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/* GNUPLOT - filters.c */

/*[

* Copyright Ethan A Merritt 2013 - 2023

* All code in this file is dual-licensed.

* Gnuplot license:

* Permission to use, copy, and distribute this software and its

* documentation for any purpose with or without fee is hereby granted,

* provided that the above copyright notice appear in all copies and

* that both that copyright notice and this permission notice appear

* in supporting documentation.

* Permission to modify the software is granted, but not the right to

* distribute the complete modified source code. Modifications are to

* be distributed as patches to the released version. Permission to

* distribute binaries produced by compiling modified sources is granted,

* provided you

* 1. distribute the corresponding source modifications from the

* released version in the form of a patch file along with the binaries,

* 2. add special version identification to distinguish your version

* in addition to the base release version number,

* 3. provide your name and address as the primary contact for the

* support of your modified version, and

* 4. retain our contact information in regard to use of the base

* software.

* Permission to distribute the released version of the source code along

* with corresponding source modifications in the form of a patch file is

* granted with same provisions 2 through 4 for binary distributions.

* This software is provided "as is" without express or implied warranty

* to the extent permitted by applicable law.

* Alternative license:

* As an alternative to distributing code in this file under the gnuplot license,

* you may instead comply with the terms below. In this case, redistribution and

* use in source and binary forms, with or without modification, are permitted

* provided that the following conditions are met:

* Redistributions of source code must retain the above copyright notice, this

* list of conditions and the following disclaimer. Redistributions in binary

* form must reproduce the above copyright notice, this list of conditions and

* the following disclaimer in the documentation and/or other materials provided

* with the distribution.

* THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"

* AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE

* IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE

* ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE

* LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR

* CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF

* SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS

* INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN

* CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)

* ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE

* POSSIBILITY OF SUCH DAMAGE.

]*/

#include "filters.h"

#include "interpol.h"

#include "alloc.h"

#include "datafile.h" /* for blank_data_line */

#include "plot2d.h" /* for cp_extend() */

#include "watch.h" /* for bisect_target() */

* local prototypes

static int do_curve_cleanup(struct coordinate *point, int npoints);

static void winnow_interior_points (struct curve_points *plot, t_cluster *cluster);

static double fpp_SG5(struct coordinate *p);

static void cluster_stats(struct coordinate *points, t_cluster *cluster);

/* Variables related to clustering

* These really belong somewhere else, but the only routines so

* far that use cluster properties are the hull routines,

* so keep them together for now.

* cluster_outlier_threshold sets criterion for finding outliers

* not currently settable

* chi_shape_default_fraction sets the default choice of chi_length to

* chi_shape_default_fraction * max(edgelengths)

* currently controlled by "set chi_shape fraction <value>"

double cluster_outlier_threshold = 0.0;

double chi_shape_default_fraction = 0.6;

* EAM December 2013

* monotonic cubic spline using the Fritsch-Carlson algorithm

* FN Fritsch & RE Carlson (1980). "Monotone Piecewise Cubic Interpolation".

* SIAM Journal on Numerical Analysis (SIAM) 17 (2): 238–246. doi:10.1137/0717021.

void

mcs_interp(struct curve_points *plot)

{

/* These track the original (pre-sorted) data points */

int N = plot->p_count;

struct coordinate *p = gp_realloc(plot->points, (N+1) * sizeof(coordinate), "mcs");

int i;

/* These will track the resulting smoothed curve (>= 3X original count) */

/* Larger number of samples gives smoother curve (no surprise!) */

int Nsamp = (samples_1 > 2*N) ? samples_1 : 2*N;

int Ntot = N + Nsamp;

struct coordinate *new_points = gp_alloc((Ntot) * sizeof(coordinate), "mcs");

double xstart = GPMAX(p[0].x, X_AXIS.min);

double xend = GPMIN(p[N-1].x, X_AXIS.max);

double xstep = (xend - xstart) / (Nsamp - 1);

/* Load output x coords for sampling */

for (i=0; i<N; i++)

new_points[i].x = p[i].x;

for ( ; i<Ntot; i++)

new_points[i].x = xstart + (i-N)*xstep;

/* Sort output x coords */

gp_qsort(new_points, Ntot, sizeof(struct coordinate), compare_x);

/* Displace any collisions */

for (i=1; i<Ntot-1; i++) {

double delta = new_points[i].x - new_points[i-1].x;

if (new_points[i+1].x - new_points[i].x < delta/1000.)

new_points[i].x -= delta/2.;

}

/* Calculate spline coefficients */

#define DX xlow

#define SLOPE xhigh

#define C1 ylow

#define C2 yhigh

#define C3 z

for (i = 0; i < N-1; i++) {

p[i].DX = p[i+1].x - p[i].x;

p[i].SLOPE = (p[i+1].y - p[i].y) / p[i].DX;

}

/* The SIAM paper only mentions setting the final slope to zero if the

* calculation is otherwise ill-behaved (how would one detect that?).

