source: trunk/kitgen/8.x/blt/generic/bltSpline.c@ 201

#include "bltInt.h"

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]

static Tcl_CmdProc SplineCmd;

/*
 * -----------------------------------------------------------------------
 *
 * 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(points, nPoints, key, foundPtr)
    Point2D points[];           /* Contains the abscissas of the data
                                 * points of interpolation. */
    int nPoints;                        /* Dimension of x. */
    double key;                 /* Value whose relative position in
                                 * x is to be located. */
    int *foundPtr;              /* (out) Returns 1 if s is found in
                                 * x and 0 otherwise. */
{
    int high, low, mid;

    low = 0;
    high = nPoints - 1;

    while (high >= low) {
        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(p, q, m1, m2, epsilon)
    Point2D *p;                 /* Coordinates of one of the points of
                                 * interpolation */
    Point2D *q;                 /* Coordinates of one of the points of
                                 * interpolation */
    double m1;                  /* Derivative condition at point P */
    double m2;                  /* Derivative condition at point Q */
    double epsilon;             /* Error tolerance used to distinguish
                                 * cases when m1 or m2 is relatively
                                 * close to the slope or twice the
                                 * slope of the line segment joining
                                 * the points P and Q.  If
                                 * epsilon is not 0.0, then epsilon
                                 * should be greater than or equal to
                                 * machine epsilon.  */
{
    double slope;

    /* Calculate the slope of the line joining P and Q. */
    slope = (q->y - p->y) / (q->x - p->x);

    if (slope != 0.0) {
        double relerr;
        double mref, mref1, mref2, prod1, prod2;

        prod1 = slope * m1;
        prod2 = slope * m2;

        /* Find the absolute values of the slopes slope, m1, and m2. */
        mref = FABS(slope);
        mref1 = FABS(m1);
        mref2 = FABS(m2);

        /*
         * If the relative deviation of m1 or m2 from slope is less than
         * epsilon, then choose case 2 or case 3.
         */
        relerr = epsilon * mref;
        if ((FABS(slope - m1) > relerr) && (FABS(slope - m2) > relerr) &&
            (prod1 >= 0.0) && (prod2 >= 0.0)) {
            double prod;

            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;
    }
}

/*
 * -----------------------------------------------------------------------
 *
 * QuadCases --
 *
 *      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(p, q, m1, m2, param, which)
    Point2D *p, *q;
    double m1, m2;
    double param[];
    int which;
{
    if ((which == 3) || (which == 4)) { /* Parameters used in both 3 and 4 */
        double mbar1, mbar2, mbar3, c1, d1, h1, j1, k1;

        c1 = p->x + (q->y - p->y) / m1;
        d1 = q->x + (p->y - q->y) / m2;
        h1 = c1 * 2.0 - p->x;
        j1 = d1 * 2.0 - q->x;
        mbar1 = (q->y - p->y) / (h1 - p->x);
        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;
            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. */
            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. */
        double ztwo;

        Z1 = (p->y - q->y + m2 * q->x - m1 * p->x) / (m2 - m1);
        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(p, q, m1, m2, epsilon, param)
    Point2D *p, *q;
    double m1, m2;
    double epsilon;
    double param[];
{
    int ncase;

    ncase = QuadChoose(p, q, m1, m2, epsilon);
    QuadCases(p, q, m1, m2, param, ncase);
    return ncase;
}

/*
 * -----------------------------------------------------------------------
 *
 * QuadGetImage --
 *
 * -----------------------------------------------------------------------
 */
INLINE static double
QuadGetImage(p1, p2, p3, x1, x2, x3)
    double p1, p2, p3;
    double x1, x2, x3;
{
    double A, B, C;
    double y;

