harvey/sys/src/cmd/map/libmap/tetra.c

#include <u.h>
#include <libc.h>
#include "map.h"

/*
 *	conformal map of earth onto tetrahedron
 *	the stages of mapping are
 *	(a) stereo projection of tetrahedral face onto
 *	    isosceles curvilinear triangle with 3 120-degree
 *	    angles and one straight side
 *	(b) map of this triangle onto half plane cut along
 *	    3 rays from the roots of unity to infinity
 *		formula (z^4+2*3^.5*z^2-1)/(z^4-2*3^.5*z^2-1)
 *	(c) do 3 times for  each sector of plane:
 *	    map of |arg z|<=pi/6, cut along z>1 into
 *	    triangle |arg z|<=pi/6, Re z<=const,
 *	    with upper side of cut going into upper half of
 *	    of vertical side of triangle and lowere into lower
 *		formula int from 0 to z dz/sqrt(1-z^3)
 *
 *	int from u to 1 3^.25*du/sqrt(1-u^3) =
		F(acos((rt3-1+u)/(rt3+1-u)),sqrt(1/2+rt3/4))
 *	int from 1 to u 3^.25*du/sqrt(u^3-1) =
 *		F(acos((rt3+1-u)/(rt3-1+u)),sqrt(1/2-rt3/4))
 *	this latter formula extends analytically down to
 *	u=0 and is the basis of this routine, with the
 *	argument of complex elliptic integral elco2
 *	being tan(acos...)
 *	the formula F(pi-x,k) = 2*F(pi/2,k)-F(x,k) is
 *	used to cross over into the region where Re(acos...)>pi/2
 *		f0 and fpi are suitably scaled complete integrals
*/

#define TFUZZ 0.00001

static struct place tpole[4];	/* point of tangency of tetrahedron face*/
static double tpoleinit[4][2] = {
	1.,	0.,
	1.,	180.,
	-1.,	90.,
	-1.,	-90.
};
static struct tproj {
	double tlat,tlon;	/* center of stereo projection*/
	double ttwist;		/* rotatn before stereo*/
	double trot;		/*rotate after projection*/
	struct place projpl;	/*same as tlat,tlon*/
	struct coord projtw;	/*same as ttwist*/
	struct coord postrot;	/*same as trot*/
} tproj[4][4] = {
{/*00*/	{0.},
 /*01*/	{90.,	0.,	90.,	-90.},
 /*02*/	{0.,	45.,	-45.,	150.},
 /*03*/	{0.,	-45.,	-135.,	30.}
},
{/*10*/	{90.,	0.,	-90.,	90.},
 /*11*/ {0.},
 /*12*/ {0.,	135.,	-135.,	-150.},
 /*13*/	{0.,	-135.,	-45.,	-30.}
},
{/*20*/	{0.,	45.,	135.,	-30.},
 /*21*/	{0.,	135.,	45.,	-150.},
 /*22*/	{0.},
 /*23*/	{-90.,	0.,	180.,	90.}
},
{/*30*/	{0.,	-45.,	45.,	-150.},
 /*31*/ {0.,	-135.,	135.,	-30.},
 /*32*/	{-90.,	0.,	0.,	90.},
 /*33*/ {0.} 
}};
static double tx[4] = {	/*where to move facet after final rotation*/
	0.,	0.,	-1.,	1.	/*-1,1 to be sqrt(3)*/
};
static double ty[4] = {
	0.,	2.,	-1.,	-1.
};
static double root3;
static double rt3inv;
static double two_rt3;
static double tkc,tk,tcon;
static double f0r,f0i,fpir,fpii;

static void
twhichp(struct place *g, int *p, int *q)
{
	int i,j,k;
	double cosdist[4];
	struct place *tp;
	for(i=0;i<4;i++) {
		tp = &tpole[i];
		cosdist[i] = g->nlat.s*tp->nlat.s +
			  g->nlat.c*tp->nlat.c*(
			  g->wlon.s*tp->wlon.s +
			  g->wlon.c*tp->wlon.c);
	}
	j = 0;
	for(i=1;i<4;i++)
		if(cosdist[i] > cosdist[j])
			j = i;
	*p = j;
	k = j==0?1:0;
	for(i=0;i<4;i++)
		if(i!=j&&cosdist[i]>cosdist[k])
			k = i;
	*q = k;
}

