1 #include <u.h>
2 #include <libc.h>
3 #include "map.h"
4
5 /*
6 * conformal map of earth onto tetrahedron
7 * the stages of mapping are
8 * (a) stereo projection of tetrahedral face onto
9 * isosceles curvilinear triangle with 3 120-degree
10 * angles and one straight side
11 * (b) map of this triangle onto half plane cut along
12 * 3 rays from the roots of unity to infinity
13 * formula (z^4+2*3^.5*z^2-1)/(z^4-2*3^.5*z^2-1)
14 * (c) do 3 times for each sector of plane:
15 * map of |arg z|<=pi/6, cut along z>1 into
16 * triangle |arg z|<=pi/6, Re z<=const,
17 * with upper side of cut going into upper half of
18 * of vertical side of triangle and lowere into lower
19 * formula int from 0 to z dz/sqrt(1-z^3)
20 *
21 * int from u to 1 3^.25*du/sqrt(1-u^3) =
22 F(acos((rt3-1+u)/(rt3+1-u)),sqrt(1/2+rt3/4))
23 * int from 1 to u 3^.25*du/sqrt(u^3-1) =
24 * F(acos((rt3+1-u)/(rt3-1+u)),sqrt(1/2-rt3/4))
25 * this latter formula extends analytically down to
26 * u=0 and is the basis of this routine, with the
27 * argument of complex elliptic integral elco2
28 * being tan(acos...)
29 * the formula F(pi-x,k) = 2*F(pi/2,k)-F(x,k) is
30 * used to cross over into the region where Re(acos...)>pi/2
31 * f0 and fpi are suitably scaled complete integrals
32 */
33
34 #define TFUZZ 0.00001
35
36 static struct place tpole[4]; /* point of tangency of tetrahedron face*/
37 static double tpoleinit[4][2] = {
38 1., 0.,
39 1., 180.,
40 -1., 90.,
41 -1., -90.
42 };
43 static struct tproj {
44 double tlat,tlon; /* center of stereo projection*/
45 double ttwist; /* rotatn before stereo*/
46 double trot; /*rotate after projection*/
47 struct place projpl; /*same as tlat,tlon*/
48 struct coord projtw; /*same as ttwist*/
49 struct coord postrot; /*same as trot*/
50 } tproj[4][4] = {
51 {/*00*/ {0.},
52 /*01*/ {90., 0., 90., -90.},
53 /*02*/ {0., 45., -45., 150.},
54 /*03*/ {0., -45., -135., 30.}
55 },
56 {/*10*/ {90., 0., -90., 90.},
57 /*11*/ {0.},
58 /*12*/ {0., 135., -135., -150.},
59 /*13*/ {0., -135., -45., -30.}
60 },
61 {/*20*/ {0., 45., 135., -30.},
62 /*21*/ {0., 135., 45., -150.},
63 /*22*/ {0.},
64 /*23*/ {-90., 0., 180., 90.}
65 },
66 {/*30*/ {0., -45., 45., -150.},
67 /*31*/ {0., -135., 135., -30.},
68 /*32*/ {-90., 0., 0., 90.},
69 /*33*/ {0.}
70 }};
71 static double tx[4] = { /*where to move facet after final rotation*/
72 0., 0., -1., 1. /*-1,1 to be sqrt(3)*/
73 };
74 static double ty[4] = {
75 0., 2., -1., -1.