    A = x1 - x2;
    B = x2 - x3;
    C = x1 - x3;

    y = (p1 * (A * A) + p2 * 2.0 * B * A + p3 * (B * B)) / (C * C);
    return y;
}

/*
 * -----------------------------------------------------------------------
 *
 * QuadSpline --
 *
 *      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).
 *
 * Results:
 *      The image of the spline at x.
 *
 * -----------------------------------------------------------------------
 */
static void
QuadSpline(intp, left, right, param, ncase)
    Point2D *intp;              /* Value at which spline is evaluated */
    Point2D *left;              /* Point to the left of the data point to
                                 * be evaluated */
    Point2D *right;             /* Point to the right of the data point to
                                 * be evaluated */
    double param[];             /* Parameters of the spline */
    int ncase;                  /* Controls the evaluation of the
                                 * spline by indicating whether one or
                                 * two knots were placed in the
                                 * interval (leftX,rightX) */
{
    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;
}

/*
 * -----------------------------------------------------------------------
 *
 * QuadSlopes --
 *
 *      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.
 *
 * Results:
 *      The output array "m" is filled with the derivates at each
 *      data point.
 *
 * -----------------------------------------------------------------------
 */
static void
QuadSlopes(points, m, nPoints)
    Point2D points[];
    double *m;                  /* (out) To be filled with the first
                                 * derivative at each data point. */
    int nPoints;                /* Number of data points (dimension of
                                 * x, y, and m). */
{
    double xbar, xmid, xhat, ydif1, ydif2;
    double yxmid;
    double m1, m2;
    double m1s, m2s;
    register int i, n, l;

    m1s = m2s = m1 = m2 = 0;
    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.
         */
        ydif1 = points[i].y - points[l].y;
        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) {
            m1s = m1, m2s = m2; /* Save slopes of starting point */
        }
        /*
         * 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.
             */
            xbar = ydif2 / m1 + points[i].x;
            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.
             */
            xbar = -ydif1 / m2 + points[i].x;
            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 {
        xmid = (points[i].x + points[n].x) / 2.0;
        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 {
        xmid = (points[0].x + points[1].x) / 2.0;
        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(origPts, nOrigPts, intpPts, nIntpPts, m, epsilon)
    Point2D origPts[];
    int nOrigPts;
    Point2D intpPts[];
    int nIntpPts;
    double *m;                  /* Slope of the spline at each point
                                 * of interpolation. */
    double epsilon;             /* Relative error tolerance (see choose) */
{
    int error;
    register int i, j;
    double param[10];
    int ncase;
    int start, end;
    int l, p;
    register int n;
    int found;

    /* Initialize indices and set error result */
    error = 0;
    l = nOrigPts - 1;
    p = l - 1;
    ncase = 1;

    /*
     * Determine if abscissas of new vector are non-decreasing.
     */
    for (j = 1; j < nIntpPts; j++) {
        if (intpPts[j].x < intpPts[j - 1].x) {
            return 2;
        }
    }
    /*
     * Determine if any of the points in xval are LESS than the
     * abscissa of the first data point.
     */
    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.
     */
    for (end = nIntpPts - 1; end >= 0; end--) {
        if (intpPts[end].x <= origPts[l].x) {
            break;
        }
    }

    if (start > 0) {
        error = 1;              /* Set error value to indicate that
                                 * extrapolation has occurred. */
        /*
         * 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 (j = 0; j < (start - 1); j++) {
            QuadSpline(intpPts + j, origPts, origPts + 1, param, ncase);
        }
        if (nIntpPts == 1) {
            return error;
        }
    }
    if ((nIntpPts == 1) && (end != (nIntpPts - 1))) {
        goto noExtrapolation;
    }
    
    /*
     * Search locates the interval in which the first in-range
     * point of evaluation lies.
     */

    i = Search(origPts, nOrigPts, intpPts[start].x, &found);
    
    n = i + 1;
    if (n >= nOrigPts) {
        n = nOrigPts - 1;
        i = 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[i].y;
            start++;
            if (start >= nIntpPts) {
                return error;
            }
        } while (intpPts[start - 1].x == intpPts[start].x);
        
