----------------------------------------------------------------------
NAME
spline - interpolate using splines under tension
SYNOPSIS
spline [file] [options]
DESCRIPTION
Without options, SPLINE reads pairs of numbers (x- and y-values) from
the standard input (or the given file), generates a smooth curve
through the points, and writes to the standard output points from the
smooth curve. The curve is a spline under tension (see references),
which is somewhat "tighter" than a cubic spline, and less
likely to have spurious inflection points. As with GRAPH, each
pair of points may optionally be followed by a comment. If the
comment is surrounded by quotes "...", the comment may contain
spaces. The given points, and their comments if any, will be
included in the output. The interpolation may optionally be
restarted after each label, so that a family of curves may be
processed together (see the -b switch).
Input lines starting with ";" are copied to the beginning of
the output file but are otherwise ignored. Blank lines are
ignored.
If the -c switch is not used, the input points must be from a function
- that is, the x values must be strictly increasing. The
output points will also be from a function. (If the -b switch
is used, this restriction applies only within each segment.)
If the -c switch is used (indicating a general curve), the
input points need not be from a function, but each pair of
points must be separated from the previous pair by a finite
distance. (If the -b switch is used, this restriction applies
only within each segment.)
options are:
-a [step [start]] Input data contains only y values - generate automatic
abscissas at intervals of step (default 1) starting at start
(default 0).
-b break the interpolation at each label. That is, the input
curve is divided into sections with the last point in
each section marked by a label (which may be empty:
""). A separate interpolating curve is to be found for
each section. In this case, the requirements on the
number of intervals (specified by the -n switch or
defaulted) and the interpolation range (specified by the
-x switch) are applied to each section independently.
-c general curve rather than function. In this case, the curve is
parameterized on the polygonal arclength from the first
to the last given point, with the whole length scaled
to be 1. Thus, the values min and max for the -x
switch should satisfy 0 <= min < max <= 1. The -s and
-c switches cannot be used together.
-n num interpolate over num intervals (default is 100)
-q Quadruple: increase the number of intervals fourfold.
-s [num [num]]
Specify slopes at beginning and end of curve. Slopes
not given are assumed to be zero. Without the -s
switch, slopes are determined from other information.
The -s and -c switches cannot be used together.
-t num Specify tension in interpolating curve. Tension of 50 gives
almost polygonal line, tension of .01 gives almost cubic
spline. Tension must be positive. Default is 1.
-x [min [max]]
Interpolate from min to max only. min and max should
be in the range of the given x values, except that if
the -c switch is used they should satisfy 0 <= min <
max <= 1.
-xl take log of x values before interpolating, take exponential
afterwards (probably necessary if -xl switch is needed for
GRAPH)
-yl take log of y values before interpolating, take exponential
afterwards (probably necessary if -yl switch is needed for
GRAPH)
NOTES
Similar to the Unix routine, except using splines under tension,
passing labels through, performing multiple interpolations from
one file and allowing general curves.
REFERENCES
A. K. Cline, "Scalar- and Planar- Valued Curve Fitting Using
Splines Under Tension", Communications of the ACM v 17 n 4 p
218-223 (Apr 74).
Schweikert, D. G. "An interpolation curve using a spline in
tension", J. Math. and Physics v 45 p 312-317 (1966).
AUTHOR
Copyright (c) 1985 James R. Van Zandt
Resale forbidden, copying for personal use encouraged.
-__--------------------------------------------------------------------
/* spline - interpolate points in a tabulated function
Usage...
spline [file][options]
options are:
-a [step [start]] automatic abscissas
-b break interpolation after each label
-c general curve
-n num interpolate over num intervals
-q increase intervals fourfold
-t num tension in interpolating curve
-x [min [max]] interpolate from min to max
-xl take logs of x values before interpolating
-yl take logs of y values before interpolating
Notes...
This program fits a smooth curve through a given set of points,
using the splines under tension introduced by Schweikert [J.
Math. and Physics v 45 pp 312-317 (1966)]. They are similar
to cubic splines, but are less likely to introduce spurious
inflection points.
