Circle On Sphere

Construct a parametrization of the circle $d$ in $3$ stages of increasing generality.

1. Suppose that $B$ is the north pole (colatitude $\varphi = 0$).

The circle is the parallel (line of constant colatitude $\varphi = \alpha$), which has $z = \cos \alpha$ and $r = \sin \alpha$; hence, it is parametrized in rectangular coordinates by $t \mapsto \big( x(t), y(t), z(t) \big)$, where $$ \left\{ \begin{align} x &= \sin \alpha \cos t \\ y &= \sin \alpha \sin t \\ z &= \cos \alpha \end{align} \right. \qquad \text{for } 0 \le t < 2\pi. $$

2. Suppose that $B$ is on the prime meridian (longitude $\theta = 0$) but colatitude has some value $\varphi = \beta$, where $0 < \beta \le \pi$.

We take the coordinates $(x, y, z)$ of the circle with center at the north pole (from 1.) and rotate them through an angle $\beta$ along the great circle that includes the prime meridian. Using standard formulas for rotating coordinates, we have $$ \begin{bmatrix} x \\ y \\ z \end{bmatrix} \mapsto \begin{bmatrix} x \cos \beta + z \sin \beta \\ y \\ -x \sin \beta + z \cos \beta \end{bmatrix}. $$

3. Suppose that $B$ is anywhere on the sphere with spherical coordinates $(\varphi, \theta) = (\beta, \gamma)$. Take the resulting coordinates from 2. and rotate them about the polar axis of the sphere through an angle of $\gamma$: $$ \begin{bmatrix} x \\ y \\ z \end{bmatrix} \mapsto \begin{bmatrix} x \cos \gamma - y \sin \gamma \\ x \sin \gamma + y \cos \gamma \\ z \end{bmatrix}. $$

Putting these together, the general case for a circle $d$ with center at $(\varphi, \theta) = (\beta, \gamma)$ and central angle $\alpha$ (that determines the radius) has rectangular coordinates: $$ \left\{ \begin{align} x &= \phantom{-}( \sin \alpha \cos \beta \cos \gamma ) \cos t + ( \sin \alpha \sin \gamma ) \sin t - (\cos \alpha \sin \beta \cos \gamma ) \\ y &= -( \sin \alpha \cos \beta \sin \gamma ) \cos t + ( \sin \alpha \cos \gamma ) \sin t + (\cos \alpha \sin \beta \sin \gamma ) \\ z &= \phantom{-}( \sin \alpha \sin \beta ) \cos t + \cos \alpha \cos \beta. \end{align} \right. $$