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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. Warning: coordinates are in degrees for
96: .I orient,
97: in radians for
98: .LR place .
99: .PP
100: .EX
101: .ta \n(xxu +\n(xxu +\n(xxu +\n(xxu +\n(xxu +\n(xxu
102: struct place pt;
103: int (*proj)() = mercator();
104: double x, y;
105: .EE
106: .PP
107: .EX
108: orient(45.0, 30.0, 180.0); /* set coordinate rotation */
109: .EE
110: .PP
111: .EX
112: . . . /* fill in the pt structure */
113: normalize(&pt); /* rotate coordinates */
114: if((*proj)(&pt, &x, &y) > 0) /* project onto x,y plane */
115: plot(x, y);
116: .EE
117: .PP
118: The projection function
119: .B (*proj)()
120: returns 1 for a good point,
121: 0 for a point on a wrong
122: sheet (e.g. the back of the world in a perspective
123: projection), and \-1 for a point that is deemed
124: unplottable (e.g. points near the poles on a Mercator projection).
125: .PP
126: Scaling may be determined from the
127: .I x-y
128: coordinates of
129: selected points.
130: Latitudes and longitudes are measured in degrees for
131: ease of specification for
132: .I orient
133: and the projection generators
134: but in radians for ease of calculation
135: for
136: .I normalize
137: and
138: .I proj.
139: In either case
140: latitude is measured positive north of the equator,
141: and longitude positive west of Greenwich.
142: Radian longitude should be limited to the range
143: .if t .I \-\(*p\(<=lon<\(*p.
144: .if n .I -pi <= lon < pi.
145: .SS Projection generators
146: Equatorial projections centered on the Prime Meridian
147: (longitude 0).
148: Parallels are straight horizontal lines.
149: .br
150: .ns
151: .IP
152: .B mercator()
153: equally spaced straight meridians, conformal,
154: straight compass courses
155: .br
156: .B sinusoidal()
157: equally spaced parallels,
158: equal-area, same as
159: .I bonne(0)
160: .br
161: .B cylequalarea(lat0)
162: equally spaced straight meridians, equal-area,
163: true scale on
164: .I lat0
165: .br
166: .B cylindrical()
167: central projection on tangent cylinder
168: .br
169: .B rectangular(lat0)
170: equally spaced parallels, equally spaced straight meridians, true scale on
171: .I lat0
172: .br
173: .B gall(lat0)
174: parallels spaced stereographically on prime meridian, equally spaced straight
175: meridians, true scale on
176: .I lat0
177: .br
178: .B mollweide()
179: (homalographic) equal-area, hemisphere is a circle
180: .PP
181: Azimuthal projections centered on the North Pole.
182: Parallels are concentric circles.
183: Meridians are equally spaced radial lines.
184: .br
185: .ns
186: .IP
187: .B azequidistant()
188: equally spaced parallels,
189: true distances from pole
190: .br
191: .B azequalarea()
192: equal-area
193: .br
194: .B gnomonic()
195: central projection on tangent plane,
196: straight great circles
197: .br
198: .B perspective(dist)
199: viewed along earth's axis
200: .I dist
201: earth radii from center of earth
202: .br
203: .B orthographic()
204: viewed from infinity
205: .br
206: .B stereographic()
207: conformal, projected from opposite pole
208: .br
209: .B laue()
210: .IR radius " = tan(2\(mu" colatitude ),
211: used in xray crystallography
212: .br
213: .B fisheye(n)
214: stereographic seen from just inside medium with refractive index n
215: .br
216: .B newyorker(r)
217: .IR radius " = log(" colatitude / r ):
218: extreme `fisheye' view from pedestal of radius
219: .I r
220: degrees
221: .PP
222: Polar conic projections symmetric about the Prime Meridian.
223: Parallels are segments of concentric circles.
224: Except in the Bonne projection,
225: meridians are equally spaced radial
226: lines orthogonal to the parallels.