* Retaining the data-derived slope makes the handling at the two ends

* of the data range consistent. See Bug #2055

/* p[N-1].SLOPE = 0; */

p[N-1].SLOPE = p[N-2].SLOPE;

p[0].C1 = p[0].SLOPE;

for (i = 0; i < N-1; i++) {

if (p[i].SLOPE * p[i+1].SLOPE <= 0) {

p[i+1].C1 = 0;

} else {

double sum = p[i].DX + p[i+1].DX;

p[i+1].C1 = (3. * sum)

/ ((sum + p[i+1].DX) / p[i].SLOPE + (sum + p[i].DX) / p[i+1].SLOPE);

}

p[N].C1 = p[N-1].SLOPE;

for (i = 0; i < N; i++) {

double temp = p[i].C1 + p[i+1].C1 - 2*p[i].SLOPE;

p[i].C2 = (p[i].SLOPE - p[i].C1 -temp) / p[i].DX;

p[i].C3 = temp / (p[i].DX * p[i].DX);

}

/* Use the coefficients C1, C2, C3 to interpolate over the requested range */

for (i = 0; i < Ntot; i++) {

double x = new_points[i].x;

double y;

TBOOLEAN exact = FALSE;

if (x == p[N-1].x) { /* Exact value for right-most point of original data */

y = p[N-1].y;

exact = TRUE;

} else {

int low = 0;

int mid;

int high = N-1;

while (low <= high) {

mid = floor((low + high) / 2);

if (p[mid].x < x)

low = mid + 1;

else if (p[mid].x > x)

high = mid - 1;

else { /* Exact value for some point in original data */

y = p[mid].y;

exact = TRUE;

break;

}

if (!exact) {

int j = GPMAX(0, high);

double diff = x - p[j].x;

y = p[j].y + p[j].C1 * diff + p[j].C2 * diff * diff + p[j].C3 * diff * diff * diff;

}

xstart = X_AXIS.min;

xend = X_AXIS.max;

if (inrange(x, xstart, xend))

new_points[i].type = INRANGE;

else

new_points[i].type = OUTRANGE;

y_axis = plot->y_axis;

store_and_update_range(&new_points[i].y, y, &new_points[i].type,

&Y_AXIS, plot->noautoscale);

}

/* Replace original data with the interpolated curve */

free(p);

plot->points = new_points;

plot->p_count = Ntot;

plot->p_max = Ntot + 1;

#undef DX

#undef SLOPE

#undef C1

#undef C2

#undef C3

}

* Binned histogram of input values.

* plot FOO using N:(1) bins{=<nbins>} {binrange=[binlow:binhigh]}

* {binwidth=<width>} with boxes

* If no binrange is given, binlow and binhigh are taken from the x range of the data.

* In either of these cases binlow is the midpoint x-coordinate of the first bin

* and binhigh is the midpoint x-coordinate of the last bin.

* Points that lie exactly on a bin boundary are assigned to the upper bin.

* Bin assignments are not affected by "set xrange".

* Notes:

* binwidth = (binhigh-binlow) / (nbins-1)

* xmin = binlow - binwidth/2

* xmax = binhigh + binwidth/2

* first bin holds points with (xmin =< x < xmin + binwidth)

* last bin holds points with (xmax-binwidth =< x < binhigh + binwidth)

* binopt = 0 (default) return sum of y values for points in each bin

* binopt = 1 return mean of y values for points in each bin

* Ethan A Merritt 2015

void

make_bins(struct curve_points *plot, int nbins,

double binlow, double binhigh, double binwidth,

int binopt)

{

int i, binno;

double *bin;

double bottom, top, range;

int *members;

struct axis *xaxis = &axis_array[plot->x_axis];

struct axis *yaxis = &axis_array[plot->y_axis];

double ymax = 0;

int N = plot->p_count;

