * in dist[i]: + */ + s1 = spline; + for (i = 0; i < nPoints - 1; i++) { + s1->x = points[i+1].x - points[i].x; + s1->y = points[i+1].y - points[i].y; + + /* + * The Norm of a linear stroke is calculated in "normal coordinates" + * and used as interval length: + */ + dx = s1->x / unitX; + dy = s1->y / unitY; + s1->t = sqrt(dx * dx + dy * dy); + + s1->x /= s1->t; /* first difference, with unit norm: */ + s1->y /= s1->t; /* || (dxdt2[i], y[i]) || = 1 */ + s1++; + } + + /* + * Setup linear System: Ax = b + */ + n = nPoints - 2; /* Without first and last point */ + if (isClosed) { + /* First and last points must be equal for CLOSED_CONTOURs */ + spline[nPoints - 1].t = spline[0].t; + spline[nPoints - 1].x = spline[0].x; + spline[nPoints - 1].y = spline[0].y; + n++; /* Add last point (= first point) */ + } + s1 = spline, s2 = s1 + 1; + for (i = 0; i < n; i++) { + /* Matrix A, mainly tridiagonal with cyclic second index + ("j = j+n mod n") + */ + A[i][0] = s1->t; /* Off-diagonal element A_{i,i-1} */ + A[i][1] = 2.0 * (s1->t + s2->t); /* A_{i,i} */ + A[i][2] = s2->t; /* Off-diagonal element A_{i,i+1} */ + + /* Right side b_x and b_y */ + s1->x = (s2->x - s1->x) * 6.0; + s1->y = (s2->y - s1->y) * 6.0; + + /* + * If the linear stroke shows a cusp of more than 90 degree, + * the right side is reduced to avoid oscillations in the + * spline: + */ + /* + * The Norm of a linear stroke is calculated in "normal coordinates" + * and used as interval length: + */ + dx = s1->x / unitX; + dy = s1->y / unitY; + norm = sqrt(dx * dx + dy * dy) / 8.5; + if (norm > 1.0) { + /* The first derivative will not be continuous */ + s1->x /= norm; + s1->y /= norm; + } + s1++, s2++; + } + + if (!isClosed) { + /* Third derivative is set to zero at both ends */ + A[0][1] += A[0][0]; /* A_{0,0} */ + A[0][0] = 0.0; /* A_{0,n-1} */ + A[n-1][1] += A[n-1][2]; /* A_{n-1,n-1} */ + A[n-1][2] = 0.0; /* A_{n-1,0} */ + } + /* Solve linear systems for dxdt2[] and y[] */ + + if (SolveCubic1(A, n)) { /* Cholesky decomposition */ + SolveCubic2(A, spline, n); /* A * dxdt2 = b_x */ + } + else { /* Should not happen, but who knows ... */ + delete [] A; + delete [] spline; + return NULL; + } + /* Shift all second derivatives one place right and update the ends. */ + s2 = spline + n, s1 = s2 - 1; + for (/* empty */; s2 > spline; s2--, s1--) { + s2->x = s1->x; + s2->y = s1->y; + } + if (isClosed) { + spline[0].x = spline[n].x; + spline[0].y = spline[n].y; + } else { + /* Third derivative is 0.0 for the first and last interval. */ + spline[0].x = spline[1].x; + spline[0].y = spline[1].y; + spline[n + 1].x = spline[n].x; + spline[n + 1].y = spline[n].y; + } + delete [] A; + return spline; +} + +// Calculate interpolated values of the spline function (defined via p_cntr +// and the second derivatives dxdt2[] and dydt2[]). The number of tabulated +// values is n. On an equidistant grid n_intpol values are calculated. +static int CubicEval(Point2d *origPts, int nOrigPts, Point2d *intpPts, + int nIntpPts, CubicSpline *spline) +{ + double t, tSkip; + Point2d q; + int count; + + /* Sum the lengths of all the segments (intervals). */ + double tMax = 0.0; + for (int ii=0; ii<nOrigPts - 1; ii++) + tMax += spline[ii].t; + + /* Need a better way of doing this... */ + + /* The distance between interpolated points */ + tSkip = (1. - 1e-7) * tMax / (nIntpPts - 1); + + t = 