Five essential spherical cartographic projections
There is a large number of different projections used to depict the spheroidal planetary surface on the flat paper map. Most of those were developed at the time when cartographic projections were used for two purposes: to provide a visual presentation of geographic objects, and, in addition, to enable computations using planar instead of spheroidal coordinates. For this, the inconvenience of limited spatial extent and breakdown of topological equivalence between limitless planar coordinate domain and finite spheroidal coordinate domain was offset by much simpler planar computational geometry. · When coordinate computations are carried out by digital computer with a basic algebraic operation with the numerical resolution provided by the 64-bit floating hardware, the motivation for computing using planar (instead of a combination of spherical and ellipsoidal coordinates) quickly evaporates. This leaves the cartographic projections with only one purpose: data visualization. · On the other hand, the requirement for computational efficiency of spheroid-to-plane coordinate transformations becomes much more important when the map, possibly consisting of millions of individual points, is created on the computer screen, than when the map is created as a hard-copy physical object. · Traditional cartographic projections invariably assume that spherical or ellipsoidal coordinates used as a source for mapping transformations are given by angular φ and λ. If the spheroidal coordinates are given in direction cosine form instead, most or many projection calculations can avoid expensive trigonometry functions, thus significantly increasing the speed of cartographic computations. · Taking into account the above, most systems that process geographic data on digital computers need only a handful of cartographic projections: · 1) Gnomonic Fastest in computation, with shapes on the map reasonably similar to the shapes in nature when the spatial extent is an order of magnitude smaller than the planetary radius. · 2) Stereographic Similar to Gnomonic, but preserving the similarity at a much greater spatial extent, at the cost of somewhat slower computations. · 3) Orthographic The projection shows a complete hemisphere inside a circle, and is useful when in addition to the objects on planetary surface the visualization includes objects above the surface, such as trajectories and positions of orbiting satellites. · 4) Conical The projection may be of use for regions with significant longitude extent in mid-latitudes. · 5) Stylindrical The projection is used to depict the whole planetary surface on a cylinder with the axis coincident with the "z" coordinate axis, "cut" along the meridian antipodal to planar coordinate origin. The shapes resemble those on the well-known Mercator map, but since projection centre is the point antipodal to the intersection of central meridian and the Equator the North/South dimension of the cylinder does not extend to infinity. · Projection geometry that relates the latitude (φ) to the planar North/South is the same as that of the Stereographic plane, except that it occurs in the meridian plane of the projected point instead of the plane defined by the projection centre, tangency point and the point to be projected; while East/West planar coordinate is equal to the longitude (λ) in angular measure. · Since the projection is a computational combination of Stereographic and "vertical" "cylindrical projections, and since it is not explicitly described or named in either of the two references given below, this document uses for it the name that is a combination of names of its two "component projections".
Description:
In all of the above:
·
S: Point on the sphere to be mapped into plane.
P: Spherical point projected ("mapped") into planar east, north
coordinate system.
C: Projection centre. (For Ortographic projection, the "center" is in
the infinity distance above the plane coordinate system origin).
T: Planar projection mapping coordinate system origin (plane/sphere
tangency point).
L: Conical projection, intersection of the parallel of latitude
at which the cone touches the sphere and the meridian plane of
the spherical point to be projected.
E: Stylindrical projection, intersection of the Equator and the
meridian plane of the point to be projected.
·
For Gnomonic, Stereographic and Orthographic projections, the geometry
is depicted in the plane defined by three points: projection centre, map
plane origin (i.e., the tangency) point and the the point to be projected.
For the other two, the geometry is depicted in the meridian plane of the
spherical point.
