Find the horizon and meridian angles

1. Connect the equator to the ecliptic

Picture two circles crossing on a sphere: the equator and the tilted ecliptic. The local meridian cuts them at generally different angle readings. In ordinary chart terminology the ecliptic meridian angle is the midheaven, often shortened to MC; the horizon intersection associated with rising is the Ascendant. We will calculate the particular branches returned by this engine rather than assuming every possible location has the same geometry.

An angle can be recovered from two component values. On a unit circle, cosine supplies the horizontal component x and sine supplies the vertical component y. The function atan2(y, x) returns the direction of that pair, retaining which quadrant contains it. For example, pairs (x, y) = (1, 1) and (−1, −1) have the same ratio y/x, but point at 45° and −135°. A single-argument inverse tangent of the ratio would lose this distinction.

Tangent, written tan, is sin divided by cos. It enters the horizon formula because tilting the observer’s horizon depends on latitude. The source formulas are below. α is the local meridian right ascension, ε the true obliquity and φ the geographic latitude. Each input is converted from degrees to radians before a trigonometric function; each atan2 result is converted back to degrees before wrapping.

To see why MC needs the tilt, take an ecliptic direction at longitude λ. Its equatorial components are x = cos λ and y = sin λ × cos ε, because the ecliptic’s sideways component projects onto the equator by cos ε. The meridian angle therefore satisfies tan α = tan λ × cos ε. Recovering λ gives the MC component pair below; atan2 retains the branch without dividing by a possibly zero component.

MC = wrap(atan2(sin α, cos α × cos ε) × 180/π)

Asc = wrap(atan2(cos α, −(sin ε × tan φ + cos ε × sin α)) × 180/π)

Work through the invented α = 30°, ε = 30°, φ = 0°. For MC, y = sin 30° = 0.5 and x = cos 30° × cos 30° = 0.75. Thus atan2(0.5, 0.75) gives 33.690067526° after conversion. For Asc, y = cos 30° = 0.866025404 and x = −(0.5 × 0 + 0.866025404 × 0.5) = −0.433012702. A positive y with a negative x lies in the second quadrant; the result is 116.565051177°.

The latitude affects Asc but does not enter MC. At the same α and ε with φ = 30°, tan φ ≈ 0.577350269, making the Asc x component approximately −0.721687836 and giving Asc ≈ 129.805571092°. This shows why changing the observer’s latitude can change the horizon direction without changing the meridian longitude.

const wrap = (x: number): number => { const r = x % 360; return r < 0 ? r + 360 : r; };
const chartAngles = (ramcDegrees: number, latitudeDegrees: number, obliquityDegrees: number) => {
  const r = Math.PI / 180;
  const a = ramcDegrees * r, phi = latitudeDegrees * r, e = obliquityDegrees * r;
  return {
    midheaven: wrap(Math.atan2(Math.sin(a), Math.cos(a) * Math.cos(e)) / r),
    ascendant: wrap(Math.atan2(Math.cos(a), -(Math.sin(e) * Math.tan(phi) + Math.cos(e) * Math.sin(a))) / r)
  };
};

2. Find the other ends of the two lines

A line through the sphere reaches the ecliptic on two opposite sides. Once the engine has chosen Asc and MC, it obtains the paired Descendant and imum coeli by adding half a turn. Imum coeli is commonly shortened to IC. These names label chart directions; they do not introduce new sampled bodies.

Descendant = wrap(Asc + 180°); IC = wrap(MC + 180°)

With invented Asc = 350° and MC = 80°, the Descendant is wrap(530°) = 170°, and IC is 260°. angle_longitudes returns [350°, 80°, 170°, 260°] in the order Asc, MC, Descendant, IC. The ordering preserves the labels when the next step zips the angles with their names.

Do not assume the four entries divide the circle into four equal pieces. Opposite pairs are separated by 180°, but Asc and MC need not be 90° apart. In our earlier nontrivial example, Asc − MC is about 82.875°, even though each opposite pair still spans exactly half a circle. House conventions decide how to subdivide intervals later.

const wrap = (x: number): number => { const r = x % 360; return r < 0 ? r + 360 : r; };
const oppositeAngles = (ascendant: number, midheaven: number): number[] =>
  [ascendant, midheaven, wrap(ascendant + 180), wrap(midheaven + 180)];

3. Append named angle records

The fifteen-entry body list from lesson 15 is now ready to receive the four angles. calculate_chart pairs the four longitudes with AscendantSymbol, Midheaven, Descendant and ImumCoeli in that order. Each becomes a ChartBody with its wire name and longitude, plus speed = 0 and declination = 0.

For invented Asc = 350°, the first appended record therefore has longitude 350°, speed 0°/day and declination 0°. These zeros are placeholders in a shared output shape. They do not report a measured stationary Ascendant or a physical direction lying on the equator. No centered speed calculation or declination transformation is performed for these four records.

15 body-list entries + 4 angle records = 19 entries in calculate_chart bodies

The resulting nineteen-entry list and the optional house-cusp list are returned as separate fields. The house calculation does not replace the four angles with twelve cusp records. Keeping those two outputs separate lets the next lessons teach house conventions without changing what these angle entries mean.