The Technique of Calculating Indicator Lines in AstroCartoGraphy

KenanYasin
2 days ago
12 min read

Updated: 2 days ago

Indicators, just as in our natal chart studies, are examined within AstroCartoGraphy according to the same fundamental principles. First of all, we need to determine at what moment and from which location we are observing the sky. This is because the angular placement of an indicator on the world map depends not only on its degree in the zodiac, but also on the date, time, and geographical location.

We need the same data in order to observe the placements in our natal chart. In AstroCartoGraphy, the process brings other coordinate systems into play a little more, but the basic logic still operates through three important pieces of data: date, time, and geographical location.

For this reason, let us first define these three basic data points.

1. Determining the time

First, we need to know the historical moment of the birth chart we are working with. The period of the year in which we were born (whether, for example, it was winter or summer) is important in terms of the Sun’s annual movement and the general arrangement of the sky. What matters here is the astronomical time corresponding to that date. This is because the positions of the transiting planets on the ecliptic can only be calculated correctly on the basis of a specific date.

2. Determining the hour

After that, the exact hour of that moment is required. Because the Earth is constantly rotating around its axis, even a difference of a few hours changes which planet is rising at which longitude, reaching the Midheaven, or setting. In other words, at different hours on the same day, the same transiting planet becomes angular on different meridians. For a natal position, this stage is essential.

3. Determining the observation location

Even more importantly, we need to know according to which geographical reference point the calculation will be made. In other words, a geographic latitude and geographic longitude must be determined. This is because we observe the sky from a specific place. When we change location, the horizon line, the meridian structure, and therefore the Ascendant and the house cusps also change. For this reason, an observation location is absolutely necessary in order to project the transits onto the map.

Once these three data points have been determined, we can convert celestial coordinates into terrestrial coordinates. This is where the mathematical foundation of AstroCartoGraphy begins. We now have a specific date, a specific time, a geographical location, and the ecliptic longitude or degree of the transiting planet belonging to that moment.

The next step is to transform the planet’s celestial position into angular lines on the Earth.

Since we will be bringing various astronomical and mathematical concepts into the process during this transformation, let us present the terms we will use here in a table.

Concept	Explanation
Planetary Degree	The ecliptic position of the natal planet on the Zodiac
Time	The local time, universal time, or sidereal time used in the calculation
Geographic Latitude (φ)	The north-south coordinate of the place of observation
Geographic Longitude (L)	The east-west coordinate of the place of observation
Right Ascension (RA)	The planet’s right ascension in the equatorial system; that is, its position on the celestial equator
Declination	The planet’s northward or southward deviation from the celestial equator
Oblique Ascension (OA)	The degree at which a planet rises obliquely from the horizon at a given latitude; especially important in Ascendant calculations
Local Sidereal Time (LST)	The sidereal time of that moment at a specific location; used in determining the meridian and house cusps
Culminating (MC)	The planet’s reaching its highest point on the local meridian
Anti-Culminating (IC)	The planet’s being at its lowest point on the opposite meridian
Rising (ASC)	The planet’s rising on the eastern horizon
Setting (DSC)	The planet’s setting on the western horizon

As can also be understood from the data in the table, we cannot calculate placements in AstroCartoGraphy without carrying out conversions between coordinate systems.

First, we convert the planet’s ecliptic degree, when necessary, into equatorial values such as RA and declination; then, by combining these data with the Earth’s rotation, local sidereal time, the latitude-longitude relationship, and the horizon-meridian system, we calculate where that planet is in a rising (ASC), setting (DSC), culminating (MC), or anti-culminating (IC) position.

In this way, we are able to transform the planet (no longer treated as an abstract degree) into specific geographical lines on the world map. The lines we see in AstroCartoGraphy are precisely the result of this entire astronomical conversion process.

The Mathematical Transformation of Indicators in AstroCartoGraphy

When we want to place a celestial body onto the world map, we must first have its celestial coordinates. For the celestial body we are observing, we may define the initial data as follows:

Ecliptic longitude: λ
Ecliptic latitude: β
Moment of observation: t
Geographic Latitude of the observation point: φ
Geographic Longitude of the observation point: L

In the first step, we convert the celestial body’s ecliptic coordinates into equatorial coordinates. This is because, in order to calculate its angular placements on the Earth, we generally need Right Ascension (RA) and Declination (δ).

Let us briefly explain why we need the equatorial coordinate system. First of all, the sign and degree placements of the planets that we express in our astrological charts (that is, their positions in the Zodiac) are expressed through ecliptic coordinates. In other words, by using ecliptic longitude (λ) and sometimes ecliptic latitude (β), we indicate where the body is located with respect to the plane of the ecliptic, which is the path followed by the Sun.

In astrology, what we call sign and degree is in fact largely the same thing. That is, when we say, “a planet is at such-and-such degree of such-and-such sign,” we are speaking in the ecliptic system.

For example, let us suppose that in our natal chart a planet is placed at 15° Leo. The expression 15° Leo is actually its ecliptic longitude on the Zodiac. When the point of 0° Aries is taken as the starting point, 15° Leo corresponds to the planet’s ecliptic longitude of 135° on the Zodiac.

