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splineutil2.c
5131 lines (4727 loc) · 165 KB
/
splineutil2.c
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/* -*- coding: utf-8 -*- */
/* Copyright (C) 2000-2012 by George Williams */
/*
* 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.
* The name of the author may not be used to endorse or promote products
* derived from this software without specific prior written permission.
* THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``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 AUTHOR 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 "fontforge.h"
#include <math.h>
#include "ustring.h"
#include "chardata.h"
#include <unistd.h>
#include <time.h>
#include <stdlib.h>
#include "views.h" /* For SCCharChangedUpdate */
int new_em_size = 1000;
int new_fonts_are_order2 = false;
int loaded_fonts_same_as_new = false;
int default_fv_row_count = 4;
int default_fv_col_count = 16;
int default_fv_font_size = 48;
int default_fv_antialias=true;
int default_fv_bbsized=false;
int snaptoint=0;
/*#define DEBUG 1*/
#if defined( FONTFORGE_CONFIG_USE_DOUBLE )
# define RE_NearZero .00000001
# define RE_Factor (1024.0*1024.0*1024.0*1024.0*1024.0*2.0) /* 52 bits => divide by 2^51 */
#else
# define RE_NearZero .00001
# define RE_Factor (1024.0*1024.0*4.0) /* 23 bits => divide by 2^22 */
#endif
int Within4RoundingErrors(bigreal v1, bigreal v2) {
bigreal temp=v1*v2;
bigreal re;
if ( temp<0 ) /* Ok, if the two values are on different sides of 0 there */
return( false ); /* is no way they can be within a rounding error of each other */
else if ( temp==0 ) {
if ( v1==0 )
return( v2<RE_NearZero && v2>-RE_NearZero );
else
return( v1<RE_NearZero && v1>-RE_NearZero );
} else if ( v1>0 ) {
if ( v1>v2 ) { /* Rounding error from the biggest absolute value */
re = v1/ (RE_Factor/4);
return( v1-v2 < re );
} else {
re = v2/ (RE_Factor/4);
return( v2-v1 < re );
}
} else {
if ( v1<v2 ) {
re = v1/ (RE_Factor/4); /* This will be a negative number */
return( v1-v2 > re );
} else {
re = v2/ (RE_Factor/4);
return( v2-v1 > re );
}
}
}
int Within16RoundingErrors(bigreal v1, bigreal v2) {
bigreal temp=v1*v2;
bigreal re;
if ( temp<0 ) /* Ok, if the two values are on different sides of 0 there */
return( false ); /* is no way they can be within a rounding error of each other */
else if ( temp==0 ) {
if ( v1==0 )
return( v2<RE_NearZero && v2>-RE_NearZero );
else
return( v1<RE_NearZero && v1>-RE_NearZero );
} else if ( v1>0 ) {
if ( v1>v2 ) { /* Rounding error from the biggest absolute value */
re = v1/ (RE_Factor/16);
return( v1-v2 < re );
} else {
re = v2/ (RE_Factor/16);
return( v2-v1 < re );
}
} else {
if ( v1<v2 ) {
re = v1/ (RE_Factor/16); /* This will be a negative number */
return( v1-v2 > re );
} else {
re = v2/ (RE_Factor/16);
return( v2-v1 > re );
}
}
}
int Within64RoundingErrors(bigreal v1, bigreal v2) {
bigreal temp=v1*v2;
bigreal re;
if ( temp<0 ) /* Ok, if the two values are on different sides of 0 there */
return( false ); /* is no way they can be within a rounding error of each other */
else if ( temp==0 ) {
if ( v1==0 )
return( v2<RE_NearZero && v2>-RE_NearZero );
else
return( v1<RE_NearZero && v1>-RE_NearZero );
} else if ( v1>0 ) {
if ( v1>v2 ) { /* Rounding error from the biggest absolute value */
re = v1/ (RE_Factor/64);
return( v1-v2 < re );
} else {
re = v2/ (RE_Factor/64);
return( v2-v1 < re );
}
} else {
if ( v1<v2 ) {
re = v1/ (RE_Factor/64); /* This will be a negative number */
return( v1-v2 > re );
} else {
re = v2/ (RE_Factor/64);
return( v2-v1 > re );
}
}
}
int RealNear(real a,real b) {
real d;
#ifdef FONTFORGE_CONFIG_USE_DOUBLE
if ( a==0 )
return( b>-1e-8 && b<1e-8 );
if ( b==0 )
return( a>-1e-8 && a<1e-8 );
d = a/(1024*1024.);
#else /* For floats */
if ( a==0 )
return( b>-1e-5 && b<1e-5 );
if ( b==0 )
return( a>-1e-5 && a<1e-5 );
d = a/(1024*64.);
#endif
a-=b;
if ( d<0 )
return( a>d && a<-d );
else
