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PROJ(3X) PROJ(3X) NAME orient, normalize - map projections SYNOPSIS orient(lat, lon, rot) double lat, lon, rot; normalize(p) struct place *p; DESCRIPTION Users of map(7) may skip to the description of `Projection generators' below. The functions orient and normalize plus a collection of map projection generators are loaded by option -lmap of ld(1). Most of them calculate maps for a spherical earth. Each map projection is available in one standard form, into which data must be normalized for transverse or nonpolar projections. Each standard projection is displayed with the Prime Meridian (longi- tude 0) being a straight vertical line, along which North is up. The orientation of nonstandard projections is specified by orient. Imagine a transparent gridded sphere around the globe. First turn the overlay about the North Pole so that the Prime Meridian (longitude 0) of the overlay coincides with meridian lon on the globe. Then tilt the North Pole of the overlay along its Prime Meridian to latitude lat on the globe. Finally again turn the overlay about its `North Pole' so that its Prime Meridian coincides with the previous position of (the over- lay's) meridian rot. Project the desired map in the standard form ap- propriate to the overlay, but presenting information from the underly- ing globe. It is not useful to use orient without using normalize. Normalize converts latitude-longitude coordinates on the globe to coor- dinates on the overlaid grid. The coordinates and their sines and cosines are input to normalize in a place structure. Transformed coor- dinates and their sines and cosines are returned in the same structure. struct place { double radianlat, sinlat, coslat; double radianlon, sinlon, coslon; }; The projection generators return a pointer to a function that converts normalized coordinates to x-y coordinates for the desired map, or 0 if the required projection is not available. The returned function is ex- emplified by proj in this example: struct place pt; int (*proj)() = mercator(); double x, y; orient(45.0, 30.0, 180.0); /* set coordinate rotation */ . . . /* fill in the pt structure */ normalize(&pt); /* rotate coordinates */ if((*proj)(&pt, &x, &y) > 0) /* project onto x,y plane */ plot(x, y); The projection function (*proj)() returns 1 for a good point, 0 for a point on a wrong sheet (e.g. the back of the world in a perspective projection), and -1 for a point that is deemed unplottable (e.g. points near the poles on a Mercator projection). Scaling may be determined from the x-y coordinates of selected points. Latitudes and longitudes are measured in degrees for ease of specifica- tion for orient and the projection generators but in radians for ease of calculation for normalize and proj. In either case latitude is mea- sured positive north of the equator, and longitude positive west of Greenwich. Radian longitude should be limited to the range -pi <= lon < pi. Projection generators Equatorial projections centered on the Prime Meridian (longitude 0). Parallels are straight horizontal lines. mercator() equally spaced straight meridians, conformal, straight compass courses sinusoidal() equally spaced parallels, equal-area, same as bonne(0) cylequalarea(lat0) equally spaced straight meridians, equal- area, true scale on lat0 cylindrical() central projection on tangent cylinder rectangular(lat0) equally spaced parallels, equally spaced straight meridians, true scale on lat0 gall(lat0) parallels spaced stereographically on prime meridian, equally spaced straight meridians, true scale on lat0 mollweide() (homalographic) equal-area, hemisphere is a circle Azimuthal projections centered on the North Pole. Parallels are con- centric circles. Meridians are equally spaced radial lines. azequidistant() equally spaced parallels, true distances from pole azequalarea() equal-area gnomonic() central projection on tangent plane, straight great circles perspective(dist) viewed along earth's axis dist earth radii from center of earth orthographic() viewed from infinity stereographic() conformal, projected from opposite pole laue() radius = tan(2xcolatitude), used in xray crystallography fisheye(n) stereographic seen from just inside medium with re- fractive index n newyorker(r) radius = log(colatitude/r): extreme `fisheye' view from pedestal of radius r degrees Polar conic projections symmetric about the Prime Meridian. Parallels are segments of concentric circles. Except in the Bonne projection, meridians are equally spaced radial lines orthogonal to the parallels. conic(lat0) central projection on cone tangent at lat0 simpleconic(lat0,lat1) equally spaced parallels, true scale on lat0 and lat1 lambert(lat0,lat1) conformal, true scale on lat0 and lat1 albers(lat0,lat1) equal-area, true scale on lat0 and lat1 bonne(lat0) equally spaced parallels, equal-area, parallel lat0 developed from tangent cone Projections with bilateral symmetry about the Prime Meridian and the equator. polyconic() parallels developed from tangent cones, equally spaced along Prime Meridian aitoff() equal-area projection of globe onto 2-to-1 ellipse, based on azequalarea lagrange() conformal, maps whole sphere into a circle bicentric(lon0) points plotted at true azimuth from two centers on the equator at longitudes +-lon0, great circles are straight lines (a stretched gnomonic projection) elliptic(lon0) points are plotted at true distance from two cen- ters on the equator at longitudes +-lon0 globular() hemisphere is circle, circular arc meridians equally spaced on equator, circular arc parallels equally spaced on 0- and 90-degree meridians vandergrinten() sphere is circle, meridians as in globular, cir- cular arc parallels resemble mercator Doubly periodic conformal projections. guyou() W and E hemispheres are square square() world is square with Poles at diagonally opposite cor- ners tetra() map on tetrahedron with edge tangent to Prime Meridian at S Pole, unfolded into equilateral triangle hex() world is hexagon centered on N Pole, N and S hemispheres are equilateral triangles Miscellaneous projections. harrison(dist,angle) oblique perspective from above the North Pole, dist earth radii from center of earth, looking along the Date Line angle degrees off vertical trapezoidal(lat0,lat1) equally spaced parallels, straight merid- ians equally spaced along parallels, true scale at lat0 and lat1 on Prime Meridian Retroazimuthal projections. At every point the angle between vertical and a straight line to `Mecca', latitude lat0 on the prime meridian, is the true bearing of Mecca. mecca(lat0) equally spaced vertical meridians homing(lat0) distances to `Mecca' are true Maps based on the spheroid. Of geodetic quality, these projections do not make sense for tilted orientations. For descriptions, see corre- sponding maps above. sp_mercator() sp_albers(lat0,lat1) SEE ALSO map(7), map(5), plot(3) BUGS Only one projection and one orientation can be active at a time. The west-longitude-positive convention betrays Yankee chauvinism. bowell PROJ(3X)