History...
3 Jun 86 -s switch allows specified slopes at ends.
2 Jun 86 Ver 1.3: Fixed several embarrassing bugs in -b option.
-q switch quadruples number of intervals.
6 Apr 86 -b switch breaks interpolation at labels.
21 Sep 85 Added -xl and -yl switches for taking logs
23 Nov 85 Passing lines starting with ';' unchanged, otherwise
ignoring them.
22 Jan 90 added missing * in FILE *file;
(suggested by Jean-Marie de Montarby)
Author...
Copyright (c) 1985, 1986 James R. Van Zandt (jrv@mbunix.mitre.org)
All rights reserved.
This program may be copied for personal, non-profit use only.
Based on algorithms by A. K. Cline [Communications of the ACM
v 17 n 4 pp 218-223 (Apr 74)].
*/
#include <stdio.h>
#include <math.h>
#define VERSION "1.3"
char *RIGHT= "Copyright (c) 1985, 1986 James R. Van Zandt";
#define ENTRIES 200
#define MAXLABELS 50
double *x, *y, *temp, *xp, *yp, *path;
double slp1=0.,slpn=0.,sigma=1.;
double abscissa=0.;
double abscissa_step=1.;
double slope1=0.,slopen=0.; /* slopes supplied by user */
double xmin=0.;
double xmax=0.;
double curv2(), mylog();
int xlog=0; /* nonzero if taking log of x values */
int ylog=0; /* nonzero if taking log of y values */
int debugging=0;
int breaking=0; /* nonzero if breaking at labels */
int labels=0; /* number of labels in input data */
int quadruple=0; /* nonzero if user asking for 4 times the resolution */
int automatic_abscissas=0;
int slopes=0; /* nonzero if slopes supplied by user */
int abscissa_arguments=0;
int curve=0; /* nonzero if general curve permitted */
int x_arguments=0;
int n; /* number of entries in x, y, yp, and temp */
int index_array[MAXLABELS]; /* indexes into x and y */
int *p_data=index_array;
int total=100;
FILE *file;
char *label_array[MAXLABELS];
char **p_text=label_array;
main(argc,argv) int argc; char **argv;
{ int nn,origin;
read_data(argc,argv);
if(breaking && labels)
{origin=0;
while(labels--)
{p_data[0] -= origin;
nn=p_data[0]+1;
if(quadruple) total=nn*4-3;
if(nn) curv0(nn,total);
origin += nn;
path += nn;
x += nn;
y += nn;
p_data++;
p_text++;
}
}
else
{if(quadruple) total=n*4-3;
curv0(n,total);
}
}
curv0(n,requested)
int n,requested;
{ int i, j, each, stop, seg=0, regressing=0;
double s,ds,xx,yy;
for(i=1; i<n; i++) if(path[i]<=path[i-1]) regressing++;
if (regressing)
{fprintf(stderr,"ERROR: independent variable not strictly increasing\n");
return;
}
if (curve && (xmin<0. || xmax>1.))