return( a>-d && a<d );
}
int RealNearish(real a,real b) {
a-=b;
return( a<.001 && a>-.001 );
}
int RealApprox(real a,real b) {
if ( a==0 ) {
if ( b<.0001 && b>-.0001 )
return( true );
} else if ( b==0 ) {
if ( a<.0001 && a>-.0001 )
return( true );
} else {
a /= b;
if ( a>=.95 && a<=1.05 )
return( true );
}
return( false );
}
int RealWithin(real a,real b,real fudge) {
return( b>=a-fudge && b<=a+fudge );
}
int RealRatio(real a,real b,real fudge) {
if ( b==0 )
return( RealWithin(a,b,fudge));
return( RealWithin(a/b,1.0,fudge));
}
static int MinMaxWithin(Spline *spline) {
extended dx, dy;
int which;
extended t1, t2;
extended w;
/* We know that this "spline" is basically one dimensional. As long as its*/
/* extrema are between the start and end points on that line then we can */
/* treat it as a line. If the extrema are way outside the line segment */
/* then it's a line that backtracks on itself */
if ( (dx = spline->to->me.x - spline->from->me.x)<0 ) dx = -dx;
if ( (dy = spline->to->me.y - spline->from->me.y)<0 ) dy = -dy;
which = dx<dy;
SplineFindExtrema(&spline->splines[which],&t1,&t2);
if ( t1==-1 )
return( true );
w = ((spline->splines[which].a*t1 + spline->splines[which].b)*t1
+ spline->splines[which].c)*t1 + spline->splines[which].d;
if ( RealNear(w, (&spline->to->me.x)[which]) || RealNear(w, (&spline->from->me.x)[which]) )
/* Close enough */;
else if ( (w<(&spline->to->me.x)[which] && w<(&spline->from->me.x)[which]) ||
(w>(&spline->to->me.x)[which] && w>(&spline->from->me.x)[which]) )
return( false ); /* Outside */
w = ((spline->splines[which].a*t2 + spline->splines[which].b)*t2
+ spline->splines[which].c)*t2 + spline->splines[which].d;
if ( RealNear(w, (&spline->to->me.x)[which]) || RealNear(w, (&spline->from->me.x)[which]) )
/* Close enough */;
else if ( (w<(&spline->to->me.x)[which] && w<(&spline->from->me.x)[which]) ||
(w>(&spline->to->me.x)[which] && w>(&spline->from->me.x)[which]) )
return( false ); /* Outside */
return( true );
}
int SplineIsLinear(Spline *spline) {
bigreal t1,t2, t3,t4;
int ret;
if ( spline->knownlinear )
return( true );
if ( spline->knowncurved )
return( false );
if ( spline->splines[0].a==0 && spline->splines[0].b==0 &&
spline->splines[1].a==0 && spline->splines[1].b==0 )
return( true );
/* Something is linear if the control points lie on the line between the */
/* two base points */
/* Vertical lines */
if ( RealNear(spline->from->me.x,spline->to->me.x) ) {
ret = RealNear(spline->from->me.x,spline->from->nextcp.x) &&
RealNear(spline->from->me.x,spline->to->prevcp.x);
if ( ! ((spline->from->nextcp.y >= spline->from->me.y &&
spline->from->nextcp.y <= spline->to->me.y &&
spline->to->prevcp.y >= spline->from->me.y &&
spline->to->prevcp.y <= spline->to->me.y ) ||
(spline->from->nextcp.y <= spline->from->me.y &&
spline->from->nextcp.y >= spline->to->me.y &&
spline->to->prevcp.y <= spline->from->me.y &&
spline->to->prevcp.y >= spline->to->me.y )) )
ret = MinMaxWithin(spline);
/* Horizontal lines */
} else if ( RealNear(spline->from->me.y,spline->to->me.y) ) {
ret = RealNear(spline->from->me.y,spline->from->nextcp.y) &&
RealNear(spline->from->me.y,spline->to->prevcp.y);
if ( ! ((spline->from->nextcp.x >= spline->from->me.x &&
spline->from->nextcp.x <= spline->to->me.x &&
spline->to->prevcp.x >= spline->from->me.x &&
spline->to->prevcp.x <= spline->to->me.x) ||
(spline->from->nextcp.x <= spline->from->me.x &&
spline->from->nextcp.x >= spline->to->me.x &&
spline->to->prevcp.x <= spline->from->me.x &&
spline->to->prevcp.x >= spline->to->me.x)) )
ret = MinMaxWithin(spline);
} else {
ret = true;
t1 = (spline->from->nextcp.y-spline->from->me.y)/(spline->to->me.y-spline->from->me.y);
t2 = (spline->from->nextcp.x-spline->from->me.x)/(spline->to->me.x-spline->from->me.x);
t3 = (spline->to->me.y-spline->to->prevcp.y)/(spline->to->me.y-spline->from->me.y);
t4 = (spline->to->me.x-spline->to->prevcp.x)/(spline->to->me.x-spline->from->me.x);
