By Farrell, Jay

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The actual Earth surface ﬂuctuates relative to the geoid. 8. The center of the ellipsoid is coincident with the center of mass of the Earth. The minor axis of the ellipse is coincident with the mean rotational axis of the Earth. 8, where φ denotes latitude, λ denotes longitude, and h denotes altitude above the reference ellipsoid. Latitude is the angle in the meridian plane from the equatorial plane to the ellipsoidal normal N . Note that the extension of the normal towards the interior of the ellipsoid will not intersect the center of the Earth except in special cases such as φ = 0 or ±90◦ .

When the relative orientation of two reference frames is known, the direction cosine matrix R12 is unique and known. Although the direction cosine matrix has nine elements, due to the three orthogonality constraints and the three normality constraints, there are only three degrees of freedom. 5 and Appendix D. Analogous to eqn. 13) v2 = I2 x2 + J2 y2 + K2 z2 where [v2 ]2 = [x2 , y2 , z2 ] Therefore, 1 [v2 ] ⎡ and by eqn. 14) ⎡ ⎤ I1 1 [v2 ] = ⎣ J1 ⎦ v2 . 4. REFERENCE FRAME TRANSFORMATIONS 39 2 = R12 [v2 ] .

Because each element of R12 is the cosine of the angle between one of the coordinate axes of φ1 and one of the coordinate axes of φ2 , the matrix R12 is referred to as a direction cosine matrix: ⎤ ⎡ cos(α1 ) cos(β1 ) cos(γ1 ) R12 = ⎣ cos(α2 ) cos(β2 ) cos(γ2 ) ⎦ . 11 depicts the angles αi for i = 1, 2, 3 that deﬁne the ﬁrst column of R12 . The βi and γi angles are deﬁned similarly. When the relative orientation of two reference frames is known, the direction cosine matrix R12 is unique and known.