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1.1 ! root 1: .TH PROJ 3X bowell ! 2: .CT 2 graphics math ! 3: .br ! 4: .SH NAME ! 5: orient, normalize \- map projections ! 6: .SH SYNOPSIS ! 7: .B orient(lat, lon, rot) ! 8: .br ! 9: .B double lat, lon, rot; ! 10: .PP ! 11: .B normalize(p) ! 12: .br ! 13: .B struct place *p; ! 14: .SH DESCRIPTION ! 15: Users of ! 16: .IR map (7) ! 17: may skip to the description of `Projection generators' ! 18: below. ! 19: .PP ! 20: The functions ! 21: .I orient ! 22: and ! 23: .I normalize ! 24: plus a collection of map projection generators ! 25: are loaded by ! 26: option ! 27: .BR -lmap ! 28: of ! 29: .IR ld (1). ! 30: Most of them ! 31: calculate maps for a spherical earth. ! 32: Each map projection is available in one standard ! 33: form, into which data must be normalized ! 34: for transverse ! 35: or nonpolar projections. ! 36: .PP ! 37: Each standard projection is displayed with the Prime ! 38: Meridian (longitude 0) being a straight vertical line, along which North ! 39: is up. ! 40: The orientation of nonstandard projections is specified by ! 41: .I orient. ! 42: Imagine a transparent gridded sphere around the globe. ! 43: First turn the overlay about the North Pole ! 44: so that the Prime Meridian (longitude 0) ! 45: of the overlay coincides with meridian ! 46: .I lon ! 47: on the globe. ! 48: Then tilt the North Pole of the ! 49: overlay along its Prime Meridian to latitude ! 50: .I lat ! 51: on the globe. ! 52: Finally again turn the ! 53: overlay about its `North Pole' so ! 54: that its Prime Meridian coincides with the previous position ! 55: of (the overlay's) meridian ! 56: .I rot. ! 57: Project the desired map in ! 58: the standard form appropriate to the overlay, but presenting ! 59: information from the underlying globe. ! 60: It is not useful to use ! 61: .I orient ! 62: without using ! 63: .IR normalize . ! 64: .PP ! 65: .I Normalize ! 66: converts latitude-longitude coordinates on the globe ! 67: to coordinates on the overlaid grid. ! 68: The coordinates and their sines and cosines ! 69: are input to ! 70: .I normalize ! 71: in a ! 72: .B place ! 73: structure. ! 74: Transformed coordinates and their sines and cosines ! 75: are returned in the same structure. ! 76: .PP ! 77: .EX ! 78: .nr xx \w'12345678' ! 79: .ta \n(xxu +\n(xxu +\n(xxu +\n(xxu +\n(xxu +\n(xxu ! 80: struct place { ! 81: double radianlat, sinlat, coslat; ! 82: double radianlon, sinlon, coslon; ! 83: }; ! 84: .EE ! 85: .PP ! 86: The projection generators ! 87: return a pointer to a function that converts normalized coordinates ! 88: to ! 89: .I x-y ! 90: coordinates for the desired map, or ! 91: 0 if the required projection ! 92: is not available. ! 93: The returned function is exemplified by ! 94: .I proj ! 95: in this example: ! 96: .PP ! 97: .EX ! 98: .ta \n(xxu +\n(xxu +\n(xxu +\n(xxu +\n(xxu +\n(xxu ! 99: struct place pt; ! 100: int (*proj)() = mercator(); ! 101: double x, y; ! 102: .EE ! 103: .PP ! 104: .EX ! 105: orient(45.0, 30.0, 180.0); /* set coordinate rotation */ ! 106: .EE ! 107: .PP ! 108: .EX ! 109: . . . /* fill in the pt structure */ ! 110: normalize(&pt); /* rotate coordinates */ ! 111: if((*proj)(&pt, &x, &y) > 0) /* project onto x,y plane */ ! 112: plot(x, y); ! 113: .EE ! 114: .PP ! 115: The projection function ! 116: .B (*proj)() ! 117: returns 1 for a good point, ! 118: 0 for a point on a wrong ! 119: sheet (e.g. the back of the world in a perspective ! 