·
Worth noting is the relative speed of sphere to plane (i.e., direct)
mapping transformation of five projections. On a AMD Ryzen 7 4700U CPU
the speed measured as millions of random spherical points projected
into the map plane per second (i.e., more is better) is as follows:
·
Gnomonic: 102.4
Stereographic: 30.6
Orthographic: 75.5
Conical: 14.3
Stylindrical: 26.3
·
The speed of the inverse (i.e. projecting the point from plane back to
sphere) will generally be 30-50 % slower (i.e., it will take a little bit
more time to "un-map" the point than it took to map it). This is rarely
a concern: while a system would usually be required to create a transient
grephical display consisting of millions of data points in sub-second
interaction time, the inverse would be required for only a single set of
display coordinates - usually the cartographic plane equivalent of the
screen position of a user-operated pointer device.
·
Following the assumption that most high volume data objects manipulated
by the system will be represented by 8-byte nemoPtU64 concise spherical
coordinates, both direct and inverse transformations will accept (or
return) the spherical coordinates in that format. The mechanism is simple:
when the nemoPtNcs * pointer is NULL, the function assumes the spherical
coordinates are given by the following nemoPtU64 item in the calling
sequence, or are expected to be returned in variable pointed to by a
nemoPtU64 * pointer that follows the NULL nemoPtNcs * in the function
calling sequence. This is only a programming convenience: the coordinates
are internally transformed into full direction cosine form before (or
after) the required cartographic computations.
·
For additional discussion of cartographic projections, see:
·
Yang, Snyder, Tobler: Map Projection Transformation, Principles
and Applications, Taylor & Francis, 2000
Snyder: Map Projections Used by the U.S. Geological Survey
Geological survey bulletin 1532, Second edition, 1984
| nemo_GnomonicInit() | Initialize Gnomonic projection |
| nemo_GnomonicDirect() | NCS to Gnomonic map plane |
| nemo_GnomonicInverse() | Gnomonic map plane to NCS |
| nemo_StereographicInit() | Initialize Stereographic projection |
| nemo_StereographicDirect() | NCS to Stereographic map plane |
| nemo_StereographicInverse() | Stereographic map plane to NCS |
| nemo_OrthographicInit() | Initialize ortographic projection |
| nemo_OrthographicDirect() | NCS to Ortographic map plane |
| nemo_OrthographicInverse() | Ortographic map plane to NCS |
| nemo_ConicalInit() | Initialize conical projection |
| nemo_ConicalDirect() | NCS to Conical map plane |
| nemo_ConicalInverse() | Conical map plane to NCS |
| nemo_StylindricalInit() | Initialize Stylindrical projection |
| nemo_StylindricalDirect() | NCS to Stylindrical map plane |
| nemo_StylindricalInverse() | Stylindrical map plane to NCS |
Synopsis:
#include <nemo.h>
int nemo_GnomonicInit(const nemoPtNcs *pPtNcsT, nemoPtU64 ptU64,
double projParms[NEMO_GNOMONIC_PCNT]);
Description:
Given the coordinates of Gnomonic projection sphere/plane tangency
point, calculate and return (in an invoker-supplied array of
NEMO_GNOMONIC_PCNT doubles) the parameters used in subsequent
direct and inverse coordinate transformations. With spherical coordinates
in the direction cosine form, Gnomonic projection has by far the quickest
direct transformation. Another useful property is that great circles
on the sphere are projected as straight lines in plane, thus many spherical
geometry propositions of local geometry (for instance, finding a point of
intersection of two great circle segments) can be readily solved using
planar coordinate geometry, thus avoiding spherical trigonometry.
The disadvantage of Gnomonic projection is the loss of similarity of shapes
as the extent of map coverage increases. What level of distortion is
acceptable depends on the application contexts. In the illustration above,
the hydrography linework was rendered in Stereographic projection in red,
before being "drawn over" in blue - in Gnomonic projection, (the same
as the rest of the map). The red "ghost" linework in the map periphery
illustrates how the visual difference between two projections depends on
the planet size, map scale and display pixel size.
