Convert geocentric latitude to geodetic latitude
The Geocentric to Geodetic Latitude block converts a geocentric latitude (λ) into geodetic latitude (μ). There are a number of geometric relationships that are used to calculate the geodetic latitude in this noniterative method. A number of angles and points are involved in the calculation, which are shown in following figure.
Given geocentric latitude (λ) and the radius (r) from the center of the planet (O) to the center of gravity (P), this noniterative method starts by computing values for the point of r that intercepts the surface of the planet (S). By rearranging the equation for an ellipse, the horizontal coordinate, is determined. When equatorial radius (R), polar radius and , are substituted for semi-major axis, semi-minor axis and vertical coordinate , the resulting equation for has the following form:
To determine the geodetic latitude at S, the equation for an ellipse with equatorial radius (R), polar radius is used again. This time it is used to define in terms of .
Additionally, the relationship between geocentric latitude at the planet's surface and geodetic latitude is used.
Using the relationship and the two equations above, the resulting equation for is obtained.
The correct sign of is determined by testing λ and if λ is less than zero changes sign accordingly.
In order to calculate the geodetic latitude of P, a number of geometric relationships are required to be calculated. These calculations follow.
The radius from the center of the planet (O) to the surface of the planet (S) is calculated by using trigonometric relationship.
The distance from S to P is defined by:
The angular difference between geocentric latitude and geodetic latitude at S(δλ) is defined by:
Using and δλ, the segment TP or the mean sea-level altitude (h) is estimated.
The equation for the radius of curvature in the Meridian at is
Using , δλ, h, and , the angular difference between geodetic latitude at S and geodetic latitude at P is defined as:
Subtracting δμ from then gives μ.
Specifies the parameter and output units:
Radius from CG to Center of Planet
This option is only available when Planet
model is set to
Specifies the planet model to use:
Specifies the flattening of the planet. This option is only
available with Planet model set to
Specifies the radius of the planet at its equator. The units
of the equatorial radius parameter should be the same as the units
for radius. This option is only available with Planet
model set to
|Scalar||Contains the geocentric latitude, in degrees. Latitude values can be any value. However, values of +90 and -90 may return unexpected values because of singularity at the poles.|
|Scalar||Contains the radius from center of the planet to the center of gravity.|
|Scalar||Contains the geodetic latitude, in degrees.|
This implementation generates a geodetic latitude that lies between ±90 degrees.
Jackson, E. B., Manual for a Workstation-based Generic Flight Simulation Program (LaRCsim) Version 1.4, NASA TM 110164, April, 1995.
Hedgley, D. R., Jr., “An Exact Transformation from Geocentric to Geodetic Coordinates for Nonzero Altitudes,” NASA TR R-458, March, 1976.
Clynch, J. R., “Radius of the Earth - Radii Used in Geodesy,” Naval Postgraduate School, 2002, http://www.oc.nps.edu/oc2902w/geodesy/radiigeo.pdf.
Stevens, B. L., and F. L. Lewis, Aircraft Control and Simulation, John Wiley & Sons, New York, 1992.
Edwards, C. H., and D. E. Penny, Calculus and Analytical Geometry 2nd Edition, Prentice-Hall, Englewood Cliffs, New Jersey, 1986.