ret = (Within16RoundingErrors(t1,t2) || (RealApprox(t1,0) && RealApprox(t2,0))) &&
(Within16RoundingErrors(t3,t4) || (RealApprox(t3,0) && RealApprox(t4,0)));
if ( ret ) {
if ( t1<0 || t2<0 || t3<0 || t4<0 ||
t1>1 || t2>1 || t3>1 || t4>1 )
ret = MinMaxWithin(spline);
}
}
spline->knowncurved = !ret;
spline->knownlinear = ret;
if ( ret ) {
/* A few places that if the spline is knownlinear then its splines[?] */
/* are linear. So give the linear version and not that suggested by */
/* the control points */
spline->splines[0].a = spline->splines[0].b = 0;
spline->splines[0].d = spline->from->me.x;
spline->splines[0].c = spline->to->me.x-spline->from->me.x;
spline->splines[1].a = spline->splines[1].b = 0;
spline->splines[1].d = spline->from->me.y;
spline->splines[1].c = spline->to->me.y-spline->from->me.y;
}
return( ret );
}
int SplineIsLinearMake(Spline *spline) {
if ( spline->islinear )
return( true );
if ( SplineIsLinear(spline)) {
spline->islinear = spline->from->nonextcp = spline->to->noprevcp = true;
spline->from->nextcp = spline->from->me;
if ( spline->from->nonextcp && spline->from->noprevcp )
spline->from->pointtype = pt_corner;
else if ( spline->from->pointtype == pt_curve || spline->from->pointtype == pt_hvcurve )
spline->from->pointtype = pt_tangent;
spline->to->prevcp = spline->to->me;
if ( spline->to->nonextcp && spline->to->noprevcp )
spline->to->pointtype = pt_corner;
else if ( spline->to->pointtype == pt_curve || spline->to->pointtype == pt_hvcurve )
spline->to->pointtype = pt_tangent;
SplineRefigure(spline);
}
return( spline->islinear );
}
static Spline *IsLinearApprox(SplinePoint *from, SplinePoint *to,
TPoint *mid, int cnt, int order2) {
bigreal vx, vy, slope;
int i;
vx = to->me.x-from->me.x; vy = to->me.y-from->me.y;
if ( vx==0 && vy==0 ) {
for ( i=0; i<cnt; ++i )
if ( mid[i].x != from->me.x || mid[i].y != from->me.y )
return( NULL );
} else if ( fabs(vx)>fabs(vy) ) {
slope = vy/vx;
for ( i=0; i<cnt; ++i )
if ( !RealWithin(mid[i].y,from->me.y+slope*(mid[i].x-from->me.x),.7) )
return( NULL );
} else {
slope = vx/vy;
for ( i=0; i<cnt; ++i )
if ( !RealWithin(mid[i].x,from->me.x+slope*(mid[i].y-from->me.y),.7) )
return( NULL );
}
from->nonextcp = to->noprevcp = true;
return( SplineMake(from,to,order2) );
}
/* This routine should almost never be called now. It uses a flawed algorithm */
/* which won't produce the best results. It gets called only when the better */
/* approach doesn't work (singular matrices, etc.) */
/* Old comment, back when I was confused... */
/* Least squares tells us that:
| S(xi*ti^3) | | S(ti^6) S(ti^5) S(ti^4) S(ti^3) | | a |
| S(xi*ti^2) | = | S(ti^5) S(ti^4) S(ti^3) S(ti^2) | * | b |
| S(xi*ti) | | S(ti^4) S(ti^3) S(ti^2) S(ti) | | c |
| S(xi) | | S(ti^3) S(ti^2) S(ti) n | | d |
and the definition of a spline tells us:
| x1 | = | 1 1 1 1 | * (a b c d)
| x0 | = | 0 0 0 1 | * (a b c d)
So we're a bit over specified. Let's use the last two lines of least squares
and the 2 from the spline defn. So d==x0. Now we've got three unknowns
and only three equations...
For order2 splines we've got
| S(xi*ti^2) | | S(ti^4) S(ti^3) S(ti^2) | | b |
| S(xi*ti) | = | S(ti^3) S(ti^2) S(ti) | * | c |
| S(xi) | | S(ti^2) S(ti) n | | d |
and the definition of a spline tells us:
| x1 | = | 1 1 1 | * (b c d)
| x0 | = | 0 0 1 | * (b c d)
=>
d = x0
b+c = x1-x0
S(ti^2)*b + S(ti)*c = S(xi)-n*x0
S(ti^2)*b + S(ti)*(x1-x0-b) = S(xi)-n*x0
[ S(ti^2)-S(ti) ]*b = S(xi)-S(ti)*(x1-x0) - n*x0
*/
static int _ApproximateSplineFromPoints(SplinePoint *from, SplinePoint *to,
TPoint *mid, int cnt, BasePoint *nextcp, BasePoint *prevcp,
int order2) {
bigreal tt, ttn;
int i, j, ret;
bigreal vx[3], vy[3], m[3][3];
bigreal ts[7], xts[4], yts[4];
BasePoint nres, pres;
int nrescnt=0, prescnt=0;
bigreal nmin, nmax, pmin, pmax, test, ptest;
bigreal bx, by, cx, cy;
memset(&nres,0,sizeof(nres)); memset(&pres,0,sizeof(pres));
/* Add the initial and end points */
ts[0] = 2; for ( i=1; i<7; ++i ) ts[i] = 1;