/* Find the range of points to be binned */

if (binlow != binhigh) {

/* Explicit binrange [min:max] in the plot command */

bottom = binlow;

top = binhigh;

} else {

/* Take binrange from the data itself */

bottom = VERYLARGE; top = -VERYLARGE;

for (i=0; i<N; i++) {

if (bottom > plot->points[i].x)

bottom = plot->points[i].x;

if (top < plot->points[i].x)

top = plot->points[i].x;

}

if (top <= bottom)

int_warn(NO_CARET, "invalid bin range [%g:%g]", bottom, top);

}

/* If a fixed binwidth was provided, find total number of bins */

if (binwidth > 0) {

double temp;

nbins = 1 + (top - bottom) / binwidth;

temp = nbins * binwidth - (top - bottom);

bottom -= temp/2.;

top += temp/2.;

}

/* otherwise we use (N-1) intervals between midpoints of bin 1 and bin N */

else {

binwidth = (top - bottom) / (nbins - 1);

bottom -= binwidth/2.;

top += binwidth/2.;

}

range = top - bottom;

bin = gp_alloc(nbins*sizeof(double), "bins");

members = gp_alloc(nbins*sizeof(int), "bins");

for (i=0; i<nbins; i++) {

bin[i] = 0;

members[i] = 0;

}

for (i=0; i<N; i++) {

if (plot->points[i].type == UNDEFINED)

continue;

binno = floor(nbins * (plot->points[i].x - bottom) / range);

if (0 <= binno && binno < nbins) {

bin[binno] += plot->points[i].y;

members[binno]++;

}

if (xaxis->autoscale & AUTOSCALE_MIN) {

if (xaxis->min > bottom)

xaxis->min = bottom;

}

if (xaxis->autoscale & AUTOSCALE_MAX) {

if (xaxis->max < top)

xaxis->max = top;

}

/* Replace the original data with one entry per bin.

* new x = midpoint of bin

* new y = sum of individual y values over all points in bin

* new z = number of points in the bin

plot->p_count = nbins;

cp_extend(plot, nbins);

for (i=0; i<nbins; i++) {

double bincent = bottom + (0.5 + (double)i) * binwidth;

double ybin = bin[i];

if ((binopt == 1) && (members[i] > 1))

ybin = bin[i]/members[i];

plot->points[i].type = INRANGE;

plot->points[i].x = bincent;

plot->points[i].xlow = bincent - binwidth/2.;

plot->points[i].xhigh = bincent + binwidth/2.;

plot->points[i].y = ybin;

plot->points[i].ylow = ybin;

plot->points[i].yhigh = ybin;

plot->points[i].z = members[i];

if (inrange(bincent, xaxis->min, xaxis->max)) {

if (ymax < ybin)

ymax = ybin;

} else {

plot->points[i].type = OUTRANGE;

}

if (yaxis->autoscale & AUTOSCALE_MIN) {

if (yaxis->min > 0)

yaxis->min = 0;

}

if (yaxis->autoscale & AUTOSCALE_MAX) {

if (yaxis->max < ymax)

yaxis->max = ymax;

}

/* Recheck range on y */

for (i=0; i<nbins; i++)

if (!inrange(plot->points[i].y, yaxis->min, yaxis->max))

plot->points[i].type = OUTRANGE;

/* Clean up */

free(bin);

free(members);

}

* spline approximation of 3D lines

* do_3d_cubic gen_2d_path_splines gen_3d_splines

* Ethan A Merritt 2019

* Replace one isocurve with a 3D natural cubic spline interpolation.

* If there are multiple isocurves, or multiple curves with isocurves,

* the caller must sort that out and call here separately for each one.

* TODO:

* number of spline samples should be independent of "set samples"

static void

do_3d_cubic(struct iso_curve *curve, enum PLOT_SMOOTH smooth_option)

{

int i, l;

int nseg = samples_1;

struct coordinate *old_points, *new_points;

double xrange, yrange, zrange;

double dx, dy, dz;

double maxdx, maxdy, maxdz;

double t, tsum, tstep;

spline_coeff *sc_x = NULL;

spline_coeff *sc_y = NULL;

spline_coeff *sc_z = NULL;

old_points = curve->points;

* Sanity check axis ranges.

* This catches curves that lie in a plane of constant x or y.