227: .br
228: .ns
229: .IP
230: .B conic(lat0)
231: central projection on cone tangent at
232: .I lat0
233: .br
234: .B simpleconic(lat0,lat1)
235: equally spaced parallels, true scale on
236: .I lat0
237: and
238: .I lat1
239: .br
240: .B lambert(lat0,lat1)
241: conformal, true scale on
242: .I lat0
243: and
244: .I lat1
245: .br
246: .B albers(lat0,lat1)
247: equal-area, true scale on
248: .I lat0
249: and
250: .I lat1
251: .br
252: .B bonne(lat0)
253: equally spaced parallels, equal-area,
254: parallel
255: .I lat0
256: developed from tangent cone
257: .PP
258: Projections with bilateral symmetry about
259: the Prime Meridian
260: and the equator.
261: .br
262: .ns
263: .IP
264: .B polyconic()
265: parallels developed from tangent cones,
266: equally spaced along Prime Meridian
267: .br
268: .B aitoff()
269: equal-area projection of globe onto 2-to-1
270: ellipse, based on
271: .I azequalarea
272: .br
273: .B lagrange()
274: conformal, maps whole sphere into a circle
275: .br
276: .B bicentric(lon0)
277: points plotted at true azimuth from two
278: centers on the equator at longitudes
279: .I \(+-lon0,
280: great circles are straight lines
281: (a stretched gnomonic projection)
282: .br
283: .B elliptic(lon0)
284: points are plotted at true distance from
285: two centers on the equator at longitudes
286: .I \(+-lon0
287: .br
288: .B globular()
289: hemisphere is circle,
290: circular arc meridians equally spaced on equator,
291: circular arc parallels equally spaced on 0- and 90-degree meridians
292: .br
293: .B vandergrinten()
294: sphere is circle,
295: meridians as in
296: .I globular,
297: circular arc parallels resemble
298: .I mercator
299: .PP
300: Doubly periodic conformal projections.
301: .br
302: .ns
303: .IP
304: .B guyou()
305: W and E hemispheres are square
306: .br
307: .B square()
308: world is square with Poles
309: at diagonally opposite corners
310: .br
311: .B tetra()
312: map on tetrahedron with edge
313: tangent to Prime Meridian at S Pole,
314: unfolded into equilateral triangle
315: .br
316: .B hex()
317: world is hexagon centered
318: on N Pole, N and S hemispheres are equilateral
319: triangles
320: .PP
321: Miscellaneous projections.
322: .br
323: .ns
324: .IP
325: .B harrison(dist,angle)
326: oblique perspective from above the North Pole,
327: .I dist
328: earth radii from center of earth, looking
329: along the Date Line
330: .I angle
331: degrees off vertical
332: .br
333: .B trapezoidal(lat0,lat1)
334: equally spaced parallels,
335: straight meridians equally spaced along parallels,
336: true scale at
337: .I lat0
338: and
339: .I lat1
340: on Prime Meridian
341: .PP
342: Retroazimuthal projections.
343: At every point the angle between vertical and a straight line to
344: `Mecca', latitude
345: .I lat0
346: on the prime meridian,
347: is the true bearing of Mecca.
348: .br
349: .ns
350: .IP
351: .B mecca(lat0)
352: equally spaced vertical meridians
353: .br
354: .B homing(lat0)
355: distances to `Mecca' are true
356: .PP
357: Maps based on the spheroid.
358: Of geodetic quality, these projections do not make sense
359: for tilted orientations.
360: For descriptions, see corresponding maps above.
361: .br
362: .ns
363: .IP
364: .B sp_mercator()
365: .br
366: .B sp_albers(lat0,lat1)
367: .SH "SEE ALSO
368: .IR map (7),
369: .IR map (5),
370: .IR plot (3)
371: .SH BUGS
372: Only one projection and one orientation can be active at a time.
373: .br
374: The west-longitude-positive convention
375: betrays Yankee chauvinism.
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