120: projection), and \-1 for a point that is deemed ! 121: unplottable (e.g. points near the poles on a Mercator projection). ! 122: .PP ! 123: Scaling may be determined from the ! 124: .I x-y ! 125: coordinates of ! 126: selected points. ! 127: Latitudes and longitudes are measured in degrees for ! 128: ease of specification for ! 129: .I orient ! 130: and the projection generators ! 131: but in radians for ease of calculation ! 132: for ! 133: .I normalize ! 134: and ! 135: .I proj. ! 136: In either case ! 137: latitude is measured positive north of the equator, ! 138: and longitude positive west of Greenwich. ! 139: Radian longitude should be limited to the range ! 140: .if t .I \-\(*p\(<=lon<\(*p. ! 141: .if n .I -pi <= lon < pi. ! 142: .SS Projection generators ! 143: Equatorial projections centered on the Prime Meridian ! 144: (longitude 0). ! 145: Parallels are straight horizontal lines. ! 146: .br ! 147: .ns ! 148: .IP ! 149: .B mercator() ! 150: equally spaced straight meridians, conformal, ! 151: straight compass courses ! 152: .br ! 153: .B sinusoidal() ! 154: equally spaced parallels, ! 155: equal-area, same as ! 156: .I bonne(0) ! 157: .br ! 158: .B cylequalarea(lat0) ! 159: equally spaced straight meridians, equal-area, ! 160: true scale on ! 161: .I lat0 ! 162: .br ! 163: .B cylindrical() ! 164: central projection on tangent cylinder ! 165: .br ! 166: .B rectangular(lat0) ! 167: equally spaced parallels, equally spaced straight meridians, true scale on ! 168: .I lat0 ! 169: .br ! 170: .B gall(lat0) ! 171: parallels spaced stereographically on prime meridian, equally spaced straight ! 172: meridians, true scale on ! 173: .I lat0 ! 174: .br ! 175: .B mollweide() ! 176: (homalographic) equal-area, hemisphere is a circle ! 177: .PP ! 178: Azimuthal projections centered on the North Pole. ! 179: Parallels are concentric circles. ! 180: Meridians are equally spaced radial lines. ! 181: .br ! 182: .ns ! 183: .IP ! 184: .B azequidistant() ! 185: equally spaced parallels, ! 186: true distances from pole ! 187: .br ! 188: .B azequalarea() ! 189: equal-area ! 190: .br ! 191: .B gnomonic() ! 192: central projection on tangent plane, ! 193: straight great circles ! 194: .br ! 195: .B perspective(dist) ! 196: viewed along earth's axis ! 197: .I dist ! 198: earth radii from center of earth ! 199: .br ! 200: .B orthographic() ! 201: viewed from infinity ! 202: .br ! 203: .B stereographic() ! 204: conformal, projected from opposite pole ! 205: .br ! 206: .B laue() ! 207: .IR radius " = tan(2\(mu" colatitude ), ! 208: used in xray crystallography ! 209: .br ! 210: .B fisheye(n) ! 211: stereographic seen from just inside medium with refractive index n ! 212: .br ! 213: .B newyorker(r) ! 214: .IR radius " = log(" colatitude / r ): ! 215: extreme `fisheye' view from pedestal of radius ! 216: .I r ! 217: degrees ! 218: .PP ! 219: Polar conic projections symmetric about the Prime Meridian. ! 220: Parallels are segments of concentric circles. ! 221: Except in the Bonne projection, ! 222: meridians are equally spaced radial ! 223: lines orthogonal to the parallels. ! 224: .br ! 225: .ns ! 226: .IP ! 227: .B conic(lat0) ! 228: central projection on cone tangent at ! 229: .I lat0 ! 230: .br ! 231: .B simpleconic(lat0,lat1) ! 232: equally spaced parallels, true scale on ! 233: .I lat0 ! 234: and ! 235: .I lat1 ! 236: .br ! 237: .B lambert(lat0,lat1) ! 238: conformal, true scale on ! 239: .I lat0 ! 240: and ! 241: .I lat1 ! 242: .br ! 