Arguments:
pPtNcsT:
pointer to nemoPtNcs structure, given projection tangency point
spherical coordinates, or NULL.
ptU64:
integer of nemoPtU64 type. Ignored if pPtNcsT is non-NULL,
otherwise tangency point coordinates.
projParms:
Array of NEMO_GNOMONIC_PCNT doubles, returned projection parameters.
The first three array elements are the coordinates of the spherical
tangency point, which the invoker may read but (like the rest of the
array) must not modify while the array is used by the two transformation
functions.
Return Value:
Integer, 1 for ill-defined tangency pont, 0 otherwise.
See Also:
nemo_OrthographicInit()
nemo_StereographicInit()
nemo_ConicalInit()
nemo_StylindricalInit()
nemo_GnomonicDirect()
nemo_GnomonicInverse()
Synopsis:
#include <nemo.h>
int nemo_GnomonicDirect(const double projParms[NEMO_GNOMONIC_PCNT],
const nemoPtNcs *pPtNcs, nemoPtU64 ptU64,
nemoPtPln *pPtPln);
Description:
Given the coordinates of a point on the sphere and an array of Gnomonic
projection parameters, return the planar coordinates of the point.
Arguments:
projParms:
Array of NEMO_GNOMONIC_PCNT doubles, given projection parameters,
previously populated by nemo_StereographicInit() function.
pPtNcs:
pointer to nemoPtNcs structure, given spherical point coordinates.
ptU64:
integer of nemoPtU64 type. Ignored if pPtNcs is non-NULL,
otherwise the given spherical point coordinates in nemoPtU64 form.
pPtPln:
Pointer to nemoPtPln structure, returned planar (projected)
point coordinates.
Return Value:
Integer, always 0.
See Also:
nemo_GnomonicInit()
nemo_GnomonicInverse()
Synopsis:
#include <nemo.h>
int nemo_GnomonicInverse(const double projParms[NEMO_GNOMONIC_PCNT],
const nemoPtPln *pPtPln,
nemoPtNcs *pPtNcs, nemoPtU64 *pPtU64);
Description:
Given the planar coordinates of a point and an array of Gnomonic projection
parameters, return the spherical coordinats of the point.
Arguments:
projParms:
Array of NEMO_GNOMONIC_PCNT doubles, given projection parameters,
previously populated by nemo_StereographicInit() function.
pPtPln:
Pointer to nemoPtPln type/structure, given planar point coordinates.
pPtNcs:
pointer to nemoPtNcs structure, returned spherical point coordinates.
Ignored if NULL.
pPtU64:
Pointer to an integer of nemoPtU64 type. Ignored if NULL, otherwise
returned spherical point coordinates in nemoPtU64 form.
Return Value:
Integer, always 0.
See Also:
nemo_GnomonicInit()
nemo_GnomonicDirect()
Synopsis:
#include <nemo.h>
int nemo_StereographicInit(const nemoPtNcs *pPtNcsT, nemoPtU64 ptU64,
double projParms[NEMO_STEREOGRAPHIC_PCNT]);
Description:
Given the coordinates of Stereographic projection sphere/plane tangency
point, calculate and return (in an invoker-supplied array of
NEMO_STEREOGRAPHIC_PCNT doubles) the parameters used in subsequent
direct and inverse coordinate transformations.
Arguments:
pPtNcsT:
pointer to nemoPtNcs structure, given projection tangency point
spherical coordinates, or NULL.
ptU64:
integer of nemoPtU64 type. Ignored if pPtNcsT is non-NULL,
otherwise tangency point coordinates.
projParms:
Array of NEMO_STEREOGRAPHIC_PCNT doubles, returned projection
parameters. The first three array elements are the coordinates of the
spherical tangency point, which the invoker may read but (like the rest
of the array) must not modify while the array is used by the two
transformation functions.
Return Value:
Integer, error indicator, non-0 in case of invalid given tangency point.
