Angle at $P$ = hour angle $H$ (for upper culmination). Angle at $Z$ = $360^\circ - A$ if azimuth measured from north westward, but conventionally we use $A$ measured from north eastward. We adopt: Angle at Z = $A$ (azimuth) only after careful quadrant check.
On a unit sphere, a spherical triangle has sides (arc lengths in radians) $a, b, c$ and opposite angles $A, B, C$. The fundamental laws: spherical astronomy problems and solutions
In this article, we will discuss some common problems and solutions in spherical astronomy. We will cover topics such as celestial coordinates, time and date, parallax and distance, and orbital mechanics. Angle at $P$ = hour angle $H$ (for upper culmination)
from equatorial via rotation matrix $R$ (latitude $\phi$): Rotation about $y$-axis by $90^\circ - \phi$: $$\beginpmatrix \cos a \cos A \ \cos a \sin A \ \sin a \endpmatrix = \beginpmatrix \sin\phi & 0 & -\cos\phi \ 0 & 1 & 0 \ \cos\phi & 0 & \sin\phi \endpmatrix \beginpmatrix \cos\delta \cos H \ \cos\delta \sin H \ \sin\delta \endpmatrix$$ On a unit sphere, a spherical triangle has
Two points on Earth (or celestial sphere) with coordinates $(\phi_1, \lambda_1)$ and $(\phi_2, \lambda_2)$ (latitude/longitude). Find: Angular distance $\sigma$ (great circle arc) and initial azimuth $\alpha_1$.
Calculate the shortest distance between Ljubljana ( ) and Rio de Janeiro ( ). Use Earth radius Step 1: Find the Angular Separation ( ) Using the Cosine Formula for distance :
Standard flat-plane geometry (the Pythagorean theorem) fails here because the "sky" is curved. Astronomers use a spherical distance formula: