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