xts[0] = from->me.x+to->me.x; yts[0] = from->me.y+to->me.y;
xts[3] = xts[2] = xts[1] = to->me.x; yts[3] = yts[2] = yts[1] = to->me.y;
nmin = pmin = 0; nmax = pmax = (to->me.x-from->me.x)*(to->me.x-from->me.x)+(to->me.y-from->me.y)*(to->me.y-from->me.y);
for ( i=0; i<cnt; ++i ) {
xts[0] += mid[i].x;
yts[0] += mid[i].y;
++ts[0];
tt = mid[i].t;
xts[1] += tt*mid[i].x;
yts[1] += tt*mid[i].y;
ts[1] += tt;
ts[2] += (ttn=tt*tt);
xts[2] += ttn*mid[i].x;
yts[2] += ttn*mid[i].y;
ts[3] += (ttn*=tt);
xts[3] += ttn*mid[i].x;
yts[3] += ttn*mid[i].y;
ts[4] += (ttn*=tt);
ts[5] += (ttn*=tt);
ts[6] += (ttn*=tt);
test = (mid[i].x-from->me.x)*(to->me.x-from->me.x) + (mid[i].y-from->me.y)*(to->me.y-from->me.y);
if ( test<nmin ) nmin=test;
if ( test>nmax ) nmax=test;
test = (mid[i].x-to->me.x)*(from->me.x-to->me.x) + (mid[i].y-to->me.y)*(from->me.y-to->me.y);
if ( test<pmin ) pmin=test;
if ( test>pmax ) pmax=test;
}
pmin *= 1.2; pmax *= 1.2; nmin *= 1.2; nmax *= 1.2;
for ( j=0; j<3; ++j ) {
if ( order2 ) {
if ( RealNear(ts[j+2],ts[j+1]) )
continue;
/* This produces really bad results!!!! But I don't see what I can do to improve it */
bx = (xts[j]-ts[j+1]*(to->me.x-from->me.x) - ts[j]*from->me.x) / (ts[j+2]-ts[j+1]);
by = (yts[j]-ts[j+1]*(to->me.y-from->me.y) - ts[j]*from->me.y) / (ts[j+2]-ts[j+1]);
cx = to->me.x-from->me.x-bx;
cy = to->me.y-from->me.y-by;
nextcp->x = from->me.x + cx/2;
nextcp->y = from->me.y + cy/2;
*prevcp = *nextcp;
} else {
vx[0] = xts[j+1]-ts[j+1]*from->me.x;
vx[1] = xts[j]-ts[j]*from->me.x;
vx[2] = to->me.x-from->me.x; /* always use the defn of spline */
vy[0] = yts[j+1]-ts[j+1]*from->me.y;
vy[1] = yts[j]-ts[j]*from->me.y;
vy[2] = to->me.y-from->me.y;
m[0][0] = ts[j+4]; m[0][1] = ts[j+3]; m[0][2] = ts[j+2];
m[1][0] = ts[j+3]; m[1][1] = ts[j+2]; m[1][2] = ts[j+1];
m[2][0] = 1; m[2][1] = 1; m[2][2] = 1;
/* Remove a terms from rows 0 and 1 */
vx[0] -= ts[j+4]*vx[2];
vy[0] -= ts[j+4]*vy[2];
m[0][0] = 0; m[0][1] -= ts[j+4]; m[0][2] -= ts[j+4];
vx[1] -= ts[j+3]*vx[2];
vy[1] -= ts[j+3]*vy[2];
m[1][0] = 0; m[1][1] -= ts[j+3]; m[1][2] -= ts[j+3];
if ( fabs(m[1][1])<fabs(m[0][1]) ) {
bigreal temp;
temp = vx[1]; vx[1] = vx[0]; vx[0] = temp;
temp = vy[1]; vy[1] = vy[0]; vy[0] = temp;
temp = m[1][1]; m[1][1] = m[0][1]; m[0][1] = temp;
temp = m[1][2]; m[1][2] = m[0][2]; m[0][2] = temp;
}
/* remove b terms from rows 0 and 2 (first normalize row 1 so m[1][1] is 1*/
vx[1] /= m[1][1];
vy[1] /= m[1][1];
m[1][2] /= m[1][1];
m[1][1] = 1;
vx[0] -= m[0][1]*vx[1];
vy[0] -= m[0][1]*vy[1];
m[0][2] -= m[0][1]*m[1][2]; m[0][1] = 0;
vx[2] -= m[2][1]*vx[1];
vy[2] -= m[2][1]*vy[1];
m[2][2] -= m[2][1]*m[1][2]; m[2][1] = 0;
vx[0] /= m[0][2]; /* This is cx */
vy[0] /= m[0][2]; /* This is cy */
/*m[0][2] = 1;*/
vx[1] -= m[1][2]*vx[0]; /* This is bx */
vy[1] -= m[1][2]*vy[0]; /* This is by */
/* m[1][2] = 0; */
vx[2] -= m[2][2]*vx[0]; /* This is ax */
vy[2] -= m[2][2]*vy[0]; /* This is ay */
/* m[2][2] = 0; */
nextcp->x = from->me.x + vx[0]/3;
nextcp->y = from->me.y + vy[0]/3;
prevcp->x = nextcp->x + (vx[0]+vx[1])/3;
prevcp->y = nextcp->y + (vy[0]+vy[1])/3;
}
test = (nextcp->x-from->me.x)*(to->me.x-from->me.x) +
(nextcp->y-from->me.y)*(to->me.y-from->me.y);
ptest = (prevcp->x-to->me.x)*(from->me.x-to->me.x) +
(prevcp->y-to->me.y)*(from->me.y-to->me.y);
if ( order2 &&
(test<nmin || test>nmax || ptest<pmin || ptest>pmax))
continue;
if ( test>=nmin && test<=nmax ) {
nres.x += nextcp->x; nres.y += nextcp->y;
++nrescnt;
}
if ( test>=pmin && test<=pmax ) {
pres.x += prevcp->x; pres.y += prevcp->y;
++prescnt;
}
if ( nrescnt==1 && prescnt==1 )
break;
}
ret = 0;
if ( nrescnt>0 ) {
ret |= 1;
nextcp->x = nres.x/nrescnt;
nextcp->y = nres.y/nrescnt;
} else
*nextcp = from->nextcp;
if ( prescnt>0 ) {
ret |= 2;
prevcp->x = pres.x/prescnt;
prevcp->y = pres.y/prescnt;
} else
*prevcp = to->prevcp;
if ( order2 && ret!=3 ) {
nextcp->x = (nextcp->x + prevcp->x)/2;
nextcp->y = (nextcp->y + prevcp->y)/2;