Ecliptic Longitude Correspondences of the Zodiac Signs

Aries	Taurus	Gemini	Cancer	Leo	Virgo
0°–29°59′	30°–59°59′	60°–89°59′	90°–119°59′	120°–149°59′	150°–179°59′
Libra	Scorpio	Sagittarius	Capricorn	Aquarius	Pisces
180°–209°59′	210°–239°59′	240°–269°59′	270°–299°59′	300°–329°59′	330°–359°59′

In astrology, a planet’s sign and degree placement is technically expressed in terms of ecliptic longitude. In the table I prepared, you can see the degree range of each sign within the 360° circle of the Zodiac. For example, when we say 15° Leo, what we actually mean is that it corresponds to a total ecliptic longitude of 120° + 15° = 135°.

Ecliptic latitude, on the other hand, does not appear in the table, because latitude shows how far north or south of the ecliptic plane the body is located. In defining sign placements, we primarily rely on ecliptic longitude.

Ecliptic values are extremely valuable for astrological interpretation; however, they are not sufficient on their own. This is because, in AstroCartoGraphy, the question we ask is no longer, “Where is the planet in the Zodiac?” We begin asking a different question:

In which places on Earth is the indicator rising, setting, culminating, or located on the lower meridian?

In order to answer this question, we need a different coordinate system. Here, ecliptic coordinates alone are not enough. This is because concepts such as rising, setting, and culmination are related to the horizon, the meridian, the Earth’s rotation, and the celestial equator. Therefore, we must convert the celestial body into the equatorial coordinate system.

At this stage, two fundamental values are used:

Right Ascension (RA / α): This is the body’s east-west-like position along the celestial equator. In a sense, it may be thought of as a kind of celestial longitude measured along the celestial equator, but the reference plane is not the ecliptic; it is the celestial equator.
Declination (δ): This shows how far north or south the body is from the celestial equator. It may be thought of as a kind of celestial latitude, but again, the reference is the celestial equator.

When we convert the indicator’s position from the place where we observe it into the equatorial coordinate system, we begin calculating directly with RA and declination.

As its name suggests, the equatorial coordinate system is based on the celestial equator. In other words, let us imagine that the Earth’s equator is projected into the sky. More precisely, let us think of the sky as being divided by meridians, or hour circles, that intersect the celestial equator at right angles.

Now let us consider that the Earth rotates around its own axis. As the Earth turns, the sky appears to slide along these meridians. Accordingly, the celestial body we observe also appears to pass from one meridian to another as the Earth rotates. This passage is what gives us time.

This is precisely why, in the equatorial system, we make use of local sidereal time. Right Ascension (RA) shows the body’s position on the celestial equator, while local sidereal time tells us which part of the sky has come to the meridian of our location at that moment. In this way, we can calculate when a body reaches the Midheaven, when it rises, or when it sets.

For example, when we want to find the MC line of a body, we investigate the geographic longitudes at which its RA coincides with the local sidereal time. When we want to find the ASC / DSC line of a body, declination also enters the calculation, because the angle at which a body reaches the horizon depends on both the latitude of the observation point and the body’s declination.

Conversion from Ecliptic Coordinates to Equatorial Coordinates

We have now identified the coordinates. From this point onward, we will convert the celestial body, whose ecliptic coordinates we know, into equatorial coordinates. In order to do this, we need to know the angular difference between the ecliptic and the celestial equator. This is called the obliquity of the ecliptic (ε). In short, it represents the inclination between the two coordinate systems and appears in the transformation formulas. The ecliptic is inclined by approximately 23.4° relative to the celestial equator. When converting a celestial body’s ecliptic longitude and latitude values into RA (α) and declination (δ), we will use this inclination as the fundamental geometric difference.

Since we will express it here as a mathematical formula, let us define the obliquity of the ecliptic as ε. The formula we use in the coordinate transformation is as follows:

α=arctan⁡((sin⁡λcos⁡ε-tan⁡βsin⁡ε)/cos⁡λ )

δ=arcsin⁡(sin⁡βcos⁡ε+cos⁡βsin⁡εsin⁡λ)

Here:

α = Right Ascension

δ = Declination

ε = obliquity of the ecliptic

λ = ecliptic longitude

β = ecliptic latitude

When we process the data through the formula, we also determine the celestial body’s position relative to the celestial equator.

In the second step, we need to calculate the Local Sidereal Time (LST) for the moment of observation. Symbolically, we may express it as follows:

LST=GST+L

Here:

GST = Greenwich Sidereal Time (because it is the principal reference meridian)

L = the geographic longitude of the observation place

LST = local sidereal time

φ = the geographic latitude of the observation point

After finding the RA value, we need to understand exactly where the body is located in the sky relative to our position. This is because RA alone gives the body’s fixed position on the celestial equator; however, what matters for us is where it is relative to the local meridian at that moment.

For this, we calculate the hour angle. The hour angle shows how far east or west the body is from the local meridian. In this way, we can understand whether the body is approaching the Midheaven, whether it has already passed the meridian, or whether it is still rising.