{fprintf(stderr,"ERROR: independent variable not in range 0 to 1\n");
return;
}
if(curve) {curv1(path,x,xp,n); curv1(path,y,yp,n);}
else {slp1=slope1; slpn=slopen; curv1(x,y,yp,n);}
s=path[0];
seg=0;
if(requested<n) requested=n;
if(x_arguments==2) /* specific range was requested */
{ds=(xmax-xmin)/requested;
if(curve)
{ds*=(path[n-1]-path[0]);
s=xmin*(path[n-1]-path[0])+path[0];
}
else s=xmin;
for(i=requested+1; i; i--)
{value(s,&xx,&yy,n);
printf("%g %g ",xx,yy);
if(j==p_data[seg]) puts(p_text[seg++]);
putchar('\n');
s+=ds;
}
}
else /* spline for entire curve was requested */
{for (i=j=0; j<n-1; j++)
{stop=requested*(j+1)/(n-1);
ds=(path[j+1]-path[j])/(stop-i);
for ( ; i<stop; i++)
{value(s,&xx,&yy,n);
printf("%g %g ",xx,yy);
if(j==p_data[seg]) puts(p_text[seg++]);
putchar('\n');
s+=ds;
}
}
xx=x[n-1]; if(xlog) xx=exp(xx);
yy=y[n-1]; if(ylog) yy=exp(yy);
printf("%g %g ",xx,yy);
if(j==p_data[seg]) puts(p_text[seg++]);
putchar('\n');
}
}
value(s,q,r,n) double s,*q,*r; int n;
{ if(curve)
{*q=curv2(path,x,xp,s,n);
*r=curv2(path,y,yp,s,n);
}
else
{*q=s;
*r=curv2(x,y,yp,s,n);
}
if(xlog) *q=exp(*q);
if(ylog) *r=exp(*r);
}
read_data(argc,argv) int argc; char **argv;
{ int i,j,length;
double xx,yy,d,*pd,sum;
char *s,*t;
#define BUFSIZE 200
static char buf[BUFSIZE];
x=path=malloc(ENTRIES*sizeof(double));
y=malloc(ENTRIES*sizeof(double));
if(x==0 || y==0) {fprintf(stderr,"can\'t allocate buffer"); exit();}
if(argc>1 && streq(argv[1],"?")) help();
if(argc<=1 || *argv[1]=='-') file=stdin;
else
{if(argc>1)
{file=fopen(argv[1],"r");
if(file==0) {printf("file %s not found\n",argv[1]); exit();}
argc--; argv++;
}
else help();
}
argc--; argv++;
while(argc>0)
{i=get_parameter(argc,argv);
argv=argv+i; argc=argc-i;
}
if(sigma<=0.)
{fprintf(stderr,"ERROR: tension must be positive\n");
exit(1);
}
if(slopes && curve)
{fprintf(stderr,"ERROR: slopes can\'t be specified for general curve\n");
exit(1);
}
sigma *= -1.;
if(xlog && !curve)
{if(x_arguments>1) xmax=mylog(xmax);
if(x_arguments>=1) xmin=mylog(xmin);
}
if(automatic_abscissas && abscissa_arguments<2 && x_arguments>=1)
abscissa=xmin;
p_data[0]=-1;
i=0;
while(i<ENTRIES)
{if(fgets(buf,BUFSIZE,file)==0)break;
t=buf;
while(*t && isspace(*t)) t++;
if(*t == 0) continue; /* skip blank lines */
buf[strlen(buf)-1]=0; /* zero out the line feed */
if(buf[0]==';') {printf("%s\n",buf); continue;} /* skip comment */
if(automatic_abscissas)
{x[i]=abscissa;
abscissa+=abscissa_step;
sscanf(buf,"%F",&y[i]);
if(ylog) y[i]=mylog(y[i]);
}
else
{sscanf(buf,"%F %F",&x[i],&y[i]);
if(xlog) x[i]=mylog(x[i]);
if(ylog) y[i]=mylog(y[i]);
}
s=buf; /* start looking for label */
while(*s==' ')s++; /* skip first number */
while(*s && (*s!=' '))s++;
if(!automatic_abscissas) /* skip second number */
{while(*s==' ')s++;
while(*s && (*s!=' '))s++;
}
while(*s==' ')s++;
if((length=strlen(s))&&(labels<MAXLABELS))
{if(*s=='\"') /* label in quotes */