0.0; /* Spline parameter value. */ + q = origPts[0]; + count = 0; + + intpPts[count++] = q; /* First point. */ + t += tSkip; + + for (int ii=0, jj=1; jj<nOrigPts; ii++, jj++) { + // Interval length + double d = spline[ii].t; + Point2d p = q; + q = origPts[ii+1]; + double hx = (q.x - p.x) / d; + double hy = (q.y - p.y) / d; + double dx0 = (spline[jj].x + 2 * spline[ii].x) / 6.0; + double dy0 = (spline[jj].y + 2 * spline[ii].y) / 6.0; + double dx01 = (spline[jj].x - spline[ii].x) / (6.0 * d); + double dy01 = (spline[jj].y - spline[ii].y) / (6.0 * d); + while (t <= spline[ii].t) { /* t in current interval ? */ + p.x += t * (hx + (t - d) * (dx0 + t * dx01)); + p.y += t * (hy + (t - d) * (dy0 + t * dy01)); + intpPts[count++] = p; + t += tSkip; + } + + // Parameter t relative to start of next interval + t -= spline[ii].t; + } + + return count; +} + +int LineElement::naturalParametricSpline(Point2d *origPts, int nOrigPts, + Region2d *extsPtr, int isClosed, + Point2d *intpPts, int nIntpPts) +{ + // Generate a cubic spline curve through the points (x_i,y_i) which are + // stored in the linked list p_cntr. + // The spline is defined as a 2d-function s(t) = (x(t),y(t)), where the + // parameter t is the length of the linear stroke. + + if (nOrigPts < 3) + return 0; + + if (isClosed) { + origPts[nOrigPts].x = origPts[0].x; + origPts[nOrigPts].y = origPts[0].y; + nOrigPts++; + } + + // Width and height of the grid is used at unit length (2d-norm) + double unitX = extsPtr->right - extsPtr->left; + double unitY = extsPtr->bottom - extsPtr->top; + if (unitX < FLT_EPSILON) + unitX = FLT_EPSILON; + if (unitY < FLT_EPSILON) + unitY = FLT_EPSILON; + + /* Calculate parameters for cubic spline: + * t = arc length of interval. + * dxdt2 = second derivatives of x with respect to t, + * dydt2 = second derivatives of y with respect to t, + */ + CubicSpline* spline = CubicSlopes(origPts, nOrigPts, isClosed, unitX, unitY); + if (spline == NULL) + return 0; + + int result= CubicEval(origPts, nOrigPts, intpPts, nIntpPts, spline); + + delete [] spline; + return result; +} + +static void CatromCoeffs(Point2d* p, Point2d* a, Point2d* b, + Point2d* c, Point2d* d) +{ + a->x = -p[0].x + 3.0 * p[1].x - 3.0 * p[2].x + p[3].x; + b->x = 2.0 * p[0].x - 5.0 * p[1].x + 4.0 * p[2].x - p[3].x; + c->x = -p[0].x + p[2].x; + d->x = 2.0 * p[1].x; + a->y = -p[0].y + 3.0 * p[1].y - 3.0 * p[2].y + p[3].y; + b->y = 2.0 * p[0].y - 5.0 * p[1].y + 4.0 * p[2].y - p[3].y; + c->y = -p[0].y + p[2].y; + d->y = 2.0 * p[1].y; +} + +int LineElement::catromParametricSpline(Point2d* points, int nPoints, + Point2d* intpPts, int nIntpPts) +{ + // The spline is computed in screen coordinates instead of data points so + // that we can select the abscissas of the interpolated points from each + // pixel horizontally across the plotting area. + + Point2d* origPts = new Point2d[nPoints + 4]; + memcpy(origPts + 1, points, sizeof(Point2d) * nPoints); + + origPts[0] = origPts[1]; + origPts[nPoints + 2] = origPts[nPoints + 1] = origPts[nPoints]; + + for (int ii=0; ii<nIntpPts; ii++) { + int interval = (int)intpPts[ii].x; + double t = intpPts[ii].y; + Point2d a, b, c, d; + CatromCoeffs(origPts + interval, &a, &b, &c, &d); + intpPts[ii].x = (d.x + t * (c.x + t * (b.x + t * a.x))) / 2.0; + intpPts[ii].y = (d.y + t * (c.y + t * (b.y + t * a.y))) / 2.0; + } + + delete [] origPts; + return 1; +} |