}
if ( order2 )
*prevcp = *nextcp;
return( ret );
}
static void TestForLinear(SplinePoint *from,SplinePoint *to) {
BasePoint off, cpoff, cpoff2;
bigreal len, co, co2;
/* Did we make a line? */
off.x = to->me.x-from->me.x; off.y = to->me.y-from->me.y;
len = sqrt(off.x*off.x + off.y*off.y);
if ( len!=0 ) {
off.x /= len; off.y /= len;
cpoff.x = from->nextcp.x-from->me.x; cpoff.y = from->nextcp.y-from->me.y;
len = sqrt(cpoff.x*cpoff.x + cpoff.y*cpoff.y);
if ( len!=0 ) {
cpoff.x /= len; cpoff.y /= len;
}
cpoff2.x = to->prevcp.x-from->me.x; cpoff2.y = to->prevcp.y-from->me.y;
len = sqrt(cpoff2.x*cpoff2.x + cpoff2.y*cpoff2.y);
if ( len!=0 ) {
cpoff2.x /= len; cpoff2.y /= len;
}
co = cpoff.x*off.y - cpoff.y*off.x; co2 = cpoff2.x*off.y - cpoff2.y*off.x;
if ( co<.05 && co>-.05 && co2<.05 && co2>-.05 ) {
from->nextcp = from->me; from->nonextcp = true;
to->prevcp = to->me; to->noprevcp = true;
} else {
Spline temp;
memset(&temp,0,sizeof(temp));
temp.from = from; temp.to = to;
SplineRefigure(&temp);
if ( SplineIsLinear(&temp)) {
from->nextcp = from->me; from->nonextcp = true;
to->prevcp = to->me; to->noprevcp = true;
}
}
}
}
/* Find a spline which best approximates the list of intermediate points we */
/* are given. No attempt is made to use fixed slope angles */
/* given a set of points (x,y,t) */
/* find the bezier spline which best fits those points */
/* OK, we know the end points, so all we really need are the control points */
/* For cubics.... */
/* Pf = point from */
/* CPf = control point, from nextcp */
/* CPt = control point, to prevcp */
/* Pt = point to */
/* S(t) = Pf + 3*(CPf-Pf)*t + 3*(CPt-2*CPf+Pf)*t^2 + (Pt-3*CPt+3*CPf-Pf)*t^3 */
/* S(t) = Pf - 3*Pf*t + 3*Pf*t^2 - Pf*t^3 + Pt*t^3 + */
/* 3*(t-2*t^2+t^3)*CPf + */
/* 3*(t^2-t^3)*CPt */
/* We want to minimize Σ [S(ti)-Pi]^2 */
/* There are four variables CPf.x, CPf.y, CPt.x, CPt.y */
/* When we take the derivative of the error term above with each of these */
/* variables, we find that the two coordinates are separate. So I shall only */
/* work through the equations once, leaving off the coordinate */
/* d error/dCPf = Σ 2*3*(t-2*t^2+t^3) * [S(ti)-Pi] = 0 */
/* d error/dCPt = Σ 2*3*(t^2-t^3) * [S(ti)-Pi] = 0 */
/* For quadratics.... */
/* CP = control point, there's only one */
/* S(t) = Pf + 2*(CP-Pf)*t + (Pt-2*CP+Pf)*t^2 */
/* S(t) = Pf - 2*Pf*t + Pf*t^2 + Pt*t^2 + */
/* 2*(t-2*t^2)*CP */
/* We want to minimize Σ [S(ti)-Pi]^2 */
/* There are two variables CP.x, CP.y */
/* d error/dCP = Σ 2*2*(t-2*t^2) * [S(ti)-Pi] = 0 */
/* Σ (t-2*t^2) * [Pf - 2*Pf*t + Pf*t^2 + Pt*t^2 - Pi + */
/* 2*(t-2*t^2)*CP] = 0 */
/* CP * (Σ 2*(t-2*t^2)*(t-2*t^2)) = Σ (t-2*t^2) * [Pf - 2*Pf*t + Pf*t^2 + Pt*t^2 - Pi] */
/* Σ (t-2*t^2) * [Pf - 2*Pf*t + Pf*t^2 + Pt*t^2 - Pi] */
/* CP = ----------------------------------------------------- */
/* Σ 2*(t-2*t^2)*(t-2*t^2) */
Spline *ApproximateSplineFromPoints(SplinePoint *from, SplinePoint *to,
TPoint *mid, int cnt, int order2) {
int ret;
Spline *spline;
BasePoint nextcp, prevcp;
int i;
if ( order2 ) {
bigreal xconst, yconst, term /* Same for x and y */;
xconst = yconst = term = 0;
for ( i=0; i<cnt; ++i ) {
bigreal t = mid[i].t, t2 = t*t;
bigreal tfactor = (t-2*t2);
term += 2*tfactor*tfactor;
xconst += tfactor*(from->me.x*(1-2*t+t2) + to->me.x*t2 - mid[i].x);
yconst += tfactor*(from->me.y*(1-2*t+t2) + to->me.y*t2 - mid[i].y);
}
if ( term!=0 ) {
BasePoint cp;
cp.x = xconst/term; cp.y = yconst/term;
from->nextcp = to->prevcp = cp;
return( SplineMake2(from,to));
}
} else {
bigreal xconst[2], yconst[2], f_term[2], t_term[2] /* Same for x and y */;
bigreal tfactor[2], determinant;
xconst[0] = xconst[1] = yconst[0] = yconst[1] =
f_term[0] = f_term[1] = t_term[0] = t_term[1] = 0;
for ( i=0; i<cnt; ++i ) {
bigreal t = mid[i].t, t2 = t*t, t3=t*t2;
bigreal xc = (from->me.x*(1-3*t+3*t2-t3) + to->me.x*t3 - mid[i].x);
bigreal yc = (from->me.y*(1-3*t+3*t2-t3) + to->me.y*t3 - mid[i].y);