* The fixup prints a warning to the user but we don't see it here.

axis_checked_extend_empty_range(FIRST_X_AXIS, "at time of spline generation");

axis_checked_extend_empty_range(FIRST_Y_AXIS, "at time of spline generation");

/* prevent gross mismatch of x/y/z units */

xrange = fabs(axis_array[FIRST_X_AXIS].max - axis_array[FIRST_X_AXIS].min);

yrange = fabs(axis_array[FIRST_Y_AXIS].max - axis_array[FIRST_Y_AXIS].min);

zrange = fabs(axis_array[FIRST_Z_AXIS].max - axis_array[FIRST_Z_AXIS].min);

/* Construct path-length vector; store it in unused slot of old_points */

t = tsum = 0.0;

maxdx = maxdy = maxdz = 0.0;

old_points[0].CRD_PATH = 0;

for (i = 1; i < curve->p_count; i++) {

dx = (old_points[i].x - old_points[i-1].x) / xrange;

dy = (old_points[i].y - old_points[i-1].y) / yrange;

dz = (old_points[i].z - old_points[i-1].z) / zrange;

tsum += sqrt( dx*dx + dy*dy + dz*dz );

old_points[i].CRD_PATH = tsum;

/* Track planarity */

if (fabs(dx) > maxdx)

maxdx = fabs(dx);

if (fabs(dy) > maxdy)

maxdy = fabs(dy);

if (fabs(dz) > maxdz)

maxdz = fabs(dz);

}

/* Normalize so that the path always runs from 0 to 1 */

for (i = 1; i < curve->p_count; i++)

old_points[i].CRD_PATH /= tsum;

tstep = old_points[curve->p_count-1].CRD_PATH / (double)(nseg - 1);

/* Create new list to hold interpolated points */

new_points = gp_alloc((nseg+1) * sizeof(struct coordinate), "3D spline");

memset( new_points, 0, (nseg+1) * sizeof(struct coordinate));

* If the curve being fitted lies entirely in one plane,

* we can do better by fitting a 2D spline rather than a 3D spline.

* This benefits the relatively common case of drawing a stack of

* 2D plots (e.g. fence plots).

* First check for a curve lying in the yz plane (x = constant).

if (maxdx < FLT_EPSILON) {

tstep = (old_points[curve->p_count-1].y - old_points[0].y) / (double)(nseg - 1);

if (smooth_option == SMOOTH_ACSPLINES)

sc_z = cp_approx_spline(curve->points, curve->p_count, 1, 2, 3);

else

sc_z = cp_tridiag(curve->points, curve->p_count, 1, 2);

for (i = 0, l = 0; i < nseg; i++) {

double temp;

t = old_points[0].y + i * tstep;

/* Move forward to the spline interval this point is in */

while ((t >= old_points[l + 1].y) && (l < curve->p_count- 2))

l++;

temp = t - old_points[l].y;

new_points[i].x = old_points[l].x; /* All the same */

new_points[i].y = t;

new_points[i].z = ((sc_z[l][3] * temp + sc_z[l][2]) * temp + sc_z[l][1])

* temp + sc_z[l][0];

}

* Check for a curve lying in the xz plane (y = constant).

else if (maxdy < FLT_EPSILON) {

tstep = (old_points[curve->p_count-1].x - old_points[0].x) / (double)(nseg - 1);

if (smooth_option == SMOOTH_ACSPLINES)

sc_z = cp_approx_spline(curve->points, curve->p_count, 0, 2, 3);

else

sc_z = cp_tridiag(curve->points, curve->p_count, 0, 2);

for (i = 0, l = 0; i < nseg; i++) {

double temp;

t = old_points[0].x + i * tstep;

/* Move forward to the spline interval this point is in */

while ((t >= old_points[l + 1].x) && (l < curve->p_count- 2))

l++;

temp = t - old_points[l].x;

new_points[i].x = t;

new_points[i].y = old_points[l].y; /* All the same */

new_points[i].z = ((sc_z[l][3] * temp + sc_z[l][2]) * temp + sc_z[l][1])

* temp + sc_z[l][0];

}

* Check for a curve lying in the xy plane (z = constant).

else if (maxdz < FLT_EPSILON) {

tstep = (old_points[curve->p_count-1].x - old_points[0].x) / (double)(nseg - 1);

if (smooth_option == SMOOTH_ACSPLINES)

sc_y = cp_approx_spline(curve->points, curve->p_count, 0, 1, 3);

else

sc_y = cp_tridiag(curve->points, curve->p_count, 0, 1);

for (i = 0, l = 0; i < nseg; i++) {

double temp;

t = old_points[0].x + i * tstep;

/* Move forward to the spline interval this point is in */

while ((t >= old_points[l + 1].x) && (l < curve->p_count- 2))

l++;

temp = t - old_points[l].x;

new_points[i].x = t;

new_points[i].y = ((sc_y[l][3] * temp + sc_y[l][2]) * temp + sc_y[l][1])

* temp + sc_y[l][0];

new_points[i].z = old_points[l].z; /* All the same */

}

* This is the general case.