        for (;;) {
            if (intpPts[start].x < origPts[n].x) {
                break;  /* Break out of for-loop */
            }
            if (intpPts[start].x == origPts[n].x) {
                do {
                    intpPts[start].y = origPts[n].y;
                    start++;
                    if (start >= nIntpPts) {
                        return error;
                    }
                } while (intpPts[start].x == intpPts[start - 1].x);
            }
            i++;
            n++;
        }
    }
    /*
     * Calculate the images of all the points which lie within
     * range of the data.
     */
    if ((i > 0) || (error != 1)) {
        ncase = QuadSelect(origPts + i, origPts + n, m[i], m[n], 
                           epsilon, param);
    }
    for (j = start; j <= end; j++) {
        /*
         * 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[j].x == origPts[n].x) {
            intpPts[j].y = origPts[n].y;
            continue;
        } else if (intpPts[j].x > origPts[n].x) {
            double delta;
            
            /* Determine that the routine is in the correct part of
               the spline. */
            do {
                i++, n++;
                delta = intpPts[j].x - origPts[n].x;
            } while (delta > 0.0);
            
            if (delta < 0.0) {
                ncase = QuadSelect(origPts + i, origPts + n, m[i], 
                           m[n], epsilon, param);
            } else if (delta == 0.0) {
                intpPts[j].y = origPts[n].y;
                continue;
            }
        }
        QuadSpline(intpPts + j, origPts + i, origPts + n, param, ncase);
    }
    
    if (end == (nIntpPts - 1)) {
        return error;
    }
    if ((n == l) && (intpPts[end].x != origPts[l].x)) {
        goto noExtrapolation;
    }

    error = 1;                  /* Set error value to indicate that
                                 * extrapolation has occurred. */
    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 (j = (end + 1); j < nIntpPts; j++) {
        QuadSpline(intpPts + j, 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
Blt_QuadraticSpline(origPts, nOrigPts, intpPts, nIntpPts)
    Point2D origPts[];
    int nOrigPts;
    Point2D intpPts[];
    int nIntpPts;
{
    double epsilon;
    double *work;
    int result;

    work = Blt_Malloc(nOrigPts * sizeof(double));
    assert(work);

    epsilon = 0.0;              /* TBA: adjust error via command-line option */
    /* allocate space for vectors used in calculation */
    QuadSlopes(origPts, work, nOrigPts);
    result = QuadEval(origPts, nOrigPts, intpPts, nIntpPts, work, epsilon);
    Blt_Free(work);
    if (result > 1) {
        return FALSE;
    }
    return TRUE;
}

/*
 * ------------------------------------------------------------------------
 *
 * Reference:
 *      Numerical Analysis, R. Burden, J. Faires and A. Reynolds.
 *      Prindle, Weber & Schmidt 1981 pp 112
 *
 * Parameters:
 *      origPts - vector of points, assumed to be sorted along x.
 *      intpPts - vector of new points.
 *
 * ------------------------------------------------------------------------
 */
int
Blt_NaturalSpline(origPts, nOrigPts, intpPts, nIntpPts)
    Point2D origPts[];
    int nOrigPts;
    Point2D intpPts[];
    int nIntpPts;
{
    Cubic2D *eq;
    Point2D *iPtr, *endPtr;
    TriDiagonalMatrix *A;
    double *dx;         /* vector of deltas in x */
    double x, dy, alpha;
    int isKnot;
    register int i, j, n;

    dx = Blt_Malloc(sizeof(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. */
    A = Blt_Malloc(sizeof(TriDiagonalMatrix) * nOrigPts);
    if (A == NULL) {
        Blt_Free(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];
    }
    eq = Blt_Malloc(sizeof(Cubic2D) * nOrigPts);

    if (eq == NULL) {
        Blt_Free(A);
        Blt_Free(dx);
        return FALSE;
    }
    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]);
    }
    Blt_Free(A);
    Blt_Free(dx);

