One who wants to comprehend the art of astrology must be well acquainted with the geometry of the celestial sphere. But for this, one needs to remember the basics of trigonometry.
This article introduces you mathematical fundamentals required to understand the primary directions calculation.
The Problem
Imagine a star on the celestial sphere. What is the easiest way to describe its coordinates? An experienced astrologer would say that the star has two spherical coordinates:
- A degree on the Zodiac circle, called celestial longitude. It is the angular distance of a star from 0° Aries along the Zodiac circle in the succession of the signs.
- And perpendicular deviation from the zodiac circle's plane, known as celestial latitude.
But here's the problem. What if I want to find the coordinates of the same star at the equatorial plane? These coordinates are necessary for calculating the primary directions.
The equatorial plane doesn't coincide with the ecliptic one - it is known as the inclination of the ecliptic. How can I subtract or add to the degrees of celestial latitude and longitude to get the equatorial coordinates of the same star?
There needs to be more than just knowing the latitude and longitude to make such a conversion. So we will set forth a reliable mathematical apparatus for working with celestial coordinates.
It All Starts With Simple Proportions
For simplicity, let's consider the Sun on the zodiac circle. By definition, the Zodiac circle is the line along which the Sun moves during the year. The Sun has no celestial latitude.
Let's look at the Zodiac circle from above (from the side of the North Pole). But first,
- Let us denote the radius of the Zodiacal circle with $R$.
- Let's direct the $X$-axis to 0° Aries and the $Y$-axis to 0° Cancer.
What is the relationship between the $x$ and $y$ coordinates of the Sun?
Let's consider triangle $ABC$ with sides $x$, $y$, and $R$. If we drop the perpendicular from point $B$ to side $R$, we'll get two small triangles with the same angles, $lon$, and $\beta = 90°$.
Let's position our small triangles 1 and 2 inside our original triangle so that all three triangles' alpha angles be in the same corner.
You see that the triangles are similar. It means that the ratio of their sides is equal.
In particular:
$$\frac{x}{R} = \frac{r_2}{x}$$ $$\frac{y}{x} = \frac{r1}{y}$$
It means that
$$x^2 = r_2 R$$ $$y^2 = r_1 R$$
Since $r_1 + r_2 = R$, we have
$$\left( \frac{x}{R} \right)^2 + \left( \frac{y}{R} \right)^2 = 1$$
The radius of the zodiac circle $R$ is constant and does not depend on the movement of the Sun. Therefore, let's denote the $\grave{x}$ and $\grave{y}$ coordinates independent of the radius.
$$\grave{x} = x / R$$ $$\grave{y} = y / R$$ $$\left(\grave{x}\right)^2 + \left(\grave{y}\right)^2 = 1$$
Sine and Cosine
In this new coordinate system, the circle's radius always equals 1. If you look closely, you can see that the Sun's $\grave{y}$-coordinate resembles the strings of a drawn bow. The $\grave{X}$-axis here is an arrow pointing to the right.
It is logical to call the coordinate y a sine or sin (from the Sanskrit word "bowstring"). Then the $\grave{x}$-coordinate will be an integral companion of the sine, so let's call it the cosine or cos, that is, going along with the bow string.
Both coordinates $\grave{x}$ and $\grave{y}$ depend on the $lon$ - the position of the Sun on the zodiacal circle.
Therefore, we write it in the form:
$$\begin{align} \sin(lon) = x / R \tag{1} \\ \cos(lon) = y / R \\ \sin(lon)^2 + \cos(lon)^2 = 1 \end{align}$$
Now back to our star. It has a celestial latitude, which we will denote by the angle $lat$. The projection of the radius $R$ of the celestial sphere onto the $xy$ plane of the zodiac circle is $R \cos(lat)$.
From ($1$) it follows:
$$\begin{cases} x = R \cos(lon)\cos(lat) \\ y = R \sin(lon)\cos(lat) \\ z = R \sin(lat) \end{cases}\tag{2}$$
$$ \begin{cases} \sin(lat) = z/R \\ \tan(lon) = y/x \end{cases} $$
The set of equations ($2$) converts cartesian coordinates $(x, y, z)$ to spherical coordinates $({lon, lat})$.
With these formulas in hand, we can move on to the next step - the transition from one coordinate system to another.
The Bottom Line
You are now familiar with the following terms:
- sine,
- cosine and
- cartesian coordinates.
You also learned the formulas for converting cartesian coordinates to spherical ones. Now you are ready to convert one spherical system to another.
