The earth rotates uniformly once per 24 hours in this frame (mean solar time). Let a point on the earth's surface P define a unit vector at the origin of the frame. The azimuth angle of this vector is measured as usual in the X-Y plane from the X-axis, so in a day, covers the complete range of 0 to 360 degrees. Generally, time zones are chosen so that at some longitude in the zone, at noon. To simplify, let P have this longitude, at latitude . Then is a measure of time of day at P with respect to noon. If we can find where vector is at sunrise, then that value of will give us the time of sunrise.
The explicit relationship between the vector xyz components and angles and is
If P were in a different place in the time zone, i.e. P had a different longitude from that assumed above, a constant shift would occur in the times of sunrise, sunset, and sun's highest elevation.
We will take the latitude of the observer to be .