    endPtr = intpPts + nIntpPts;
    /* Now calculate the new values */
    for (iPtr = intpPts; iPtr < endPtr; iPtr++) {
        iPtr->y = 0.0;
        x = iPtr->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) {
            iPtr->y = origPts[i].y;
        } else {
            i--;
            x -= origPts[i].x;
            iPtr->y = origPts[i].y + 
                x * (eq[i].b + x * (eq[i].c + x * eq[i].d));
        }
    }
    Blt_Free(eq);
    return TRUE;
}

static Blt_OpSpec splineOps[] =
{
    {"natural", 1, (Blt_Op)Blt_NaturalSpline, 6, 6, "x y splx sply",},
    {"quadratic", 1, (Blt_Op)Blt_QuadraticSpline, 6, 6, "x y splx sply",},
};
static int nSplineOps = sizeof(splineOps) / sizeof(Blt_OpSpec);

/*ARGSUSED*/
static int
SplineCmd(clientData, interp, argc, argv)
    ClientData clientData;      /* Not used. */
    Tcl_Interp *interp;
    int argc;
    char **argv;
{
    Blt_Op proc;
    Blt_Vector *x, *y, *splX, *splY;
    double *xArr, *yArr;
    register int i;
    Point2D *origPts, *intpPts;
    int nOrigPts, nIntpPts;
    
    proc = Blt_GetOp(interp, nSplineOps, splineOps, BLT_OP_ARG1, argc, argv,0);
    if (proc == NULL) {
        return TCL_ERROR;
    }
    if ((Blt_GetVector(interp, argv[2], &x) != TCL_OK) ||
        (Blt_GetVector(interp, argv[3], &y) != TCL_OK) ||
        (Blt_GetVector(interp, argv[4], &splX) != TCL_OK)) {
        return TCL_ERROR;
    }
    nOrigPts = Blt_VecLength(x);
    if (nOrigPts < 3) {
        Tcl_AppendResult(interp, "length of vector \"", argv[2], "\" is < 3",
            (char *)NULL);
        return TCL_ERROR;
    }
    for (i = 1; i < nOrigPts; i++) {
        if (Blt_VecData(x)[i] < Blt_VecData(x)[i - 1]) {
            Tcl_AppendResult(interp, "x vector \"", argv[2],
                "\" must be monotonically increasing", (char *)NULL);
            return TCL_ERROR;
        }
    }
    /* Check that all the data points aren't the same. */
    if (Blt_VecData(x)[i - 1] <= Blt_VecData(x)[0]) {
        Tcl_AppendResult(interp, "x vector \"", argv[2],
         "\" must be monotonically increasing", (char *)NULL);
        return TCL_ERROR;
    }
    if (nOrigPts != Blt_VecLength(y)) {
        Tcl_AppendResult(interp, "vectors \"", argv[2], "\" and \"", argv[3],
            " have different lengths", (char *)NULL);
        return TCL_ERROR;
    }
    nIntpPts = Blt_VecLength(splX);
    if (Blt_GetVector(interp, argv[5], &splY) != TCL_OK) {
        /*
         * If the named vector to hold the ordinates of the spline
         * doesn't exist, create one the same size as the vector
         * containing the abscissas.
         */
        if (Blt_CreateVector(interp, argv[5], nIntpPts, &splY) != TCL_OK) {
            return TCL_ERROR;
        }
    } else if (nIntpPts != Blt_VecLength(splY)) {
        /*
         * The x and y vectors differ in size. Make the number of ordinates
         * the same as the number of abscissas.
         */
        if (Blt_ResizeVector(splY, nIntpPts) != TCL_OK) {
            return TCL_ERROR;
        }
    }
    origPts = Blt_Malloc(sizeof(Point2D) * nOrigPts);
    if (origPts == NULL) {
        Tcl_AppendResult(interp, "can't allocate \"", Blt_Itoa(nOrigPts), 
                "\" points", (char *)NULL);
        return TCL_ERROR;
    }
    intpPts = Blt_Malloc(sizeof(Point2D) * nIntpPts);
    if (intpPts == NULL) {
        Tcl_AppendResult(interp, "can't allocate \"", Blt_Itoa(nIntpPts), 
                "\" points", (char *)NULL);
        Blt_Free(origPts);
        return TCL_ERROR;
    }
    xArr = Blt_VecData(x);
    yArr = Blt_VecData(y);
    for (i = 0; i < nOrigPts; i++) {
        origPts[i].x = xArr[i];
        origPts[i].y = yArr[i];
    }
    xArr = Blt_VecData(splX);
    yArr = Blt_VecData(splY);
    for (i = 0; i < nIntpPts; i++) {
        intpPts[i].x = xArr[i];
        intpPts[i].y = yArr[i];
    }
    if (!(*proc) (origPts, nOrigPts, intpPts, nIntpPts)) {
        Tcl_AppendResult(interp, "error generating spline for \"", 
                Blt_NameOfVector(splY), "\"", (char *)NULL);
        Blt_Free(origPts);
        Blt_Free(intpPts);
        return TCL_ERROR;
    }
    yArr = Blt_VecData(splY);
    for (i = 0; i < nIntpPts; i++) {
        yArr[i] = intpPts[i].y;
    }
    Blt_Free(origPts);
    Blt_Free(intpPts);

