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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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