int
Xtetra(struct place *place, double *x, double *y)
{
	int i,j;
	struct place pl;
	register struct tproj *tpp;
	double vr, vi;
	double br, bi;
	double zr,zi,z2r,z2i,z4r,z4i,sr,si,tr,ti;
	twhichp(place,&i,&j);
	copyplace(place,&pl);
	norm(&pl,&tproj[i][j].projpl,&tproj[i][j].projtw);
	Xstereographic(&pl,&vr,&vi);
	zr = vr/2;
	zi = vi/2;
	if(zr<=TFUZZ)
		zr = TFUZZ;
	csq(zr,zi,&z2r,&z2i);
	csq(z2r,z2i,&z4r,&z4i);
	z2r *= two_rt3;
	z2i *= two_rt3;
	cdiv(z4r+z2r-1,z4i+z2i,z4r-z2r-1,z4i-z2i,&sr,&si);
	csqrt(sr-1,si,&tr,&ti);
	cdiv(tcon*tr,tcon*ti,root3+1-sr,-si,&br,&bi);
	if(br<0) {
		br = -br;
		bi = -bi;
		if(!elco2(br,bi,tk,1.,1.,&vr,&vi))
			return 0;
		vr = fpir - vr;
		vi = fpii - vi;
	} else 
		if(!elco2(br,bi,tk,1.,1.,&vr,&vi))
			return 0;
	if(si>=0) {
		tr = f0r - vi;
		ti = f0i + vr;
	} else {
		tr = f0r + vi;
		ti = f0i - vr;
	}
	tpp = &tproj[i][j];
	*x = tr*tpp->postrot.c +
	     ti*tpp->postrot.s + tx[i];
	*y = ti*tpp->postrot.c -
	     tr*tpp->postrot.s + ty[i];
	return(1);
}

int
tetracut(struct place *g, struct place *og, double *cutlon)
{
	int i,j,k;
	if((g->nlat.s<=-rt3inv&&og->nlat.s<=-rt3inv) && 
	   (ckcut(g,og,*cutlon=0.)==2||ckcut(g,og,*cutlon=PI)==2))
		return(2);
	twhichp(g,&i,&k);
	twhichp(og,&j,&k);
	if(i==j||i==0||j==0)
		return(1);
	return(0);
}

proj
tetra(void)
{
	register i;
	int j;
	register struct place *tp;
	register struct tproj *tpp;
	double t;
	root3 = sqrt(3.);
	rt3inv = 1/root3;
	two_rt3 = 2*root3;
	tkc = sqrt(.5-.25*root3);
	tk = sqrt(.5+.25*root3);
	tcon = 2*sqrt(root3);
	elco2(tcon/(root3-1),0.,tkc,1.,1.,&f0r,&f0i);
	elco2(1.e15,0.,tk,1.,1.,&fpir,&fpii);
	fpir *= 2;
	fpii *= 2;
	for(i=0;i<4;i++) {
		tx[i] *= f0r*root3;
		ty[i] *= f0r;
		tp = &tpole[i];
		t = tp->nlat.s = tpoleinit[i][0]/root3;
		tp->nlat.c = sqrt(1 - t*t);
		tp->nlat.l = atan2(tp->nlat.s,tp->nlat.c);
		deg2rad(tpoleinit[i][1],&tp->wlon);
		for(j=0;j<4;j++) {
			tpp = &tproj[i][j];
			latlon(tpp->tlat,tpp->tlon,&tpp->projpl);
			deg2rad(tpp->ttwist,&tpp->projtw);
			deg2rad(tpp->trot,&tpp->postrot);
		}
	}
	return(Xtetra);
}