    /* Finally update the vector. The size of the vector hasn't
     * changed, just the data. Reset the vector using TCL_STATIC to
     * indicate this. */
    if (Blt_ResetVector(splY, Blt_VecData(splY), Blt_VecLength(splY),
            Blt_VecSize(splY), TCL_STATIC) != TCL_OK) {
        return TCL_ERROR;
    }
    return TCL_OK;
}

int
Blt_SplineInit(interp)
    Tcl_Interp *interp;
{
    static Blt_CmdSpec cmdSpec =
    {"spline", SplineCmd,};

    if (Blt_InitCmd(interp, "blt", &cmdSpec) == NULL) {
        return TCL_ERROR;
    }
    return TCL_OK;
}


#define SQR(x)  ((x)*(x))

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(A, n)
    TriDiagonalMatrix A[];
    int n;
{
    int i;
    double m_ij, m_n, m_nn, d;

    if (n < 1) {
        return FALSE;           /* Dimension should be at least 1 */
    }
    d = A[0][1];                /* D_{0,0} = A_{0,0} */
    if (d <= 0.0) {
        return FALSE;           /* 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 FALSE;       /* 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 FALSE;
        }
    }
    return TRUE;
}

/*
 * 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(A, spline, nIntervals)
    TriDiagonalMatrix A[];
    CubicSpline spline[];
    int nIntervals;
{
    int i;
    double x, y;
    int n, m;

    n = nIntervals - 2;
    m = nIntervals - 1;

    /* Division by transpose of C : b = C^{-T} * b */
    x = spline[m].x;
    y = spline[m].y;
    for (i = 0; i < n; i++) {
        spline[i + 1].x -= A[i][2] * spline[i].x; /* C_{i,i+1} * x(i) */
        spline[i + 1].y -= A[i][2] * spline[i].y; /* C_{i,i+1} * x(i) */
        x -= A[i][0] * spline[i].x;     /* C_{i,n-1} * x(i) */
        y -= A[i][0] * spline[i].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 (i = 0; i < nIntervals; i++) {
        spline[i].x /= A[i][1];
        spline[i].y /= A[i][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 (i = (n - 1); i >= 0; i--) {
        /* C_{i,i+1} * x_{i+1} + C_{i,n-1} * x_{n-1} */
        spline[i].x -= A[i][2] * spline[i + 1].x + A[i][0] * x;
        spline[i].y -= A[i][2] * spline[i + 1].y + A[i][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(points, nPoints, isClosed, unitX, unitY)
    Point2D points[];
    int nPoints;                /* Number of points (nPoints>=3) */
    int isClosed;               /* CLOSED_CONTOUR or OPEN_CONTOUR  */
    double unitX, unitY;        /* Unit length in x and y (norm=1) */
{
    CubicSpline *spline;
    register CubicSpline *s1, *s2;
    int n, i;
    double norm, dx, dy;
    TriDiagonalMatrix *A;       /* The tri-diagonal matrix is saved here. */
    
