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authorJacob McDonnell <jacob@jacobmcdonnell.com>2026-04-26 16:38:00 -0400
committerJacob McDonnell <jacob@jacobmcdonnell.com>2026-04-26 16:38:00 -0400
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-.TH PROJ 3X bowell
-.CT 2 graphics math
-.br
-.SH NAME
-orient, normalize \- map projections
-.SH SYNOPSIS
-.B orient(lat, lon, rot)
-.br
-.B float lat, lon, rot;
-.PP
-.B normalize(p)
-.br
-.B struct place *p;
-.SH DESCRIPTION
-Users of
-.IR map (7)
-may skip to the description of `Projection generators'
-below.
-.PP
-The functions
-.I orient
-and
-.I normalize
-plus a collection of map projection generators
-are loaded by
-option
-.BR -lmap
-of
-.IR 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.
-.PP
-Each standard projection is displayed with the Prime
-Meridian (longitude 0) being a straight vertical line, along which North
-is up.
-The orientation of nonstandard projections is specified by
-.I 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
-.I lon
-on the globe.
-Then tilt the North Pole of the
-overlay along its Prime Meridian to latitude
-.I 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 overlay's) meridian
-.I rot.
-Project the desired map in
-the standard form appropriate to the overlay, but presenting
-information from the underlying globe.
-It is not useful to use
-.I orient
-without using
-.IR normalize .
-.PP
-.I Normalize
-converts latitude-longitude coordinates on the globe
-to coordinates on the overlaid grid.
-The coordinates and their sines and cosines
-are input to
-.I normalize
-in a
-.B place
-structure.
-Transformed coordinates and their sines and cosines
-are returned in the same structure.
-.PP
-.EX
-.nr xx \w'12345678'
-.ta \n(xxu +\n(xxu +\n(xxu +\n(xxu +\n(xxu +\n(xxu
- struct place {
- float radianlat, sinlat, coslat;
- float radianlon, sinlon, coslon;
- };
-.EE
-.PP
-The projection generators
-return a pointer to a function that converts normalized coordinates
-to
-.I x-y
-coordinates for the desired map, or
-0 if the required projection
-is not available.
-The returned function is exemplified by
-.I proj
-in this example:
-.PP
-.EX
-.ta \n(xxu +\n(xxu +\n(xxu +\n(xxu +\n(xxu +\n(xxu
- struct place pt;
- int (*proj)() = mercator();
- float x, y;
-.EE
-.PP
-.EX
- orient(45.0, 30.0, 180.0); /* set coordinate rotation */
-.EE
-.PP
-.EX
- . . . /* fill in the pt structure */
- normalize(&pt); /* rotate coordinates */
- if((*proj)(&pt, &x, &y) > 0) /* project onto x,y plane */
- plot(x, y);
-.EE
-.PP
-The projection function
-.B (*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).
-.PP
-Scaling may be determined from the
-.I x-y
-coordinates of
-selected points.
-Latitudes and longitudes are measured in degrees for
-ease of specification for
-.I orient
-and the projection generators
-but in radians for ease of calculation
-for
-.I normalize
-and
-.I proj.
-In either case
-latitude is measured positive north of the equator,
-and longitude positive west of Greenwich.
-Radian longitude should be limited to the range
-.if t .I \-\(*p\(<=lon<\(*p.
-.if n .I -pi <= lon < pi.
-.SS Projection generators
-Equatorial projections centered on the Prime Meridian
-(longitude 0).
-Parallels are straight horizontal lines.
-.br
-.ns
-.IP
-.B mercator()
-equally spaced straight meridians, conformal,
-straight compass courses
-.br
-.B sinusoidal()
-equally spaced parallels,
-equal-area, same as
-.I bonne(0)
-.br
-.B cylequalarea(lat0)
-equally spaced straight meridians, equal-area,
-true scale on
-.I lat0
-.br
-.B cylindrical()
-central projection on tangent cylinder
-.br
-.B rectangular(lat0)
-equally spaced parallels, equally spaced straight meridians, true scale on
-.I lat0
-.br
-.B gall(lat0)
-parallels spaced stereographically on prime meridian, equally spaced straight
-meridians, true scale on
-.I lat0
-.br
-.B mollweide()
-(homalographic) equal-area, hemisphere is a circle
-.PP
-Azimuthal projections centered on the North Pole.
-Parallels are concentric circles.