* Calculate spline coefficients for each dimension x, y, z

else {

if (smooth_option == SMOOTH_ACSPLINES) {

sc_x = cp_approx_spline(curve->points, curve->p_count, PATHCOORD, 0, 3);

sc_y = cp_approx_spline(curve->points, curve->p_count, PATHCOORD, 1, 3);

sc_z = cp_approx_spline(curve->points, curve->p_count, PATHCOORD, 2, 3);

} else {

sc_x = cp_tridiag( curve->points, curve->p_count, PATHCOORD, 0);

sc_y = cp_tridiag( curve->points, curve->p_count, PATHCOORD, 1);

sc_z = cp_tridiag( curve->points, curve->p_count, PATHCOORD, 2);

}

for (i = 0, l=0; i < nseg; i++) {

double temp;

t = i * tstep;

/* Move forward to the spline interval this point is in */

while ((t >= old_points[l + 1].CRD_PATH) && (l < curve->p_count- 2))

l++;

temp = t - old_points[l].CRD_PATH;

new_points[i].x = ((sc_x[l][3] * temp + sc_x[l][2]) * temp + sc_x[l][1])

* temp + sc_x[l][0];

new_points[i].y = ((sc_y[l][3] * temp + sc_y[l][2]) * temp + sc_y[l][1])

* temp + sc_y[l][0];

new_points[i].z = ((sc_z[l][3] * temp + sc_z[l][2]) * temp + sc_z[l][1])

* temp + sc_z[l][0];

}

/* We're done with the spline coefficients */

free(sc_x);

free(sc_y);

free(sc_z);

/* Replace original data with spline approximation */

free(curve->points);

curve->points = new_points;

curve->p_count = nseg;

curve->p_max = nseg+1; /* not sure why we asked for 1 extra */

}

* Generate 2D splines along a path for each set of points in the plot,

* smoothing option SMOOTH_PATH.

* TODO:

* - number of spline samples should be controlled by something other

* than "set samples"

* - spline weights from an additional column

void

gen_2d_path_splines( struct curve_points *plot )

{

int i;

int ic, first_point; /* indexes for original data */

int is = 0; /* index for new (splined) data */

struct coordinate *old_points = NULL;

struct coordinate *splined_points;

spline_coeff *sc_x = NULL;

spline_coeff *sc_y = NULL;

double xrange = fabs(axis_array[plot->x_axis].max - axis_array[plot->x_axis].min);

double yrange = fabs(axis_array[plot->y_axis].max - axis_array[plot->y_axis].min);

int curves = num_curves(plot);

/* Allocate space to hold the interpolated points */

splined_points = gp_alloc( (plot->p_count + samples_1 * curves) * sizeof(struct coordinate), NULL );

memset( splined_points, 0, (plot->p_count + samples_1 * curves) * sizeof(struct coordinate));

first_point = 0;

for (ic = 0; ic < curves; ic++) {

double t, tstep, tsum;

double dx, dy;

int l;

int nold;

int num_points = next_curve(plot, &first_point);

TBOOLEAN closed = FALSE;

/* Make a copy of the original points so that we don't corrupt the

* list by adding up to three new ones.

old_points = gp_realloc( old_points, (num_points + 3) * sizeof(struct coordinate),

"spline points");

memcpy( &old_points[1], &plot->points[first_point], num_points * sizeof(struct coordinate));

/* Remove any unusable points (NaN, missing, duplicates) before fitting a spline.

* If that leaves fewer than 3 points, skip it.

nold = do_curve_cleanup(&old_points[1], num_points);

if (nold < 3) {

first_point += num_points;

continue;

}

/* We expect one of two cases. Either this really is a closed

* curve (end point matches start point) or it is an open-ended

* path that may not be monotonic on x.