tfactor[0] = (t-2*t2+t3); tfactor[1]=(t2-t3);
xconst[0] += tfactor[0]*xc;
xconst[1] += tfactor[1]*xc;
yconst[0] += tfactor[0]*yc;
yconst[1] += tfactor[1]*yc;
f_term[0] += 3*tfactor[0]*tfactor[0];
f_term[1] += 3*tfactor[0]*tfactor[1];
/*t_term[0] += 3*tfactor[1]*tfactor[0];*/
t_term[1] += 3*tfactor[1]*tfactor[1];
}
t_term[0] = f_term[1];
determinant = f_term[1]*t_term[0] - f_term[0]*t_term[1];
if ( determinant!=0 ) {
to->prevcp.x = -(xconst[0]*f_term[1]-xconst[1]*f_term[0])/determinant;
to->prevcp.y = -(yconst[0]*f_term[1]-yconst[1]*f_term[0])/determinant;
if ( f_term[0]!=0 ) {
from->nextcp.x = (-xconst[0]-t_term[0]*to->prevcp.x)/f_term[0];
from->nextcp.y = (-yconst[0]-t_term[0]*to->prevcp.y)/f_term[0];
} else {
from->nextcp.x = (-xconst[1]-t_term[1]*to->prevcp.x)/f_term[1];
from->nextcp.y = (-yconst[1]-t_term[1]*to->prevcp.y)/f_term[1];
}
to->noprevcp = from->nonextcp = false;
return( SplineMake3(from,to));
}
}
if ( (spline = IsLinearApprox(from,to,mid,cnt,order2))!=NULL )
return( spline );
ret = _ApproximateSplineFromPoints(from,to,mid,cnt,&nextcp,&prevcp,order2);
if ( ret&1 ) {
from->nextcp = nextcp;
from->nonextcp = false;
} else {
from->nextcp = from->me;
from->nonextcp = true;
}
if ( ret&2 ) {
to->prevcp = prevcp;
to->noprevcp = false;
} else {
to->prevcp = to->me;
to->noprevcp = true;
}
TestForLinear(from,to);
spline = SplineMake(from,to,order2);
return( spline );
}
static bigreal ClosestSplineSolve(Spline1D *sp,bigreal sought,bigreal close_to_t) {
/* We want to find t so that spline(t) = sought */
/* find the value which is closest to close_to_t */
/* on error return closetot */
extended ts[3];
int i;
bigreal t, best, test;
_CubicSolve(sp,sought,ts);
best = 9e20; t= close_to_t;
for ( i=0; i<3; ++i ) if ( ts[i]>-.0001 && ts[i]<1.0001 ) {
if ( (test=ts[i]-close_to_t)<0 ) test = -test;
if ( test<best ) {
best = test;
t = ts[i];
}
}
return( t );
}
struct dotbounds {
BasePoint unit;
BasePoint base;
bigreal len;
bigreal min,max; /* If min<0 || max>len the spline extends beyond its endpoints */
};
static bigreal SigmaDeltas(Spline *spline,TPoint *mid, int cnt, DBounds *b, struct dotbounds *db) {
int i;
bigreal xdiff, ydiff, sum, temp, t;
SplinePoint *to = spline->to, *from = spline->from;
extended ts[2], x,y;
struct dotbounds db2;
bigreal dot;
int near_vert, near_horiz;
if ( (xdiff = to->me.x-from->me.x)<0 ) xdiff = -xdiff;
if ( (ydiff = to->me.y-from->me.y)<0 ) ydiff = -ydiff;
near_vert = ydiff>2*xdiff;
near_horiz = xdiff>2*ydiff;
sum = 0;
for ( i=0; i<cnt; ++i ) {
if ( near_vert ) {
t = ClosestSplineSolve(&spline->splines[1],mid[i].y,mid[i].t);
} else if ( near_horiz ) {
t = ClosestSplineSolve(&spline->splines[0],mid[i].x,mid[i].t);
} else {
t = (ClosestSplineSolve(&spline->splines[1],mid[i].y,mid[i].t) +
ClosestSplineSolve(&spline->splines[0],mid[i].x,mid[i].t))/2;
}
temp = mid[i].x - ( ((spline->splines[0].a*t+spline->splines[0].b)*t+spline->splines[0].c)*t + spline->splines[0].d );
sum += temp*temp;
temp = mid[i].y - ( ((spline->splines[1].a*t+spline->splines[1].b)*t+spline->splines[1].c)*t + spline->splines[1].d );
sum += temp*temp;
}
/* Ok, we've got distances from a set of points on the old spline to the */
/* new one. Let's do the reverse: find the extrema of the new and see how*/
/* close they get to the bounding box of the old */
/* And get really unhappy if the spline extends beyond the end-points */
db2.min = 0; db2.max = db->len;
SplineFindExtrema(&spline->splines[0],&ts[0],&ts[1]);
for ( i=0; i<2; ++i ) {
if ( ts[i]!=-1 ) {
x = ((spline->splines[0].a*ts[i]+spline->splines[0].b)*ts[i]+spline->splines[0].c)*ts[i] + spline->splines[0].d;
y = ((spline->splines[1].a*ts[i]+spline->splines[1].b)*ts[i]+spline->splines[1].c)*ts[i] + spline->splines[1].d;
if ( x<b->minx )
sum += (x-b->minx)*(x-b->minx);
else if ( x>b->maxx )
sum += (x-b->maxx)*(x-b->maxx);
dot = (x-db->base.x)*db->unit.x + (y-db->base.y)*db->unit.y;
if ( dot<db2.min ) db2.min = dot;
