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/*
* 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 <assert.h>
#include <float.h>
#include <math.h>
#include "bltInt.h"
#include "bltSpline.h"
#include "bltOp.h"
typedef int (SplineProc)(Point2d origPts[], int nOrigPts, Point2d intpPts[],
int nIntpPts);
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[], /* 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(
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(Point2d *p, Point2d *q, double m1, double 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(Point2d *p, Point2d *q, double m1, double 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(double p1, double p2, double p3, double x1, double x2, double 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(
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(Point2d *points, double *m, int nPoints)
{
double xbar, xmid, xhat, ydif1, ydif2;
double yxmid;
double m1, m2;
double m1s, m2s;
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(
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;
int i, j;
double param[10];
int ncase;
int start, end;
int l, p;
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(Point2d *origPts, int nOrigPts, Point2d *intpPts,
int nIntpPts)
{
double epsilon;
double *work;
int result;
work = malloc(nOrigPts * sizeof(double));
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);
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(Point2d *origPts, int nOrigPts, Point2d *intpPts,
int nIntpPts)
{
Cubic2D *eq;
Point2d *ip, *iend;
TriDiagonalMatrix *A;
double *dx; /* vector of deltas in x */
double x, dy, alpha;
int isKnot;
int i, j, n;
dx = 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 = malloc(sizeof(TriDiagonalMatrix) * nOrigPts);
if (A == NULL) {
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 = malloc(sizeof(Cubic2D) * nOrigPts);
if (eq == NULL) {
free(A);
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]);
}
free(A);
free(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));
}
}
free(eq);
return TRUE;
}
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 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(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(
Point2d points[],
int nPoints, /* Number of points (nPoints>=3) */
int isClosed, /* CLOSED_CONTOUR or OPEN_CONTOUR */
double unitX,
double unitY) /* Unit length in x and y (norm=1) */
{
CubicSpline *spline;
CubicSpline *s1, *s2;
int n, i;
double norm, dx, dy;
TriDiagonalMatrix *A; /* The tri-diagonal matrix is saved here. */
spline = malloc(sizeof(CubicSpline) * nPoints);
if (spline == NULL) {
return NULL;
}
A = malloc(sizeof(TriDiagonalMatrix) * nPoints);
if (A == NULL) {
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 ... */
free(A);
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;
}
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(Point2d *origPts, int nOrigPts, Point2d *intpPts, int nIntpPts,
CubicSpline *spline)
{
double t, tSkip, tMax;
Point2d q;
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++) {
Point2d p;
double d, hx, dx0, dx01, hy, dy0, dy01;
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(Point2d *origPts, int nOrigPts, Region2d *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);
free(spline);
return result;
}
static INLINE 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;
}
/*
*---------------------------------------------------------------------------
*
* Blt_ParametricCatromSpline --
*
* Computes a spline based upon the data points, returning a new (larger)
* coordinate array of points.
*
* Results:
* None.
*
*---------------------------------------------------------------------------
*/
int
Blt_CatromParametricSpline(Point2d *points, int nPoints, Point2d *intpPts,
int nIntpPts)
{
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 = 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;
}
free(origPts);
return 1;
}
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