243: .B albers(lat0,lat1) ! 244: equal-area, true scale on ! 245: .I lat0 ! 246: and ! 247: .I lat1 ! 248: .br ! 249: .B bonne(lat0) ! 250: equally spaced parallels, equal-area, ! 251: parallel ! 252: .I lat0 ! 253: developed from tangent cone ! 254: .PP ! 255: Projections with bilateral symmetry about ! 256: the Prime Meridian ! 257: and the equator. ! 258: .br ! 259: .ns ! 260: .IP ! 261: .B polyconic() ! 262: parallels developed from tangent cones, ! 263: equally spaced along Prime Meridian ! 264: .br ! 265: .B aitoff() ! 266: equal-area projection of globe onto 2-to-1 ! 267: ellipse, based on ! 268: .I azequalarea ! 269: .br ! 270: .B lagrange() ! 271: conformal, maps whole sphere into a circle ! 272: .br ! 273: .B bicentric(lon0) ! 274: points plotted at true azimuth from two ! 275: centers on the equator at longitudes ! 276: .I \(+-lon0, ! 277: great circles are straight lines ! 278: (a stretched gnomonic projection) ! 279: .br ! 280: .B elliptic(lon0) ! 281: points are plotted at true distance from ! 282: two centers on the equator at longitudes ! 283: .I \(+-lon0 ! 284: .br ! 285: .B globular() ! 286: hemisphere is circle, ! 287: circular arc meridians equally spaced on equator, ! 288: circular arc parallels equally spaced on 0- and 90-degree meridians ! 289: .br ! 290: .B vandergrinten() ! 291: sphere is circle, ! 292: meridians as in ! 293: .I globular, ! 294: circular arc parallels resemble ! 295: .I mercator ! 296: .PP ! 297: Doubly periodic conformal projections. ! 298: .br ! 299: .ns ! 300: .IP ! 301: .B guyou() ! 302: W and E hemispheres are square ! 303: .br ! 304: .B square() ! 305: world is square with Poles ! 306: at diagonally opposite corners ! 307: .br ! 308: .B tetra() ! 309: map on tetrahedron with edge ! 310: tangent to Prime Meridian at S Pole, ! 311: unfolded into equilateral triangle ! 312: .br ! 313: .B hex() ! 314: world is hexagon centered ! 315: on N Pole, N and S hemispheres are equilateral ! 316: triangles ! 317: .PP ! 318: Miscellaneous projections. ! 319: .br ! 320: .ns ! 321: .IP ! 322: .B harrison(dist,angle) ! 323: oblique perspective from above the North Pole, ! 324: .I dist ! 325: earth radii from center of earth, looking ! 326: along the Date Line ! 327: .I angle ! 328: degrees off vertical ! 329: .br ! 330: .B trapezoidal(lat0,lat1) ! 331: equally spaced parallels, ! 332: straight meridians equally spaced along parallels, ! 333: true scale at ! 334: .I lat0 ! 335: and ! 336: .I lat1 ! 337: on Prime Meridian ! 338: .PP ! 339: Retroazimuthal projections. ! 340: At every point the angle between vertical and a straight line to ! 341: `Mecca', latitude ! 342: .I lat0 ! 343: on the prime meridian, ! 344: is the true bearing of Mecca. ! 345: .br ! 346: .ns ! 347: .IP ! 348: .B mecca(lat0) ! 349: equally spaced vertical meridians ! 350: .br ! 351: .B homing(lat0) ! 352: distances to `Mecca' are true ! 353: .PP ! 354: Maps based on the spheroid. ! 355: Of geodetic quality, these projections do not make sense ! 356: for tilted orientations. ! 357: For descriptions, see corresponding maps above. ! 358: .br ! 359: .ns ! 360: .IP ! 361: .B sp_mercator() ! 362: .br ! 363: .B sp_albers(lat0,lat1) ! 364: .SH "SEE ALSO ! 365: .IR map (7), ! 366: .IR map (5), ! 367: .IR plot (3) ! 368: .SH BUGS ! 369: Only one projection and one orientation can be active at a time. ! 370: .br ! 371: The west-longitude-positive convention ! 372: betrays Yankee chauvinism.
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