Question

Given \( h_P = H_P + N_P \), where \( h_P \) is the ellipsoidal/geodetic height at point \( P \), \( H_P \) is the orthometric height and \( N_P \) is the geoid undulation along the ellipsoidal normal. Which one of the following statements is correct?

• A. Two points on the Earth’s surface that have the same ellipsoidal height will be on the same equipotential surface
• B. The geoid undulation is the separation of the equipotential surface at the ground surface with respect to the ellipsoid
• C. The points on an equipotential surface will not have the same orthometric height
• D. The orthometric height of the instantaneous sea level is zero

Hint:

Ellipsoidal height is geometric, orthometric height is based on gravity (above geoid), and geoid undulation is the difference between the two. Points on equipotential surfaces may have varying terrain heights.

Answer 1

The Correct Option is C

In geodesy, the ellipsoidal height \( h_P \) is related to the orthometric height \( H_P \) and geoid undulation \( N_P \) by:
\[ h_P = H_P + N_P \]

(A) is incorrect because two points can have the same ellipsoidal height but still lie on different equipotential surfaces (i.e., have different gravitational potential energy).

(B) is a misstatement. The geoid undulation \( N_P \) is the separation between the geoid (a particular equipotential surface) and the reference ellipsoid — not the ground surface.

(C) is correct because equipotential surfaces (like the geoid) are based on gravitational potential, and orthometric height is the height above the geoid. Thus, points on the same equipotential surface (e.g., the geoid) can be at different elevations with respect to Earth's surface and still have different orthometric heights.

(D) is false since the orthometric height of mean sea level is considered zero, not the instantaneous sea level which fluctuates.

\[ \boxed{\text{Therefore, option (C) is the correct statement.}} \]