See Also:
nemo_GnomonicInit()
nemo_OrthographicInit()
nemo_ConicalInit()
nemo_StylindricalInit()
nemo_StereographicDirect()
nemo_StereographicInverse()
Synopsis:
#include <nemo.h>
int nemo_StereographicDirect(const double projParms[NEMO_STEREOGRAPHIC_PCNT],
const nemoPtNcs *, nemoPtU64 ptU64,
nemoPtPln *pPtPln);
Description:
Given the coordinates of a point on the sphere and an array of Stereographic
projection parameters, return the planar coordinates of the point.
Arguments:
projParms:
Array of NEMO_STEREOGRAPHIC_PCNT doubles, given projection parameters,
previously populated by nemo_StereographicInit() function.
pPtNcs:
pointer to nemoPtNcs structure, given spherical point coordinates.
ptU64:
integer of nemoPtU64 type. Ignored if pPtNcs is non-NULL,
otherwise the given spherical point coordinates in nemoPtU64 form.
pPtPln:
Pointer to nemoPtPln structure, returned planar (projected)
point coordinates.
Return Value:
Integer, error indicator, non-0 in case of invalid given point.
See Also:
nemo_StereographicInit()
nemo_StereographicInverse()
Synopsis:
#include <nemo.h>
int nemo_StereographicInverse(const double projParms[NEMO_STEREOGRAPHIC_PCNT],
const nemoPtPln *, nemoPtNcs *pPtNcs,
nemoPtU64 *pPtU64);
Description:
Given the planar coordinates of a point and an array of Stereographic
projection parameters, return the spherical coordinates of the point.
Arguments:
projParms:
Array of NEMO_STEREOGRAPHIC_PCNT doubles, given projection parameters,
previously populated by nemo_StereographicInit() function.
pPtPln:
Pointer to nemoPtPln type/structure, given planar point coordinates.
pPtNcs:
pointer to nemoPtNcs structure, returned spherical point coordinates.
Ignored if NULL.
pPtU64:
Pointer to an integer of nemoPtU64 type. Ignored if NULL, otherwise
returned spherical point coordinates in nemoPtU64 form.
Return Value:
Integer, error indicator, non-0 in case of invalid return specifications.
See Also:
nemo_StereographicInit()
nemo_StereographicDirect()
Synopsis:
#include <nemo.h>
int nemo_OrthographicInit(const nemoPtNcs *pPtNcsPc, nemoPtU64 pPtU64Pc,
const nemoPtNcs *pPtNcsUp, nemoPtU64 pPtU64Up,
double ppft,
double projParms[NEMO_ORTHOGRAPHIC_PCNT]);
Description:
Given the coordinates of Orthograpgic projection sphere point that
corresponds to the origin of the planar east, north coordinate system,
calculate and return (in an invoker-supplied array of
NEMO_ORTHOGRAPHIC_PCNT doubles) the parameters used in subsequent
direct and inverse coordinate transformations.
Arguments:
pPtNcsT:
pointer to nemoPtNcs structure, given spherical point corresponding
to the origin of the planar coordinate system. The argument can be NULL,
in which case the coordinates are assumed to be given in nemoPtU64
form, in the argument that follows.
ptU64:
integer of nemoPtU64 type. Ignored if pPtNcsT is non-NULL,
otherwise spherical point coordinates as defined above.
projParms:
Array of NEMO_ORTHOGRAPHIC_PCNT doubles, returned projection
parameters. The first three array elements are the coordinates of the
spherical center-point (see above), which the invoker may read but
(like the rest of the array) must not modify while the array is used
by the two transformation functions.
Return Value:
Integer, 1 for ill-defined center point, 0 otherwise.
