Vertex and Anti-Vertex: How They Are Calculated

The vertex is a chart point from modern Western astrology rather than from jyotisha: no kundli on this site prints it, and it is not one of the four angles. It is easy to define and slightly awkward to compute. This page sets out the geometry, computes the point two independent ways and checks that they agree, works it through for the same imaginary birth used on the lagna and midheaven pages, and then follows it through one full turn of the sky at six Indian cities. The count turns up one fact worth knowing before trusting any vertex figure: its behaviour changes sharply at the latitude equal to the tilt of the Earth’s axis, about 23.4° N, which runs through India.

The Prime Vertical and Its Two Crossings

Three great circles are drawn around any birthplace. The horizon gives the ascendant and descendant where the ecliptic crosses it. The meridian, running north–south through the point overhead, gives the midheaven and imum coeli, as the midheaven page explains. The third is the prime vertical: the circle that runs from the due-east point of the horizon, up through the zenith, down to the due-west point and round underneath. It cuts the meridian at right angles.

The ecliptic crosses the prime vertical at two opposite points. The crossing on the western side is the vertex; the one on the eastern side, exactly 180° away, is the anti-vertex. Either may be above or below the horizon. In the census further down the vertex stands above the horizon for half of the sidereal day at every city and below it for the other half.

Two Ways to Compute It, Checked Against Each Other

By scanning. Take each ecliptic longitude in turn, convert it to right ascension and declination, and find where it stands in the sky for the given sidereal time and latitude. A point on the prime vertical is neither north nor south of the observer, so the site’s code walks round the ecliptic in half-degree steps until the north–south component changes sign, then halves the interval until the crossing is pinned down. Of the two crossings, the one with a westward component is the vertex.

By a shortcut. The ascendant formula from the lagna-by-hand page can be reused with two inputs changed: put in the co-latitude (90° minus the birthplace’s latitude) instead of the latitude, and the local sidereal time plus 180° instead of the sidereal time. The “ascendant” that comes out is the vertex. This was not taken on trust: several variants were tested against the scan, and only this one matched. The version without the 180° turn, for instance, came out as much as 179.8° wrong.

Over 2,756 test cases, every 2.5° of latitude from 5° to 35° N and sidereal times every 1.7° round the circle, the scan and the shortcut agree to better than a millionth of a degree. The anti-vertex needs no separate calculation: add 180° to the vertex.

At the Equator, the Vertex Sits on an Equinox

At the equator the point overhead lies on the celestial equator, and so do the due-east and due-west points. The prime vertical there is the celestial equator itself, and the ecliptic meets the celestial equator only at the two equinox points, 0° Aries and 0° Libra in the tropical zodiac. So at the equator the vertex is always one of those two degrees, whatever the time. The code confirms it: at a latitude of 0.01°, across the whole sidereal day, the vertex never strays more than 0.03° from an equinox point. Moving away from the equator tilts the prime vertical away from the celestial equator, and the vertex starts to swing.

Worked Example: The 1992 Delhi Birth

The birth is 09:20 IST on 8 July 1992 at Delhi (28.61° N), the imaginary chart of the lagna and midheaven pages. Its local sidereal time is 61.0600°, so the shortcut uses a sidereal time of 241.0600° and a co-latitude of 61.39°. The tropical vertex comes out at 8°40′ Capricorn. Taking off the Lahiri ayanamsa of 23.7523° gives 14°54′ Sagittarius in the sidereal zodiac, so the tropical and sidereal vertex fall in different signs, just as the two forms of the lagna do.

Counted as whole signs from the Leo lagna, the vertex is in the 5th sign, not the 7th, and the anti-vertex in the 11th. Along the zodiac the vertex lies 55.9° behind the descendant, more than a whole sign away from it, and at the moment of birth it stands 55.2° below the western horizon. Tropically it is 81.3° from the nearer equinox point.

Six Cities Through One Sidereal Day

The census follows the sky through one sidereal day from 1 January 2026 at six cities on the site’s list, from Thiruvananthapuram, the southernmost, to Jammu, the northernmost. It samples 720 evenly spaced instants and computes the vertex at each by both methods (they agree throughout). The house columns are shares of the day, counted as whole signs from the sidereal lagna; they describe clock time, not people.

Because the vertex is the western crossing, it is tempting to picture it sitting beside the descendant; the count shows how loose that picture is. It occupies the 7th sign, the descendant’s own, for no more than 15% of the day at any of the six cities, and at none of them is the 7th its most frequent sign. At Delhi it spreads across the 5th to the 9th sign and comes as far as 57.4° from the descendant; at Jammu the limit is a little tighter, 50.1°. In the whole sample it never came a full right angle from the descendant.

The Line at 23.4° North

The six cities fall into two groups, and the dividing latitude is the obliquity, 23.44° in the census year. South of it lie Thiruvananthapuram, Chennai, Mumbai and Kolkata. There the Sun’s path can pass straight overhead, and when the ecliptic runs through the zenith it meets the prime vertical right there, so the vertex climbs to the point overhead (or sinks to the point underfoot) and sits a quarter-circle from the horizon. At each of these four cities it came within a degree of the zenith or nadir and within a degree of 90° from the descendant, and it reached the 4th and the 10th signs from the lagna for part of the day.

North of the line, at Delhi and Jammu, the ecliptic never reaches the zenith, the vertex never rose or sank more than 56.2° from the horizon, and it never left the 5th to 9th signs. The tropical swing does the opposite: it widens going north. At Thiruvananthapuram the vertex stays within 21.9° of an equinox point, still less than a sign; the reach grows city by city towards the line, and beyond it the vertex reaches the solstice points; at both northern cities it fell in each of the twelve tropical signs at some point of the census day. A tropical vertex worked out for a birth at Thiruvananthapuram and one for a birth at Jammu are therefore not drawn from the same range.

Finding Your Own Vertex

The site has no vertex calculator, and none of its tools prints the point. It can be done by hand with the shortcut above. Work out the local sidereal time of the birth exactly as the lagna page does, add 180°, and feed that into the same ascendant formula with the co-latitude in place of the latitude; the answer is the tropical vertex. Subtract the ayanamsa for the sidereal figure, and count its sign from your lagna. For the lagna itself, the ascendant calculator gives the sign and degree with the tropical value and the ayanamsa beside it. How whole-sign counting works is set out in the guide to the 12 houses and on the houses pages, and the part of fortune is another Western derived point worked through by hand on this site.

What This Page Does Not Give

Writing on the vertex mostly concerns fated meetings and what the point means in a given sign or house. Nothing of that kind appears here. The geometry fixes the degree; it offers no test of any reading built on that degree, so the page stops at the degree.

In Brief

• Definition: the vertex is the western crossing of the ecliptic and the prime vertical; the anti-vertex is the eastern one, 180° away
• Shortcut: the ascendant formula with the co-latitude and the sidereal time plus 180°, matched against a numerical scan over 2,756 cases
• Equator: the vertex is always an equinox point there
• Worked birth: vertex 14°54′ Sagittarius sidereal (8°40′ Capricorn tropical), the 5th sign from a Leo lagna, 55.9° from the descendant
• Near the descendant? in the 7th sign for at most 15% of the day at the six cities
• The 23.4° line: south of it the vertex can reach the zenith and the 4th or 10th sign; north of it, neither