76 };
77 static double root3;
78 static double rt3inv;
79 static double two_rt3;
80 static double tkc,tk,tcon;
81 static double f0r,f0i,fpir,fpii;
82
83 static void
twhichp(struct place * g,int * p,int * q)84 twhichp(struct place *g, int *p, int *q)
85 {
86 int i,j,k;
87 double cosdist[4];
88 struct place *tp;
89 for(i=0;i<4;i++) {
90 tp = &tpole[i];
91 cosdist[i] = g->nlat.s*tp->nlat.s +
92 g->nlat.c*tp->nlat.c*(
93 g->wlon.s*tp->wlon.s +
94 g->wlon.c*tp->wlon.c);
95 }
96 j = 0;
97 for(i=1;i<4;i++)
98 if(cosdist[i] > cosdist[j])
99 j = i;
100 *p = j;
101 k = j==0?1:0;
102 for(i=0;i<4;i++)
103 if(i!=j&&cosdist[i]>cosdist[k])
104 k = i;
105 *q = k;
106 }
107
108 int
Xtetra(struct place * place,double * x,double * y)109 Xtetra(struct place *place, double *x, double *y)
110 {
111 int i,j;
112 struct place pl;
113 register struct tproj *tpp;
114 double vr, vi;
115 double br, bi;
116 double zr,zi,z2r,z2i,z4r,z4i,sr,si,tr,ti;
117 twhichp(place,&i,&j);
118 copyplace(place,&pl);
119 norm(&pl,&tproj[i][j].projpl,&tproj[i][j].projtw);
120 Xstereographic(&pl,&vr,&vi);
121 zr = vr/2;
122 zi = vi/2;
123 if(zr<=TFUZZ)
124 zr = TFUZZ;
125 csq(zr,zi,&z2r,&z2i);
126 csq(z2r,z2i,&z4r,&z4i);
127 z2r *= two_rt3;
128 z2i *= two_rt3;
129 cdiv(z4r+z2r-1,z4i+z2i,z4r-z2r-1,z4i-z2i,&sr,&si);
130 csqrt(sr-1,si,&tr,&ti);
131 cdiv(tcon*tr,tcon*ti,root3+1-sr,-si,&br,&bi);
132 if(br<0) {
133 br = -br;
134 bi = -bi;
135 if(!elco2(br,bi,tk,1.,1.,&vr,&vi))
136 return 0;
137 vr = fpir - vr;
138 vi = fpii - vi;
139 } else
140 if(!elco2(br,bi,tk,1.,1.,&vr,&vi))
141 return 0;
142 if(si>=0) {
143 tr = f0r - vi;
144 ti = f0i + vr;
145 } else {
146 tr = f0r + vi;
147 ti = f0i - vr;
148 }
149 tpp = &tproj[i][j];
150 *x = tr*tpp->postrot.c +
151 ti*tpp->postrot.s + tx[i];
152 *y = ti*tpp->postrot.c -
153 tr*tpp->postrot.s + ty[i];
154 return(1);
155 }
156
157 int
tetracut(struct place * g,struct place * og,double * cutlon)158 tetracut(struct place *g, struct place *og, double *cutlon)
159 {
160 int i,j,k;
161 if((g->nlat.s<=-rt3inv&&og->nlat.s<=-rt3inv) &&
162 (ckcut(g,og,*cutlon=0.)==2||ckcut(g,og,*cutlon=PI)==2))
163 return(2);
164 twhichp(g,&i,&k);
165 twhichp(og,&j,&k);
166 if(i==j||i==0||j==0)
167 return(1);
168 return(0);
169 }
170
171 proj
tetra(void)172 tetra(void)
173 {
174 register i;
175 int j;
176 register struct place *tp;
177 register struct tproj *tpp;
178 double t;
179 root3 = sqrt(3.);
180 rt3inv = 1/root3;
181 two_rt3 = 2*root3;
182 tkc = sqrt(.5-.25*root3);
183 tk = sqrt(.5+.25*root3);
184 tcon = 2*sqrt(root3);
185 elco2(tcon/(root3-1),0.,tkc,1.,1.,&f0r,&f0i);
186 elco2(1.e15,0.,tk,1.,1.,&fpir,&fpii);
187 fpir *= 2;
188 fpii *= 2;
189 for(i=0;i<4;i++) {
190 tx[i] *= f0r*root3;
191 ty[i] *= f0r;
192 tp = &tpole[i];
193 t = tp->nlat.s = tpoleinit[i][0]/root3;
194 tp->nlat.c = sqrt(1 - t*t);
195 tp->nlat.l = atan2(tp->nlat.s,tp->nlat.c);
196 deg2rad(tpoleinit[i][1],&tp->wlon);
197 for(j=0;j<4;j++) {
198 tpp = &tproj[i][j];
199 latlon(tpp->tlat,tpp->tlon,&tpp->projpl);
200 deg2rad(tpp->ttwist,&tpp->projtw);
201 deg2rad(tpp->trot,&tpp->postrot);
202 }
203 }
204 return(Xtetra);
205 }
206
207