See Also:
nemo_GnomonicInit()
nemo_StereographicInit()
nemo_ConicalInit()
nemo_StylindricalInit()
nemo_OrthographicDirect()
nemo_OrthographicInverse()
Synopsis:
#include <nemo.h>
int nemo_OrthographicDirect(const double projParms[NEMO_ORTHOGRAPHIC_PCNT],
const nemoPtNcs *pPtNcs, nemoPtU64 ptU64,
nemoPtPln *pPtPln);
Description:
Given the coordinates of a point on the sphere and an array of Orthographic
projection parameters, return the planar coordinates of the point.
Arguments:
projParms:
Array of NEMO_ORTHOGRAPHIC_PCNT doubles, given projection parameters,
previously populated by nemo_OrthographicInit() function.
pPtNcs:
pointer to nemoPtNcs structure, given spherical point coordinates.
ptU64:
integer of nemoPtU64 type. Ignored if pPtNcs is non-NULL,
otherwise the given spherical point coordinates in nemoPtU64 form.
pPtPln:
Pointer to nemoPtPln structure, returned planar (projected)
point coordinates.
Return Value:
Integer, always 0;
See Also:
nemo_OrthographicInit()
nemo_OrthographicInverse()
Synopsis:
#include <nemo.h>
int nemo_OrthographicInverse(const double projParms[NEMO_ORTHOGRAPHIC_PCNT],
const nemoPtPln *pPtPln,
nemoPtNcs *pPtNcsT, nemoPtU64 *pPtU64);
Description:
Given the planar coordinates of a point and an array of Orthographic
projection parameters, return the spherical coordinates of the point.
Arguments:
projParms:
Array of NEMO_ORTHOGRAPHIC_PCNT doubles, given projection parameters,
previously populated by nemo_OrthographicInit() function.
pPtPln:
Pointer to nemoPtPln type/structure, given planar point coordinates.
pPtNcs:
pointer to nemoPtNcs structure, returned spherical point coordinates.
Ignored if NULL.
pPtU64:
Pointer to an integer of nemoPtU64 type. Ignored if NULL, otherwise
returned spherical point coordinates in nemoPtU64 form.
Return Value:
Integer, error indicator, non-0 in case of invalid return specifications.
See Also:
nemo_OrthographicInit()
nemo_OrthographicDirect()
Synopsis:
#include <nemo.h>
int nemo_ConicalInit(const nemoPtNcs *pPtNcs, nemoPtU64 ptU64,
double projParms[NEMO_CONICAL_PCNT]);
See Also: nemo_GnomonicInit()
nemo_OrthographicInit()
nemo_StereographicInit()
nemo_StylindricalInit()
nemo_ConicalDirect()
nemo_ConicalInverse()
Synopsis:
#include <nemo.h>
int nemo_ConicalDirect(const double projParms[NEMO_CONICAL_PCNT],
const nemoPtNcs *pPtNcsT, nemoPtU64 ptU64,
nemoPtPln *pPtPln);
See Also:
nemo_ConicalInit()
nemo_ConicalInverse()
Synopsis:
#include <nemo.h>
int nemo_ConicalInverse(const double projParms[NEMO_CONICAL_PCNT],
const nemoPtPln *pPtPln,
nemoPtNcs *pPtNcsT, nemoPtU64 *pPtU64);
See Also:
nemo_ConicalInit()
nemo_ConicalDirect()
Synopsis:
#include <nemo.h>
int nemo_StylindricalInit(const nemoPtNcs *pPtNcsCm, nemoPtU64 ptU64,
double projParms[NEMO_STYLINDRICAL_PCNT]);
Description:
Given the coordinates of a point on "stylindrical" projection central
meridian, calculate and return (in an invoker-supplied array of
STYLINDRICAL_PCNT doubles) the parameters used in subsequent direct
and inverse coordinate transformations.
Arguments:
pPtNcsT:
pointer to nemoPtNcs structure, given point that defines the central
half-meridian of the projection. Any point on the half-meridian can be
given, as long as it if not in very high latitudes, where the spherical
vector form of coordinated might not define it's longitude with
sufficient resolution. (it will commonly be a point on the Equator).