-Meridians are equally spaced radial lines.
-.br
-.ns
-.IP
-.B azequidistant()
-equally spaced parallels,
-true distances from pole
-.br
-.B azequalarea()
-equal-area
-.br
-.B gnomonic()
-central projection on tangent plane,
-straight great circles
-.br
-.B perspective(dist)
-viewed along earth's axis
-.I dist
-earth radii from center of earth
-.br
-.B orthographic()
-viewed from infinity
-.br
-.B stereographic()
-conformal, projected from opposite pole
-.br
-.B laue()
-.IR radius " = tan(2\(mu" colatitude ),
-used in xray crystallography
-.br
-.B fisheye(r)
-.IR radius " = log(" colatitude / r ):
-.I New Yorker
-map from viewing pedestal of radius
-.I r
-degrees
-.PP
-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.
-.br
-.ns
-.IP
-.B conic(lat0)
-central projection on cone tangent at
-.I lat0
-.br
-.B simpleconic(lat0,lat1)
-equally spaced parallels, true scale on
-.I lat0
-and
-.I lat1
-.br
-.B lambert(lat0,lat1)
-conformal, true scale on
-.I lat0
-and
-.I lat1
-.br
-.B albers(lat0,lat1)
-equal-area, true scale on
-.I lat0
-and
-.I lat1
-.br
-.B bonne(lat0)
-equally spaced parallels, equal-area,
-parallel
-.I lat0
-developed from tangent cone
-.PP
-Projections with bilateral symmetry about
-the Prime Meridian
-and the equator.
-.br
-.ns
-.IP
-.B polyconic()
-parallels developed from tangent cones,
-equally spaced along Prime Meridian
-.br
-.B aitoff()
-equal-area projection of globe onto 2-to-1
-ellipse, based on
-.I azequalarea
-.br
-.B lagrange()
-conformal, maps whole sphere into a circle
-.br
-.B bicentric(lon0)
-points plotted at true azimuth from two
-centers on the equator at longitudes
-.I \(+-lon0,
-great circles are straight lines
-(a stretched gnomonic projection)
-.br
-.B elliptic(lon0)
-points are plotted at true distance from
-two centers on the equator at longitudes
-.I \(+-lon0
-.br
-.B globular()
-hemisphere is circle,
-circular arc meridians equally spaced on equator,
-circular arc parallels equally spaced on 0- and 90-degree meridians
-.br
-.B vandergrinten()
-sphere is circle,
-meridians as in
-.I globular,
-circular arc parallels resemble
-.I mercator
-.PP
-Doubly periodic conformal projections.
-.br
-.ns
-.IP
-.B guyou()
-W and E hemispheres are square
-.br
-.B square()
-world is square with Poles
-at diagonally opposite corners
-.br
-.B tetra()
-map on tetrahedron with edge
-tangent to Prime Meridian at S Pole,
-unfolded into equilateral triangle
-.br
-.B hex()
-world is hexagon centered
-on N Pole, N and S hemispheres are equilateral
-triangles
-.PP
-Miscellaneous projections.
-.br
-.ns
-.IP
-.B harrison(dist,angle)
-oblique perspective from above the North Pole,
-.I dist
-earth radii from center of earth, looking
-along the Date Line
-.I angle
-degrees off vertical
-.br
-.B trapezoidal(lat0,lat1)
-equally spaced parallels,
-straight meridians equally spaced along parallels,
-true scale at
-.I lat0
-and
-.I lat1
-on Prime Meridian
-.PP
-Retroazimuthal projections.
-At every point the angle between vertical and a straight line to
-`Mecca', latitude
-.I lat0
-on the prime meridian,
-is the true bearing of Mecca.
-.br
-.ns
-.IP
-.B mecca(lat0)
-equally spaced vertical meridians
-.br
-.B homing(lat0)
-distances to `Mecca' are true
-.PP
-Maps based on the spheroid.
-Of geodetic quality, these projections do not make sense
-for tilted orientations.
-For descriptions, see corresponding maps above.
-.br
-.ns
-.IP
-.B sp_mercator()
-.br
-.B sp_albers(lat0,lat1)
-.SH "SEE ALSO
-.IR map (7),
-.IR map (5),
-.IR plot (3)
-.SH BUGS
-Only one projection and one orientation can be active at a time.
-.br
-The west-longitude-positive convention
-betrays Yankee chauvinism.