# Inconsistent coordinates of centroid were found using the given value of radius of curvature in the prime vertical and in the meridian

I am currently writing a function on geodetic datum transformation in Hong Kong by referring to the Explanatory Notes on Geodetic Datums in Hong Kong by the Survey & Mapping Office of the Lands Department.

On p. C7,

upsilon_s = radius of curvature in the prime vertical = a / (1 - e^2 * (sin(phi))^2)^(1/2)

rho_s = radius of curvature in the meridian = a * (1 - e^2) / (1 - e^2 * (sin(phi))^2)^(3/2)

a = semi-major axis of the reference ellipsoid

e^2 = the first eccentricity of the reference ellipsoid

On p. C9, you can find the value of these parameters in the table "Parameters for Projection Formulae", and below the table, there is a note that

upsilon_s, rho_s and psi_s are parameters of a point near the centroid of Hong Kong and given to simplify the transformation.

What I am interested is to find the phi (the latitude) of the centroid, the default value of my function calculating the parameters.

Using the radius of curvature in the prime vertical = 6381480.500, I got phi = 0.38942 (in radian);

While using the radius of curvature in the meridian = 6359840.760, I obtained phi = 0.6638568 (in radian, corrected).

My calculation:

asin(sqrt((1 - (6378388 / 6381480.500)^2) / (2/297 - 1/297^2))) = 0.38942

asin(sqrt((1 - (6378388 * (1 - (2/297 - 1/297^2)) / 6359840.760)^(2/3)) / (2/297 - 1/297^2))) = 0.6638568

The difference is considerable. So, I am wondering if I mis-calculated the centroid or the wrong values are put in the document.

• That looks like a factor of 2 different, did you miss a divide by 2? – Ian Turton Mar 14 '16 at 9:01
• @iant Thanks for your response. There is no * 2 or / 2 in the formula, so I don't think I had miss a divide by 2; also, 0.38942 * 2 = 0.77884 that is not close to 0.78267 (the rounding error should not be that large). – kitman0804 Mar 14 '16 at 10:28
• I am having trouble determining what you are trying to calculate: no equation on p. C7 has anything to do with "calculating the centroid" of anything (in fact, the word "centroid" does not even appear on that page). Could you explain what your input values are, describe what you are attempting to calculate, and show examples of your calculations? – whuber Mar 14 '16 at 16:38
• @whuber The word "centroid" appears on p. C9 below the parameter table. And on p. C7, there are equations linking the radius of curvature in the prime vertical and in the meridian to "upsilon_s" and "rho_s" to "phi" (the latitude) of the centroid. What I have done is plugging in the parameter (a and e^2 in the parameter table) and solve for "phi_centroid". – kitman0804 Mar 15 '16 at 1:59
• @whuber Thanks. I have asked the land department, they said they would investigate why this discrepancy happen. – kitman0804 Mar 17 '16 at 15:51