{t=s+1;
while(*t && (*t!='\"')) t++;
t++;
}
else /* label without quotes */
{t=s;
while(*t && (*t!=' '))t++;
}
*t=0;
length=t-s;
p_data[labels]=i;
p_text[labels]=malloc(length+1);
if(p_text[labels]) strcpy(p_text[labels++],s);
}
i++;
}
n=i;
if(breaking && (!labels || p_data[labels-1]!=n-1))
{p_data[labels]=i-1;
if(p_text[labels]=malloc(1)) *p_text[labels++]=0;
}
yp=malloc(n*sizeof(double));
temp=malloc(n*sizeof(double));
if(temp==0 || yp==0) {fprintf(stderr,"can\'t allocate buffer"); exit();}
if(curve)
{xp=malloc(n*sizeof(double));
path=malloc(n*sizeof(double));
if(xp==0|| path==0) {fprintf(stderr,"can\'t allocate buffer"); exit();}
path[0]=sum=0.;
for (i=1; i<n; i++)
{xx=x[i]-x[i-1];
yy=y[i]-y[i-1];
path[i]=(sum+=sqrt(xx*xx + yy*yy));
}
/* for(i=0; i<n; i++)
printf("path[%d]=%g x[%d]=%g \n",i,path[i],i,x[i]); */
}
}
/* get_parameter - process one command line option
(return # parameters used) */
get_parameter(argc,argv) int argc; char **argv;
{ int i;
if(streq(*argv,"-a"))
{i=get_double(argc,argv,2,&abscissa_step,&abscissa,&abscissa);
abscissa_arguments=i-1;
automatic_abscissas=1;
return i;
}
else if(streq(*argv,"-b")) {breaking=1; return 1;}
else if(streq(*argv,"-c")) {curve=1; return 1;}
else if(streq(*argv,"-d")) {debugging=1; return 1;}
else if(streq(*argv,"-n"))
{if((argc>1) && numeric(argv[1])) total=atoi(argv[1]);
return 2;
}
else if(streq(*argv,"-q")) {quadruple=1; return 1;}
else if(streq(*argv,"-t"))
{return(get_double(argc,argv,1,&sigma,&abscissa,&abscissa));
}
else if(streq(*argv,"-s"))
{slopes++;
return(get_double(argc,argv,2,&slope1,&slopen,&slopen));
}
else if(streq(*argv,"-x"))
{i=get_double(argc,argv,2,&xmin,&xmax,&xmax);
x_arguments=i-1;
return i;
}
else if(streq(*argv,"-xl")) {xlog++; return 1;}
else if(streq(*argv,"-yl")) {ylog++; return 1;}
else gripe(argv);
}
get_double(argc,argv,permitted,a,b,c)
int argc,permitted; char **argv; double *a,*b,*c;
{ int i=1;
if((permitted--)>0 && (argc>i) && numeric(argv[i])) *a=atof(argv[i++]);
if((permitted--)>0 && (argc>i) && numeric(argv[i])) *b=atof(argv[i++]);
if((permitted--)>0 && (argc>i) && numeric(argv[i])) *c=atof(argv[i++]);
return i;
}
int streq(a,b) char *a,*b;
{ while(*a)
{if(*a!=*b)return 0;
a++; b++;
}
return 1;
}
gripe_arg(s) char *s;
{ fprintf(stderr,"argument missing for switch %s",s);
help();
}
gripe(argv) char **argv;
{ fprintf(stderr,*argv); fprintf(stderr," isn\'t a legal argument \n\n");
help();
}
help()
{ fprintf(stderr,"spline version %s",VERSION);
fprintf(stderr,"\nusage: spline [file][options]\n");
fprintf(stderr,"options are:\n");
fprintf(stderr," -a [step [start]] automatic abscissas \n");
fprintf(stderr," -b break interpolation after each label \n");
fprintf(stderr," -c general curve \n");
fprintf(stderr," -n num interpolate over num intervals \n");
fprintf(stderr," -q increase intervals fourfold \n");
fprintf(stderr," -s [num [num]] specify slopes at beginning and end \n");