Our formula for finding the hour angle of the celestial body is:

H=LST-α

Here, H shows how far east or west the body is relative to the meridian at the place of observation. The letter H is the universally used abbreviation for hour angle. As the hour angle, it expresses the body’s angular distance from the meridian of the observation place.

Once we have found the hour angle, we can move on to calculating the body’s apparent position in the local sky. In other words, as observers, we must determine where that celestial body is positioned relative to our own horizon. In this way, I can calculate the times at which the celestial body will rise and set on my own horizon, that is, the ASC / DSC points.

When finding the MC and IC lines, we mainly look at the body’s position relative to the local meridian; for this reason, RA, local sidereal time, and hour angle are often sufficient. However, in the case of the ASC and DSC lines, the issue is the body’s relationship to the horizon. Therefore, at this stage, we move into the local horizon coordinate system and calculate the body’s altitude (h).

Now we can calculate the body’s position in the horizon system. Our formula for altitude (h) is:

sin⁡h=sin⁡ϕsin⁡δ+cos⁡ϕcos⁡δcos⁡H

Here:

h = altitude

φ = the geographic latitude of the observation place

δ = the declination of the body

H = the hour angle

Formula Explanation
H ≈ 0	For a body to be on the MC, its hour angle must be approximately zero.
LST ≈ α	When this equality is satisfied, the celestial body is on the local meridian of that place and is at the point of culmination.
H ≈ 180°	For a body to be on the IC, its hour angle must be 180°.
H = 12h	Or, equivalently, we may express it as 12h; in this case, the body is on the opposite meridian and is in the position of anti-culmination.

Although the values seen in practical AstroCartoGraphy drawings may appear approximate because of map projection, rounding differences, or the calculation precision of the software, in the fundamental definition we take H = 0 for the MC and H = 180° for the IC.

Up to this point, we have calculated the values for culmination (MC) and anti-culmination (IC). However, the matter becomes slightly more complex for the Ascendant (ASC / rising) and Descendant (DSC / setting) lines. This is because here we must examine the intersection with the horizon. We will observe the indicators that rise and set on the horizon. If we think in terms of the horizon we observe from our own location -which we will call the local position- then, for a body to be exactly on the horizon, its altitude must be zero. In other words, our formula will be:

h=0

Now we will proceed with a new equation. In order to observe at which hour angle a celestial body rises or sets, we will need the latitude of our observation place. If we express geographic latitude by ϕ and declination by δ, then our formula is:

Since h = 0, then:

0=sin⁡ϕsin⁡δ+cos⁡ϕcos⁡δcos⁡H

From this, our formula becomes:

cos⁡H=-tan⁡ϕtan⁡δ

If a solution exists, then we obtain the ASC lines on the eastern horizon and the DSC lines on the western horizon. However, there is one more concept to consider. The concept of Oblique Ascension (OA) also comes into play. This is because a celestial body does not rise perpendicular to the equator; rather, it rises at an oblique angle depending on the geographic latitude at which we are located. For this reason, especially in the rising and setting lines, RA alone will not be sufficient. We must also take the effect of latitude into account.

Symbolically, we will evaluate the oblique ascension of a given body according to the following logic:

OA=f(α,δ,ϕ)

That is, oblique ascension appears as a function depending on the celestial body’s Right Ascension (α), declination (δ), and the geographic latitude of the observation place (ϕ).

As you will notice, the data we have obtained so far apply only to a single place of observation. In other words, we have examined them using the coordinates of only one city. In the final stage, however, we must repeat the calculations not for a single city, but for all suitable latitude-longitude points on Earth, so that in this way we may determine the placements of the celestial body’s other lines as well. Here too, you may follow the calculation of the placements through a short table.

table.

Line	At which longitudes / combinations	Condition
MC line	At which geographic longitudes	LST = α
IC line	At which geographic longitudes	LST = α + 180°
ASC line	At which geographic latitude-longitude combinations	h = 0 and the body is in the east
DSC line	At which geographic latitude-longitude combinations	h = 0 and the body is in the west

As can also be understood from the table, the celestial coordinates of a celestial body can be converted, through mathematical transformations, into angular lines on the Earth (MC, IC, ASC, DSC). In this way, we are also able to observe the body’s other lines.

Today, all of these complex procedures are calculated in the background by simple astrology software, and what remains for us as astrologers is to interpret the lines. Even so, I believe that it is not sound to make interpretations without understanding the logic of the calculation. We are no longer obliged to deal with all the trigonometric formulas ourselves, but it is still important to have at least some idea of how the stages of the calculation proceed.

Up to this point, I will briefly summarize the sequence of the operations we have carried out:

We take the celestial body’s ecliptic coordinates: ecliptic longitude (λ) and ecliptic latitude (β)
We convert these values into equatorial coordinates (RA, declination): α, δ
Using the moment of observation and the location data, we calculate the Local Sidereal Time: LST
We find the hour angle for rising and setting: H = LST - α
By using the horizon and meridian equations, we determine the places where the celestial body is located on the ASC, DSC, MC, and IC
We transform the points obtained into cartographic lines on the world map