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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 float 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: float radianlat, sinlat, coslat;
82: float 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: float 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(r)
211: .IR radius " = log(" colatitude / r ):
212: .I New Yorker
213: map from viewing pedestal of radius
214: .I r
215: degrees
216: .PP
217: Polar conic projections symmetric about the Prime Meridian.
218: Parallels are segments of concentric circles.
219: Except in the Bonne projection,
220: meridians are equally spaced radial
221: lines orthogonal to the parallels.
222: .br
223: .ns
224: .IP
225: .B conic(lat0)
226: central projection on cone tangent at
227: .I lat0
228: .br
229: .B simpleconic(lat0,lat1)
230: equally spaced parallels, true scale on
231: .I lat0
232: and
233: .I lat1
234: .br
235: .B lambert(lat0,lat1)
236: conformal, true scale on
237: .I lat0
238: and
239: .I lat1
240: .br
241: .B albers(lat0,lat1)
242: equal-area, true scale on
243: .I lat0
244: and
245: .I lat1
246: .br
247: .B bonne(lat0)
248: equally spaced parallels, equal-area,
249: parallel
250: .I lat0
251: developed from tangent cone
252: .PP
253: Projections with bilateral symmetry about
254: the Prime Meridian
255: and the equator.
256: .br
257: .ns
258: .IP
259: .B polyconic()
260: parallels developed from tangent cones,
261: equally spaced along Prime Meridian
262: .br
263: .B aitoff()
264: equal-area projection of globe onto 2-to-1
265: ellipse, based on
266: .I azequalarea
267: .br
268: .B lagrange()
269: conformal, maps whole sphere into a circle
270: .br
271: .B bicentric(lon0)
272: points plotted at true azimuth from two
273: centers on the equator at longitudes
274: .I \(+-lon0,
275: great circles are straight lines
276: (a stretched gnomonic projection)
277: .br
278: .B elliptic(lon0)
279: points are plotted at true distance from
280: two centers on the equator at longitudes
281: .I \(+-lon0
282: .br
283: .B globular()
284: hemisphere is circle,
285: circular arc meridians equally spaced on equator,
286: circular arc parallels equally spaced on 0- and 90-degree meridians
287: .br
288: .B vandergrinten()
289: sphere is circle,
290: meridians as in
291: .I globular,
292: circular arc parallels resemble
293: .I mercator
294: .PP
295: Doubly periodic conformal projections.
296: .br
297: .ns
298: .IP
299: .B guyou()
300: W and E hemispheres are square
301: .br
302: .B square()
303: world is square with Poles
304: at diagonally opposite corners
305: .br
306: .B tetra()
307: map on tetrahedron with edge
308: tangent to Prime Meridian at S Pole,
309: unfolded into equilateral triangle
310: .br
311: .B hex()
312: world is hexagon centered
313: on N Pole, N and S hemispheres are equilateral
314: triangles
315: .PP
316: Miscellaneous projections.
317: .br
318: .ns
319: .IP
320: .B harrison(dist,angle)
321: oblique perspective from above the North Pole,
322: .I dist
323: earth radii from center of earth, looking
324: along the Date Line
325: .I angle
326: degrees off vertical
327: .br
328: .B trapezoidal(lat0,lat1)
329: equally spaced parallels,
330: straight meridians equally spaced along parallels,
331: true scale at
332: .I lat0
333: and
334: .I lat1
335: on Prime Meridian
336: .PP
337: Retroazimuthal projections.
338: At every point the angle between vertical and a straight line to
339: `Mecca', latitude
340: .I lat0
341: on the prime meridian,
342: is the true bearing of Mecca.
343: .br
344: .ns
345: .IP
346: .B mecca(lat0)
347: equally spaced vertical meridians
348: .br
349: .B homing(lat0)
350: distances to `Mecca' are true
351: .PP
352: Maps based on the spheroid.
353: Of geodetic quality, these projections do not make sense
354: for tilted orientations.
355: For descriptions, see corresponding maps above.
356: .br
357: .ns
358: .IP
359: .B sp_mercator()
360: .br
361: .B sp_albers(lat0,lat1)
362: .SH "SEE ALSO
363: .IR map (7),
364: .IR map (5),
365: .IR plot (3)
366: .SH BUGS
367: Only one projection and one orientation can be active at a time.
368: .br
369: The west-longitude-positive convention
370: betrays Yankee chauvinism.
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