We are familiar with the antiscia as a point on the ecliptic, a mirror reflection of the planet relative to the 0° Cancer - 0° Capricorn axis.
The antiscia is the intersection of a circle parallel to the celestial equator drawn through the planet.
But the situation changes if the planet has a non-zero latitude on the ecliptic. Then the intersection of the circle drawn through the planet will give not one but two antiscias.
Moreover, in some cases, as shown in the figure below, there may not be such intersections at all, so a planet with a non-zero latitude may have from zero to two antiscias.
Antiscia Equation
Let's derive an equation to find antiscia's longitude on the ecliptic. From eq. (1) of the ecliptic, it follows that
$$ \sin D = \sin\epsilon \sin\lambda $$
Here
- $D$ is the declination of the planet
- $\epsilon$ is the inclination of the ecliptic, and
- $\lambda$ is the celestial longitude of the planet
Since no right ascension is specified, this equation is true for any planet with a given $D$, including the antiscia.
The second antiscia's longitude is equal to $180° - \lambda$, i.e., we have
$$\begin{cases} \lambda_1 = \arcsin(\sin D / \sin\epsilon) \\ \lambda_2 = 180° - \lambda_1 \end{cases}$$
According to the equations of ecliptic-equator conversion, we can then convert these coordinates $(\lambda_i, 0)$ to equatorial $(RA_i, D)$ and use them as promittor's coordinate in primary direction..
