[GIS] Converting Geocentric to Geodetic

datumelevationlatitude longitude

I am only looking for a confirmation of what I believe to be true. If my explaination or conclusion is not accurate I would appreciate it if the reason can be explained to me.

Background Information:

I am attempting to research a method to calculate the Latitude, Longitude, and Height of a point on the planet earth given Range, Azimuth, and Elevation of a point relative to another point where the Lat, Long, and Altitude is known. I am going to be using a geophysical library that allows me to provide it a geocentric value and it will return a Latitude and Longitude and the Geodetic Datum.

Is WGS-84 a valid datum for a Geocentric datum?
If I provide it a WGS-84 (X,Y,Z) point should it return a WGS-84 Latitude and Longitude?

Best Answer

Any geodetic datum is suitable for expressing coordinates in both geocentric and geodetic coordinate systems, so you should be fine using WGS 84.

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[GIS] locating latitude-longitude point in a piste-map with matrix transforms

If the image is in the least realistic--and it appears to be--then it represents a projection from a portion of Euclidean three-space onto a portion of two-space. We cannot expect this projection to have a nice mathematical formula, because it is likely the artist has introduced irrregular small distortions, but we can expect its formula to somewhat approximate formulas for such projections. These formulas tend to be linear or projective and they also tend to be simpler when Cartesian coordinates are used.

Therefore, what you should consider doing at the outset is this:

Convert all (lat, lon, elevation) values to geocentric (x, y, z) values.
Associated with each (x, y, z) value is a pair of image coordinates (u, v). We probably lose little by estimating u and v independently of each other, which reduces the problem to one of interpolating samples of a scalar function. The previous considerations suggest that you seek interpolating functions that locally are of the form
```
u = a x + b y + c z + d
```
(linear) or even
```
u = (a x + b y + c z + d) / (a' x + b' y + c' z + 1)
```
(projective) and similarly for v. This reduces the problem to one of performing an interpolation in three dimensions, which is sufficiently routine to give you access to a large body of methods and software.

Alternatively, you might consider triangulating the (x,y,z) points and interpolating their associated (u,v) values separately across the triangles; even a linear interpolation across each triangular face could do a pretty good job. This is a 3D analog of what ESRI's 3D Analyst does in two dimensions with TINs.

Please note, though, that unless you have the elevation of a given location, you cannot hope to plot it accurately on this map: changing the elevation will create a nearly vertical line on the map, whence your location could be plotted practically anywhere along this line. With considerable work you could solve this problem for points on the ground by first digitizing the evident discontinuities in the inverse projection--that is, representing the outlines of mountain ridges as polylines--and constructing a local inverse interpolator (respecting these breaklines) that associates to each (u,v) on the image a location (x,y,z) in the world. You could exploit this to identify which point along any vertical line has a (lat,lon) close to the one you started with.

NED Latitude – Does the NED Use Geocentric or Geodetic Latitude?

One could use either kind of latitude to locate points on the WGS 84 ellipsoid (used by the NED) or any other ellipsoid, but "everybody knows" that the values will always be given as geodetic latitudes. However, it is surprisingly hard to find an authoritative statement to that effect!

Before we go on, it helps to understand that although a datum like the WGS 84 ellipsoid describes a reference surface to locate everything on, above, and within the Earth, it is compatible with many different ways to associate coordinates with the geometric points on that ellipsoid. Two of those ways are the geocentric and geodetic systems.

The geocentric latitude of a point is the angle it makes with the Equatorial plane.
The geodetic latitude of a point is the angle made between the Equatorial plane and a perpendicular from that point to the surface of the datum beneath it.

The two angles are the same only when the point is located over the Equator (when they equal zero) or over one of the poles (when they equal -90 or +90 degrees). Otherwise the points denoted by the same numerical values of latitude and longitude can differ by as much as several tens of kilometers in the two systems (as I recall--I haven't rechecked that figure recently).

Both types of latitude can in principle be used, even though WGS 84 is referred to as a "geodetic" datum. (Indeed, both of them, as well as a "reduced latitude," are used in the governing NIMA technical report TR8350.2, available at http://earth-info.nga.mil/GandG/publications/tr8350.2/wgs84fin.pdf. See p. 4-4 and section 5.2 (on Gravity Potential) et seq.)

In the recent article Using NHDPlus as the Land Base for the Noah-distributed Model (David, Maidment, et al., Transactions in GIS 2009; 13(4): 363-377) a group of GIS luminaries writes

To distinguish between a spherical and spheroidal Earth, two types of latitudes are needed: geocentric and geodetic as shown in Figure 4.

Longitudes are not affected by this difference in Earth shape because it only involves North-South flattening. The geocentric latitude \Phi is the acute angle measured between the equatorial plane and a line joining the center of the Earth and a point on the surface of the sphere or spheroid. The geodetic latitude \phi′ is the acute angle between the equatorial plane and a line drawn perpendicular to the tangent plane of a point on the reference sphere or spheroid. Normal map coordinates are given in longitude and geodetic latitude.

(Emphasis added.)

In the 2007 book Unmanned Aerial Vehicle Real-time Guidance System Via State-space Heuristic Search, Manuel Soto makes a similarly clear statement about which latitude is used by the NED (and employs similar figures to illustrate the distinction). The relevant pages (85-88) are available on Google books.

Best Answer

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[GIS] locating latitude-longitude point in a piste-map with matrix transforms

NED Latitude – Does the NED Use Geocentric or Geodetic Latitude?

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