* For plot style "with filledcurves closed" we add an extra

* point at the end if it is not already there.

if (old_points[1].x == old_points[nold].x

&& old_points[1].y == old_points[nold].y)

closed = TRUE;

if ((plot->plot_style == FILLEDCURVES) && !closed) {

old_points[++nold] = old_points[1];

closed = TRUE;

}

if (closed) {

/* Wrap around to one point before and one point after the path closure */

nold += 2;

old_points[0] = old_points[nold-3];

old_points[nold-1] = old_points[2];

} else {

/* Dummy up an extension at either end */

nold += 2;

old_points[0].x = old_points[1].x + old_points[1].x - old_points[2].x;

old_points[nold-1].x = old_points[nold-2].x + old_points[nold-2].x - old_points[nold-3].x;

old_points[0].y = old_points[1].y + old_points[1].y - old_points[2].y;

old_points[nold-1].y = old_points[nold-2].y + old_points[nold-2].y - old_points[nold-3].y;

}

/* Construct path-length vector; store it in an unused slot of old_points */

t = tsum = 0.0;

old_points[0].CRD_PATH = 0;

if (xrange == 0)

xrange = 1.;

if (yrange == 0)

yrange = 1.;

for (i = 1; i < nold; i++) {

dx = (old_points[i].x - old_points[i-1].x) / xrange;

dy = (old_points[i].y - old_points[i-1].y) / yrange;

tsum += sqrt( dx*dx + dy*dy );

old_points[i].CRD_PATH = tsum;

}

/* Normalize so that the path fraction always runs from 0 to 1 */

for (i = 1; i < nold; i++)

old_points[i].CRD_PATH /= tsum;

tstep = 1.0 / (double)(samples_1 - 1);

/* Calculate spline coefficients for x and for y as a function of path */

sc_x = cp_tridiag( old_points, nold, PATHCOORD, 0);

sc_y = cp_tridiag( old_points, nold, PATHCOORD, 1);

/* First output point is the same as the original first point */

splined_points[is++] = old_points[1];

/* Skip the points in the overlap region */

for (i = 0; i * tstep < old_points[1].CRD_PATH; i++)

;

/* Use spline coefficients to generate a new point at each sample interval. */

for (l=0; i < samples_1; i++) {

double temp;

t = i * tstep;

/* Stop before wrapping around. Copy the original end point. */

if (t > old_points[nold-2].CRD_PATH) {

splined_points[is++] = old_points[nold-2];

break;

}

/* Move forward to the spline interval this point is in */

while ((t >= old_points[l + 1].CRD_PATH) && (l < nold- 2))

l++;

temp = t - old_points[l].CRD_PATH;

splined_points[is].x = ((sc_x[l][3] * temp + sc_x[l][2]) * temp + sc_x[l][1])

* temp + sc_x[l][0];

splined_points[is].y = ((sc_y[l][3] * temp + sc_y[l][2]) * temp + sc_y[l][1])

* temp + sc_y[l][0];

is++;

}

/* Done with spline coefficients */

free(sc_x);

free(sc_y);

/* Add a seperator point after this set of splined points */

splined_points[is++].type = UNDEFINED;

first_point += num_points;

}

/* Replace original data with splined approximation */

free(old_points);

free(plot->points);

plot->points = splined_points;

plot->p_max = curves * samples_1;

plot->p_count = is;

return;

}

* Externally callable interface to 3D spline routines

void

gen_3d_splines( struct surface_points *plot )

{

struct iso_curve *curve = plot->iso_crvs;

while (curve) {

/* Remove any unusable points before fitting a spline */

curve->p_count = do_curve_cleanup(curve->points, curve->p_count);

if (curve->p_count > 3)

do_3d_cubic(curve, plot->plot_smooth);

curve = curve->next;

}

static int

do_curve_cleanup( struct coordinate *point, int npoints )

{

int i, keep;

/* Step through points in curve keeping only the usable ones.

* Discard duplicates

keep = 0;

for (i = 0; i < npoints; i++) {

if (point[i].type == UNDEFINED)

continue;

if (isnan(point[i].x) || isnan(point[i].y) || isnan(point[i].z))

continue;

if (i != keep)

point[keep] = point[i];

/* FIXME: should probably check fabs(this-prev) < EPS */

if ((keep > 0) && (point[keep].x == point[keep-1].x)

&& (point[keep].y == point[keep-1].y)

&& (point[keep].z == point[keep-1].z))

continue;

keep++;

}

return keep;

}

* convex_hull() replaces the original set of points with a subset that

* delimits the convex hull of the original points.

* The hull is found using Graham's algorithm

* RL Graham (1972), Information Processing Letters 1: 132–133.

