6

The formulas above assume that the angular velocity vector and the position vector are provided in Cartesian coordinates. However, usually they are specified in terms of latitude and longitude. Thus one must transform both vectors. The usual case is to calculate the relative velocity between two plates somewhere along their common boundary. Table 1 (next page) lists the pole position and rates of rotation for relative motion between plate pairs shown in the figure. The Cartesian position of a point along the plate boundary is

x = acosθ cosφ y = acosθ sinφ z = asinθ

(6)

where θ is latitude and φ is longitude. It is helpful to memorize the conversion from latitude longitude to the Cartesian co-ordinate system where the x-axis runs from the center of the earth, to a point at 0˚ latitude and 0˚ longitude (i.e. the Greenwich meridian), the y-axis runs through a point at 0˚ latitude and 90˚ east longitude and the z-axis runs along the spin-axis to the north pole.

S i m i l a r l y t h e p o l e p o s i t i o n s m u s t b e c o n v e r t e d f r o m g e o g r a p h i c c o - o r d i n a t e s ( θ p , φ into the Cartesian system p )

ω x = ω c o s θ p c o s φ p ω y = ω c o s θ p s i n ω_{z = ω sinθp }φ p

(7)

where |ω| is the magnitude of the rotation vector provided in Table 1. There are two ways to compute the magnitude of the velocity. One could compute the cross product of the rotation vector and the position vector (equation 2). Then the magnitude of the velocity is

v = v x 2 + v y 2 + v z 2 ( ) 1 / 2

(8)

A second approach is to calculate the angle Δ between the position vector and the angular velocity vector using equation 5 and then use that value in equation 4 to calculate the magnitude of the velocity. Indeed, both Fowler and Turcotte & Schubert use this second approach. However, they use the rather cumbersome spherical trigonometry to calculate the angle Δ. Since I don't remember the spherical trigonometry formulas, I prefer to use equation 4 above after converting everything to Cartesian coordinates.