    spline = Blt_Malloc(sizeof(CubicSpline) * nPoints);
    if (spline == NULL) {
        return NULL;
    }
    A = Blt_Malloc(sizeof(TriDiagonalMatrix) * nPoints);
    if (A == NULL) {
        Blt_Free(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 ... */
        Blt_Free(A);
        Blt_Free(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;
    }
    Blt_Free( 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(origPts, nOrigPts, intpPts, nIntpPts, spline)
    Point2D origPts[];
    int nOrigPts;
    Point2D intpPts[];
    int nIntpPts;
    CubicSpline spline[];
{
    double t, tSkip, tMax;
    Point2D p, q;
    double d, hx, dx0, dx01, hy, dy0, dy01;
    register int i, j, count;

    /* Sum the lengths of all the segments (intervals). */
    tMax = 0.0;
    for (i = 0; i < nOrigPts - 1; i++) {
        tMax += spline[i].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 (i = 0, j = 1; j < nOrigPts; i++, j++) {
        d = spline[i].t;        /* Interval length */
        p = q;
        q = origPts[i+1];
        hx = (q.x - p.x) / d;
        hy = (q.y - p.y) / d;
        dx0 = (spline[j].x + 2 * spline[i].x) / 6.0;
        dy0 = (spline[j].y + 2 * spline[i].y) / 6.0;
        dx01 = (spline[j].x - spline[i].x) / (6.0 * d);
        dy01 = (spline[j].y - spline[i].y) / (6.0 * d);
        while (t <= spline[i].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[i].t;
    }
    return count;
}

/*
 * 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.
 */
int
Blt_NaturalParametricSpline(origPts, nOrigPts, extsPtr, isClosed, 
        intpPts, nIntpPts)
    Point2D origPts[];
    int nOrigPts;
    Extents2D *extsPtr;
    int isClosed;
    Point2D *intpPts;
    int nIntpPts;
{
    double unitX, unitY;        /* To define norm (x,y)-plane */
    CubicSpline *spline;
    int result;

    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) */
    unitX = extsPtr->right - extsPtr->left;
    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, 
     */
    spline = CubicSlopes(origPts, nOrigPts, isClosed, unitX, unitY);
    if (spline == NULL) {
        return 0;
    }
    result= CubicEval(origPts, nOrigPts, intpPts, nIntpPts, spline);
    Blt_Free(spline);
    return result;
}

static void
CatromCoeffs(p, a, b, c, d)
    Point2D *p;
    Point2D *a, *b, *c, *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;
}

/*
 *----------------------------------------------------------------------
 *
 * Blt_ParametricCatromSpline --
 *
 *      Computes a spline based upon the data points, returning a new
 *      (larger) coordinate array or points.
 *
 * Results:
 *      None.
 *
 *----------------------------------------------------------------------
 */
int
Blt_CatromParametricSpline(points, nPoints, intpPts, nIntpPts)
    Point2D *points;
    int nPoints;
    Point2D *intpPts;
    int nIntpPts;
{
    register int i;
    Point2D *origPts;
    double t;
    int interval;
    Point2D a, b, c, d;

    assert(nPoints > 0);
    /*
     * 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.
     */
    origPts = Blt_Malloc((nPoints + 4) * sizeof(Point2D));
    memcpy(origPts + 1, points, sizeof(Point2D) * nPoints);

    origPts[0] = origPts[1];
    origPts[nPoints + 2] = origPts[nPoints + 1] = origPts[nPoints];

    for (i = 0; i < nIntpPts; i++) {
        interval = (int)intpPts[i].x;
        t = intpPts[i].y;
        assert(interval < nPoints);
        CatromCoeffs(origPts + interval, &a, &b, &c, &d);
        intpPts[i].x = (d.x + t * (c.x + t * (b.x + t * a.x))) / 2.0;
        intpPts[i].y = (d.y + t * (c.y + t * (b.y + t * a.y))) / 2.0;
    }
    Blt_Free(origPts);
    return 1;
}