* winnow_interior_points() is a helper routine that can greatly reduce

* processing time for large data sets but is otherwise not necessary.

* expand_hull() pushes the hull segments away from the interior to

* produce a bounding curve exterior to all points.

* - Ethan A Merritt 2021

#define CROSS(p1,p2,p3) \

( ((p2)->x - (p1)->x) * ((p3)->y - (p2)->y) \

- ((p2)->y - (p1)->y) * ((p3)->x - (p2)->x) )

void

convex_hull(struct curve_points *plot)

{

int i;

struct coordinate *points = plot->points;

struct coordinate **stack = NULL;

int np = plot->p_count;

int ntop;

/* cluster is used to hold x/y ranges and centroid */

t_cluster cluster;

cluster.npoints = plot->p_count;

/* Special cases */

if (np < 3)

return;

if (np == 3) {

cp_extend(plot, 4);

plot->points[3] = plot->points[0];

plot->p_count = 4;

return;

}

/* Find x and y limits of points in the cluster. */

cluster_stats(plot->points, &cluster);

/* This is not strictly necessary, but greatly reduces the number

* of points to be sorted and tested for the hull boundary.

winnow_interior_points(plot, &cluster);

/* Sort the remaining points (probably only need to sort on x?) */

gp_qsort(plot->points, plot->p_count,

sizeof(struct coordinate), compare_xyz);

/* Find hull points using a variant of Graham's algorithm.

* The path through the points is accumulated on a stack.

stack = gp_alloc( (np+1)*sizeof(void *), "Hull" );

/* Initialize stack with known start of top arc and first candidate point */

stack[0] = &points[0];

stack[1] = &points[1];

np = 2;

for (i=2; i<plot->p_count; i++) {

while ((np >= 2) && CROSS( stack[np-2], stack[np-1], &points[i] ) >= 0)

np--;

stack[np++] = &points[i];

}

/* push onto stack the first candidate point for lower arc */

i -= 2;

stack[np++] = &points[i];

ntop = np;

for (i--; i>=0; i--) {

while ( (np >= ntop) && CROSS( stack[np-2], stack[np-1], &points[i] ) >= 0)

np--;

stack[np++] = &points[i];

}

/* Replace the original list of points with the ordered path */

points = gp_alloc( np * sizeof(struct coordinate), "Hull" );

for (i=0; i<np; i++)

points[i] = *(stack[i]);

cp_extend(plot, 0);

free(stack);

plot->points = points;

plot->p_count = np;

plot->p_max = np;

}

* winnow_interior_points() is an optional helper routine for convex_hull.

* It reduces the number of points to be sorted and processed by removing

* points in a quadrilateral bounded by the four points with max/min x/y.

static void

winnow_interior_points (struct curve_points *plot, t_cluster *cluster)

{

#define TOLERANCE -1.e-10

struct coordinate *p, *pp1, *pp2, *pp3, *pp4;

struct coordinate *points = plot->points;

double area;

int i, np;

/* Find points defining maximal extent on x and y */

pp1 = pp2 = pp3 = pp4 = plot->points;

for (p = plot->points; p < &(plot->points[plot->p_count]); p++) {

if (p->x == cluster->xmin) pp1 = p;

if (p->x == cluster->xmax) pp2 = p;

if (p->y == cluster->ymin) pp3 = p;

if (p->y == cluster->ymax) pp4 = p;

}

/* Ignore any points that lie inside the clockwise triangle bounded by pp1 pp2 pp3 */

area = fabs(-pp2->y*pp3->x + pp1->y*(-pp2->x + pp3->x)

+ pp1->x*(pp2->y - pp3->y) + pp2->x*pp3->y);

area += TOLERANCE;

for (i=0; i<plot->p_count; i++) {

double px = points[i].x;

double py = points[i].y;

double s = (pp1->y*pp3->x - pp1->x*pp3->y + (pp3->y - pp1->y)*px + (pp1->x - pp3->x)*py);

double t = (pp1->x*pp2->y - pp1->y*pp2->x + (pp1->y - pp2->y)*px + (pp2->x - pp1->x)*py);

if ( (s < TOLERANCE) && (t < TOLERANCE) && (fabs(s+t) < area) )

points[i].type = EXCLUDEDRANGE;

}

/* Also ignore points in the clockwise triangle bounded by pp3 pp4 pp1 */

area = fabs(-pp4->y*pp1->x + pp3->y*(-pp4->x + pp1->x)