if ( dot>db2.max ) db2.max = dot;
}
}
SplineFindExtrema(&spline->splines[1],&ts[0],&ts[1]);
for ( i=0; i<2; ++i ) {
if ( ts[i]!=-1 ) {
x = ((spline->splines[0].a*ts[i]+spline->splines[0].b)*ts[i]+spline->splines[0].c)*ts[i] + spline->splines[0].d;
y = ((spline->splines[1].a*ts[i]+spline->splines[1].b)*ts[i]+spline->splines[1].c)*ts[i] + spline->splines[1].d;
if ( y<b->miny )
sum += (y-b->miny)*(y-b->miny);
else if ( y>b->maxy )
sum += (y-b->maxy)*(y-b->maxy);
dot = (x-db->base.x)*db->unit.x + (y-db->base.y)*db->unit.y;
if ( dot<db2.min ) db2.min = dot;
if ( dot>db2.max ) db2.max = dot;
}
}
/* Big penalty for going beyond the range of the desired spline */
if ( db->min==0 && db2.min<0 )
sum += 10000 + db2.min*db2.min;
else if ( db2.min<db->min )
sum += 100 + (db2.min-db->min)*(db2.min-db->min);
if ( db->max==db->len && db2.max>db->len )
sum += 10000 + (db2.max-db->max)*(db2.max-db->max);
else if ( db2.max>db->max )
sum += 100 + (db2.max-db->max)*(db2.max-db->max);
return( sum );
}
static void ApproxBounds(DBounds *b,TPoint *mid, int cnt, struct dotbounds *db) {
int i;
bigreal dot;
b->minx = b->maxx = mid[0].x;
b->miny = b->maxy = mid[0].y;
db->min = 0; db->max = db->len;
for ( i=1; i<cnt; ++i ) {
if ( mid[i].x>b->maxx ) b->maxx = mid[i].x;
if ( mid[i].x<b->minx ) b->minx = mid[i].x;
if ( mid[i].y>b->maxy ) b->maxy = mid[i].y;
if ( mid[i].y<b->miny ) b->miny = mid[i].y;
dot = (mid[i].x-db->base.x)*db->unit.x + (mid[i].y-db->base.y)*db->unit.y;
if ( dot<db->min ) db->min = dot;
if ( dot>db->max ) db->max = dot;
}
}
static int GoodCurve(SplinePoint *sp, int check_prev ) {
bigreal dx, dy, lenx, leny;
if ( sp->pointtype!=pt_curve && sp->pointtype!=pt_hvcurve )
return( false );
if ( check_prev ) {
dx = sp->me.x - sp->prevcp.x;
dy = sp->me.y - sp->prevcp.y;
} else {
dx = sp->me.x - sp->nextcp.x;
dy = sp->me.y - sp->nextcp.y;
}
/* If the cp is very close to the base point the point might as well be a corner */
if ( dx<0 ) dx = -dx;
if ( dy<0 ) dy = -dy;
if ( dx+dy<1 )
return( false );
if ( check_prev ) {
if ( sp->prev==NULL )
return( true );
lenx = sp->me.x - sp->prev->from->me.x;
leny = sp->me.y - sp->prev->from->me.y;
} else {
if ( sp->next==NULL )
return( true );
lenx = sp->me.x - sp->next->to->me.x;
leny = sp->me.y - sp->next->to->me.y;
}
if ( lenx<0 ) lenx = -lenx;
if ( leny<0 ) leny = -leny;
if ( 50*(dx+dy) < lenx+leny )
return( false );
return( true );
}
/* pf == point from (start point) */
/* Δf == slope from (cp(from) - from) */
/* pt == point to (end point, t==1) */
/* Δt == slope to (cp(to) - to) */
/* A spline from pf to pt with slope vectors rf*Δf, rt*Δt is: */
/* p(t) = pf + [ 3*rf*Δf ]*t + 3*[pt-pf+rt*Δt-2*rf*Δf] *t^2 + */
/* [2*pf-2*pt+3*rf*Δf-3*rt*Δt]*t^3 */
/* So I want */
/* d Σ (p(t(i))-p(i))^2/ d rf == 0 */
/* d Σ (p(t(i))-p(i))^2/ d rt == 0 */
/* now... */
/* d Σ (p(t(i))-p(i))^2/ d rf == 0 */
/* => Σ 3*t*Δf*(1-2*t+t^2)*
* [pf-pi+ 3*(pt-pf)*t^2 + 2*(pf-pt)*t^3] +
* 3*[t - 2*t^2 + t^3]*Δf*rf +
* 3*[t^2-t^3]*Δt*rt */
/* and... */
/* d Σ (p(t(i))-p(i))^2/ d rt == 0 */
/* => Σ 3*t^2*Δt*(1-t)*
* [pf-pi+ 3*(pt-pf)*t^2 + 2*(pf-pt)*t^3] +
* 3*[t - 2*t^2 + t^3]*Δf*rf +
* 3*[t^2-t^3]*Δt*rt */
/* Now for a long time I looked at that and saw four equations and two unknowns*/
/* That was I was trying to solve for x and y separately, and that doesn't work. */
/* There are really just two equations and each sums over both x and y components */
/* Old comment: */
/* I used to do a least squares aproach adding two more to the above set of equations */
/* which held the slopes constant. But that didn't work very well. So instead*/
/* Then I tried doing the approximation, and then forcing the control points */
/* to be in line (witht the original slopes), getting a better approximation */
/* to "t" for each data point and then calculating an error array, approximating*/
/* it, and using that to fix up the final result */
/* Then I tried checking various possible cp lengths in the desired directions*/