The other half of the selected meridian will be the half-meridian
along which the cylinder is "cut" when unfurled into the plane.
ptU64:
integer of nemoPtU64 type. Ignored if pPtNcsT is non-NULL,
otherwise tangency point coordinates.
projParms:
Array of NEMO_STYLINDRICAL_PCNT doubles, returned projection
parameters. The first three array elements are the the spherical
coordinates of central meridian equatorial point, which the invoker
may read but (like the rest of the array) not modify while the
array is used by the two transformation functions.
Return Value:
Integer, error indicator, non-0 in case of invalid given central
meridian definition point.
Example:
...
/* The longitude 11° 30' East (for instance, Munich, Bavaria) will be used
as the central half-meridian. In addition to (unavoidable) "Antarctica
incision", the only landmass its antipodal half-meridian will cut in
two pieces is the remote Aleutian island of Nikolski (population 39).
(There is no "Antarctica-incision-only" half-meridian on planet Earth).
*/
double projParms[NEMO_STYLINDRICAL_PCNT]; /* projection parameters */
nemoPtEll cmeEll = {0, NEMO_DEG2RAD * 11.5}; /* central meridian... */
nemoPtNcs cmeNcs; /* ...equatorial point, on ellipsoid and NCS. */
int iRtcd;
...
nemo_EllToNcs(nemo_ElrWgs84(), &cmeEll, &cmeNcs);
iRtcs = nemo_StylindricalInit(&cmeNcs, NEMO_U64_UNDEF, projParms);
...
See Also:
nemo_GnomonicInit()
nemo_OrthographicInit()
nemo_StereographicInit()
nemo_ConicalInit()
nemo_StylindricalDirect()
nemo_StylindricalInverse()
Synopsis:
#include <nemo.h>
int nemo_StylindricalDirect(const double projParms[NEMO_STYLINDRICAL_PCNT],
const nemoPtNcs *pPtNcs, nemoPtU64 ptU64,
nemoPtPln *pPtPln);
Description:
Given the coordinates of a point on the sphere and an array of Stylindrical
projection parameters, return the planar coordinats of the point.
Arguments:
projParms:
Array of NEMO_STYLINDRICAL_PCNT doubles, given projection parameters,
previously populated by nemo_StylindricalInit() function.
pPtNcs:
pointer to nemoPtNcs structure, given spherical point coordinates.
ptU64:
integer of nemoPtU64 type. Ignored if pPtNcs is non-NULL,
otherwise the given spherical point coordinates in nemoPtU64 form.
pPtPln:
Pointer to nemoPtPln type/structure, returned planar (projected)
point coordinates.
Return Value:
Integer, error indicator, non-0 in case of invalid given point.
See Also:
nemo_StylindricalInit()
nemo_StylindricalInverse()
Synopsis:
#include <nemo.h>
int nemo_StylindricalInverse(const double projParms[NEMO_STYLINDRICAL_PCNT],
const nemoPtPln *pPtPln,
nemoPtNcs *pPtNcs, nemoPtU64 *pPtU64);
Description:
Given the planar coordinates of a point and an array of Stylindrical
projection parameters, return the spherical coordinats of the point.
Arguments:
pPtPln:
Pointer to nemoPtPln type/structure, given planar point coordinates.
projParms:
Array of NEMO_STYLINDRICAL_PCNT doubles, given projection parameters,
previously populated by nemo_StylindricalInit() function.
pPtNcs:
pointer to nemoPtNcs structure, returned spherical point coordinates.
Ignored if NULL.
pPtU64:
Pointer to an integer of nemoPtU64 type. Ignored if NULL, otherwise
returned spherical point coordinates in nemoPtU64 form.
Return Value:
Integer, error indicator, non-0 in case of invalid return coordinates
specifications.
See Also:
nemo_StylindricalInit()
nemo_StylindricalDirect()