+ pp3->x*(pp4->y - pp1->y) + pp4->x*pp1->y);

area += TOLERANCE;

for (i=0; i<plot->p_count; i++) {

double px = points[i].x;

double py = points[i].y;

double s = (pp3->y*pp1->x - pp3->x*pp1->y + (pp1->y - pp3->y)*px + (pp3->x - pp1->x)*py);

double t = (pp3->x*pp4->y - pp3->y*pp4->x + (pp3->y - pp4->y)*px + (pp4->x - pp3->x)*py);

if ( (s < TOLERANCE) && (t < TOLERANCE) && (fabs(s+t) < area) )

points[i].type = EXCLUDEDRANGE;

}

/* Discard the interior points and outliers */

np = 0;

p = points;

for (i=0; i<plot->p_count; i++) {

if (points[i].type == UNDEFINED)

continue;

if (points[i].type != EXCLUDEDRANGE)

p[np++] = points[i];

}

plot->p_count = np;

#undef TOLERANCE

}

* expand_hull() "inflates" a convex or concave hull by displacing

* each edge away from the interior along its normal vector

* by requested distance d.

void

expand_hull(struct curve_points *plot)

{

struct coordinate *newpoints = NULL;

struct coordinate *points = plot->points;

double scale = plot->smooth_parameter;

double d = fabs(scale);

int N = plot->p_count;

struct coordinate *v1, *v2, *v3;

t_cluster cluster;

double winding; /* +1 for clockwise -1 for anticlockwise */

int ni;

/* Determine whether the hull points are ordered clockwise or

* anticlockwise. This allows easy determination of the interior

* side of each edge and the concave/convex status of each vertex.

cluster.npoints = N;

cluster_stats(points, &cluster);

v1 = &points[ (cluster.pin == 0) ? N-2 : cluster.pin-1 ];

v2 = &points[cluster.pin];

v3 = &points[cluster.pin + 1];

if (CROSS(v2,v1,v3) > 0)

winding = 1.0;

else

winding = -1.0;

/* Each edge of the hull is displaced outward by a constant amount d.

* At each convex vertex this replaces the original point with two

* points that are the vertices of a beveled join.

* At each concave vertex this displaces the original point toward

* the mouth of the concavity.

newpoints = gp_alloc(2*N * sizeof(struct coordinate), "expand hull");

ni = 0;

for (int i = 0; i < N; i++) {

double m1, m2, d2norm;

double dx1, dy1, dx2, dy2;

v1 = &points[ (i == 0) ? N-2 : i-1 ];

v2 = &points[i];

v3 = &points[ (i == N-1) ? 1 : i+1 ];

m1 = (v2->y - v1->y) / (v2->x - v1->x); /* slope of v1v2 */

d2norm = d*d / (1/(m1*m1) + 1);

dx1 = copysign( sqrt(d2norm), -winding * (v2->y - v1->y) );

dy1 = copysign( sqrt(d*d - d2norm), winding * (v2->x - v1->x) );

m2 = (v3->y - v2->y) / (v3->x - v2->x); /* slope of v2v3 */

d2norm = d*d / (1/(m2*m2) + 1);

dx2 = copysign( sqrt(d2norm), -winding * (v3->y - v2->y) );

dy2 = copysign( sqrt(d*d - d2norm), winding * (v3->x - v2->x) );

/* convex vertex */

if (winding * CROSS(v1,v2,v3) < 0) {

newpoints[ni] = points[i];

newpoints[ni].x = v2->x + dx1;

newpoints[ni].y = v2->y + dy1;

ni++;

newpoints[ni] = points[i];

newpoints[ni].x = v2->x + dx2;

newpoints[ni].y = v2->y + dy2;

ni++;

/* concave vertex (over-emphasizes steep holes) */

} else {

double dnorm = d / sqrt((dx1+dx2)*(dx1+dx2) + (dy1+dy2)*(dy1+dy2));

newpoints[ni] = points[i];

newpoints[ni].x = v2->x + dnorm * (dx1 + dx2);

newpoints[ni].y = v2->y + dnorm * (dy1 + dy2);

ni++;

}

/* Replace original point list with the new one */

cp_extend(plot, 0);

plot->points = newpoints;

plot->p_count = ni;

plot->p_max = 2*N;

}

* Find min, max, center of mass points in cluster held in plot->points.

* All points not marked EXCLUDEDRANGE are assumed to be in a single cluster.

static void

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