/* finding the best one or two, and doing a 2D binary search using that as a */
/* starting point. */
/* And sometimes a least squares approach will give us the right answer, so */
/* try that too. */
/* This still isn't as good as I'd like it... But I haven't been able to */
/* improve it further yet */
#define TRY_CNT 2
#define DECIMATION 5
Spline *ApproximateSplineFromPointsSlopes(SplinePoint *from, SplinePoint *to,
TPoint *mid, int cnt, int order2) {
BasePoint tounit, fromunit, ftunit;
bigreal flen,tlen,ftlen,dot;
Spline *spline, temp;
BasePoint nextcp;
int bettern, betterp;
bigreal offn, offp, incrn, incrp, trylen;
int nocnt = 0, totcnt;
bigreal curdiff, bestdiff[TRY_CNT];
int i,j,besti[TRY_CNT],bestj[TRY_CNT],k,l;
bigreal fdiff, tdiff, fmax, tmax, fdotft, tdotft;
DBounds b;
struct dotbounds db;
bigreal offn_, offp_, finaldiff;
bigreal pt_pf_x, pt_pf_y, determinant;
bigreal consts[2], rt_terms[2], rf_terms[2];
/* If all the selected points are at the same spot, and one of the */
/* end-points is also at that spot, then just copy the control point */
/* But our caller seems to have done that for us */
/* If the two end-points are corner points then allow the slope to vary */
/* Or if one end-point is a tangent but the point defining the tangent's */
/* slope is being removed then allow the slope to vary */
/* Except if the slope is horizontal or vertical then keep it fixed */
if ( ( !from->nonextcp && ( from->nextcp.x==from->me.x || from->nextcp.y==from->me.y)) ||
(!to->noprevcp && ( to->prevcp.x==to->me.x || to->prevcp.y==to->me.y)) )
/* Preserve the slope */;
else if ( ((from->pointtype == pt_corner && from->nonextcp) ||
(from->pointtype == pt_tangent &&
((from->nonextcp && from->noprevcp) || !from->noprevcp))) &&
((to->pointtype == pt_corner && to->noprevcp) ||
(to->pointtype == pt_tangent &&
((to->nonextcp && to->noprevcp) || !to->nonextcp))) ) {
from->pointtype = to->pointtype = pt_corner;
return( ApproximateSplineFromPoints(from,to,mid,cnt,order2) );
}
/* If we are going to honour the slopes of a quadratic spline, there is */
/* only one possibility */
if ( order2 ) {
if ( from->nonextcp )
from->nextcp = from->next->to->me;
if ( to->noprevcp )
to->prevcp = to->prev->from->me;
from->nonextcp = to->noprevcp = false;
fromunit.x = from->nextcp.x-from->me.x; fromunit.y = from->nextcp.y-from->me.y;
tounit.x = to->prevcp.x-to->me.x; tounit.y = to->prevcp.y-to->me.y;
if ( !IntersectLines(&nextcp,&from->nextcp,&from->me,&to->prevcp,&to->me) ||
(nextcp.x-from->me.x)*fromunit.x + (nextcp.y-from->me.y)*fromunit.y < 0 ||
(nextcp.x-to->me.x)*tounit.x + (nextcp.y-to->me.y)*tounit.y < 0 ) {
/* If the slopes don't intersect then use a line */
/* (or if the intersection is patently absurd) */
from->nonextcp = to->noprevcp = true;
from->nextcp = from->me;
to->prevcp = to->me;
TestForLinear(from,to);
} else {
from->nextcp = to->prevcp = nextcp;
from->nonextcp = to->noprevcp = false;
}
return( SplineMake2(from,to));
}
/* From here down we are only working with cubic splines */
if ( cnt==0 ) {
/* Just use what we've got, no data to improve it */
/* But we do sometimes get some cps which are too big */
bigreal len = sqrt((to->me.x-from->me.x)*(to->me.x-from->me.x) + (to->me.y-from->me.y)*(to->me.y-from->me.y));
if ( len==0 ) {
from->nonextcp = to->noprevcp = true;
from->nextcp = from->me;
to->prevcp = to->me;
} else {
BasePoint noff, poff;
bigreal nlen, plen;
noff.x = from->nextcp.x-from->me.x; noff.y = from->nextcp.y-from->me.y;
poff.x = to->me.x-to->prevcp.x; poff.y = to->me.y-to->prevcp.y;
nlen = sqrt(noff.x*noff.x + noff.y+noff.y);
plen = sqrt(poff.x*poff.x + poff.y+poff.y);
if ( nlen>len/3 ) {
noff.x *= len/3/nlen; noff.y *= len/3/nlen;
from->nextcp.x = from->me.x + noff.x;
from->nextcp.y = from->me.y + noff.y;
}
if ( plen>len/3 ) {
poff.x *= len/3/plen; poff.y *= len/3/plen;
to->prevcp.x = to->me.x + poff.x;
to->prevcp.y = to->me.y + poff.y;
}
}