fprintf(stderr," -t num tension in interpolating curve \n");
fprintf(stderr," -x [min [max]] interpolate from min to max \n");
fprintf(stderr," -xl take logs of x values before interpolating \n");
fprintf(stderr," -yl take logs of y values before interpolating \n");
exit();
}
numeric(s) char *s;
{ char c;
while(c=*s++)
{if((c<='9' && c>='0') || c=='+' || c=='-' || c=='.') continue;
return 0;
}
return 1;
}
curv1(x,y,yp,n) double *x,*y,*yp; int n;
{ int i;
double c1,c2,c3,deln,delnm1,delnn,dels,delx1,delx2,delx12;
double diag1,diag2,diagin,dx1,dx2,exps;
double sigmap,sinhs,sinhin,slpp1,slppn,spdiag;
delx1=x[1] - x[0];
dx1=(y[1] - y[0])/delx1;
if(slopes) {slpp1=slp1; slppn=slpn;}
else
{if(n!=2)
{delx2= x[2] - x[1];
delx12= x[2] - x[0];
c1= -(delx12 + delx1)/delx12/delx1;
c2= delx12/delx1/delx2;
c3= -delx1/delx12/delx2;
slpp1= c1*y[0] + c2*y[1] + c3*y[2];
deln= x[n-1] - x[n-2];
delnm1= x[n-2] - x[n-3];
delnn= x[n-1] - x[n-3];
c1= (delnn + deln)/delnn/deln;
c2= -delnn/deln/delnm1;
c3= deln/delnn/delnm1;
slppn= c3*y[n-3] + c2*y[n-2] + c1*y[n-1];
}
else yp[0]=yp[1]=0.;
}
/* denormalize tension factor */
sigmap=fabs(sigma)*(n-1)/(x[n-1]-x[0]);
/* set up right hand side and tridiagonal system for
yp and perform forward elimination */
dels=sigmap*delx1;
exps=exp(dels);
sinhs=.5*(exps-1./exps);
sinhin=1./(delx1*sinhs);
diag1=sinhin*(dels*.5*(exps+1./exps)-sinhs);
diagin=1./diag1;
yp[0]=diagin*(dx1-slpp1);
spdiag=sinhin*(sinhs-dels);
temp[0]=diagin*spdiag;
if(n!=2)
{for(i=1; i<=n-2; i++)
{delx2=x[i+1]-x[i];
dx2=(y[i+1]-y[i])/delx2;
dels=sigmap*delx2;
exps=exp(dels);
sinhs=.5*(exps-1./exps);
sinhin=1./(delx2*sinhs);
diag2=sinhin*(dels*(.5*(exps+1./exps))-sinhs);
diagin=1./(diag1+diag2-spdiag*temp[i-1]);
yp[i]=diagin*(dx2-dx1-spdiag*yp[i-1]);
spdiag=sinhin*(sinhs-dels);
temp[i]=diagin*spdiag;
dx1=dx2;
diag1=diag2;
}
}
diagin=1./(diag1-spdiag*temp[n-2]);
yp[n-1]=diagin*(slppn-dx2-spdiag*yp[n-2]);
/* perform back substitution */
for (i=n-2; i>=0; i--) yp[i] -= temp[i]*yp[i+1];
}
double curv2(x,y,yp,t,n) double *x,*y,*yp,t; int n;
{ int i,j;
static int i1=1;
double del1,del2,dels,exps,exps1,s,sigmap,sinhd1,sinhd2,sinhs;
s=x[n-1]-x[0];
sigmap=fabs(sigma)*(n-1)/s;
#ifdef WORK
for (j=2; j; j--) /* want: x[i-1] <= t < x[i], 0 < i <= n */
{for (i=i1; i<n; i++)
{if(x[i]>t) break;
}
if(i==n) i=n-1;
if(x[i-1]<=t || t<=x[0]) break;
i1=1;
}
#endif
i=i1;
if(i>n) i=1;
while(i<n && t>=x[i]) i++;
while(i>1 && x[i-1]>t) i--;
i1=i;
del1=t-x[i-1];
del2=x[i]-t;
dels=x[i] - x[i-1];
exps1=exp(sigmap*del1); sinhd1=.5*(exps1-1./exps1);
exps= exp(sigmap*del2); sinhd2=.5*(exps-1./exps);
exps=exps1*exps; sinhs=.5*(exps-1./exps);
return ((yp[i]*sinhd1 + yp[i-1]*sinhd2)/sinhs +
((y[i] - yp[i])*del1 + (y[i-1] - yp[i-1])*del2)/dels);
}
double mylog(x) double x;
{ if(x>0.) return log(x);
fprintf(stderr,"can%'t take log of nonpositive number");
exit(1);
return 0.;
}