# Appendix I.4

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## Appendix I.4: Conversion Between Geodetic and Geocentric Latitude

Figure I.3-1 represents a cross section of the earth with polar radius rp and equatorial radius re. A feature is represented at height hf above Mean Sea Level. The height is measured along a line through the feature and perpendicular to the earth ellipsoid's surface. The distance of the feature from the earth's center is rf.

φgcf ≡ geocentric (or geometric) latitude of the feature.

-900 ≤φgcf ≤ 90 0. North is positive.

φgdf ≡ geodetic latitude of the feature.

-900 ≤φgdf ≤ 90 0. North is positive.

The boundary of the earth's cross section is assumed to be an ellipse formed by the equation

p>when the cross section in the xz plane.

A. Conversion from Geodetic to Geocentric Latitude

Suppose that φgdf is known. Then the slope of the line tangent to the earth's cross section at φgdf is

-tan(900 - φgdf) = 1 /tan(φgdf).

This slope is equal to the derivative dzs/dxs. From the equation for the earth's cross section,

So

or

where (xs, zs) is the point on the cross section boundary at geodetic latitude φgdf

Solving for xs gives

Substitution in the equation for the earth's cross section gives

or

Substitution of this back in the equation for xs gives

These equations for xs and zs can be put in the forms

(I-31)

where θf is the East longitude of the feature,

(I-32)

and

(I-33)

because xs and ys will always have the same signs as cos(θf) and sin(θf), respectively, and zs will always have the same sign as sin(φgdf). The coordinates (xf, yf, zf) of the feature can now be calculated.

and

Finally, so

(I-34)

and the distance to the feature from the earth's center is

B. Conversion from Geocentric to Geodetic Latitude

Assume that φgcf, the geocentric latitude of a feature is known. If the features geodetic latitude is to be found, its height above Mean Sea Level must also be known. The features height above Mean Sea Level can be calculated if its distance from the center of the earth is known.

Case 1 - hf the height of the feature above Mean Sea Level, is known

Since φgdf will not be known initially for substitution into the right side of the above equation, substitute a guess, starting with φ gdf ≈ φgcf, to compute a better guess. Interate until the guesses converge.

Case 2 - rf, the distance of the feature from the center of the earth, is known.

Since ,

Define

then , and this equation could be solved for hf if φgdf were known.

Since φgdf is not known initially, guess that φgdf ≈ φgcf, solve for the corresponding hf, use this as a guess height together with the guess φgdf is the same as the most recent guess. Then use φgdf to find a new guess for hf. Interate this procedure until both φgdf and hf converge to stable values.

Amended May 8, 2006