All positional astronomy begins with the celestial sphere — an imaginary sphere of infinite radius centred on the observer. Every star, planet, and point is projected onto this sphere, providing the coordinate framework for measuring directions in the sky.
The meridian and horizon intersect the ecliptic at four points that define every chart:
As Earth rotates, every celestial body traces a diurnal circle parallel to the equator. The hour angle measures how far a body has traveled past the meridian:
HA = (RAMC − RA) mod 360°HA = 0° at the MC (body culminating), increasing westward. HA = 180° at the IC. The four quadrants are defined by HA ranges:
The Polich & Page Topocentric system (1964) defines house cusps by linearly interpolating an astronomical quantity called the pole from the meridian (pole = 0°) to the horizon (pole = observer's latitude φ). This produces a house division tied to the observer's actual position on Earth's surface.
Instead of dividing time (Placidus) or projecting from a reference circle (Regiomontanus, Campanus), the topocentric system divides the pole linearly. Each quadrant is trisected by computing intermediate poles:
polen = arctan(tan(φ) × n/3)where n = 1, 2, 3 counts the cusps from the meridian toward the horizon. At n=0 (MC/IC) the pole is 0°. At n=3 (ASC/DSC) the pole equals φ.
Once the pole for each cusp is known, the ecliptic degree on that cusp is found by solving for the ecliptic longitude whose oblique ascension (under the cusp's pole) equals the required RAMC offset. The four cardinal cusps (MC, IC, ASC, DSC) are computed by standard spherical trigonometry; the eight intermediate cusps use the interpolated poles.
At low latitudes (φ < 10°), topocentric cusps are virtually identical to Placidus. The systems diverge most above φ = 50°, where pole variation across houses becomes substantial.Every house cusp and every planet in the topocentric system has a pole — the geographic latitude at which that point's ecliptic degree would rise exactly on the horizon. The pole is the bridge between the equatorial and local-horizon coordinate frames.
The 12 cusp poles follow a symmetric pattern. Starting from the MC (pole = 0°) and moving toward the ASC (pole = φ):
Cusp 11: pole = arctan(tan(φ) × 1/3) Cusp 12: pole = arctan(tan(φ) × 2/3) ASC: pole = arctan(tan(φ) × 3/3) = φThe same pattern applies to cusps 2 and 3 (from ASC back toward IC), and mirrors in the western quadrants for cusps 5, 6, 8, and 9.
A planet inherits a pole from its DMO position by the same interpolation. If a planet's DMO places it exactly on a cusp, it gets that cusp's pole. Between cusps, the pole is interpolated from the planet's fractional position within its 30° house segment.
Planet pole = arctan(tan(φ) × DMO / 90°)The pole determines how a planet's OA and OD are computed, which in turn determines all equatorial aspects and primary directions. Two planets at the same ecliptic longitude but different DMO positions will have different poles, different OA/OD values, and different aspect patterns. This is the core distinction from Placidus, where all bodies use the observer's latitude φ directly.
The pole concept makes the topocentric system inherently three-dimensional. Each planet occupies not just a position on the ecliptic or equator, but a specific mundane location that determines its geometric relationship to the observer.Every celestial body traces a diurnal circle as Earth rotates — a small circle parallel to the equator. Geometrically, this path forms the surface of a cone whose apex is at the celestial pole and whose half-angle equals the body's co-declination (90° − |δ|).
The horizon plane slices through the cone, creating two arcs:
Each half-arc, measured from the horizon to the meridian, is a semi-arc:
Ascensional Difference: AD = arcsin(tan(δ) × tan(φ)) Diurnal Semi-Arc: DSA = 90° + AD Nocturnal Semi-Arc: NSA = 90° − ADThe meridian distance (MD) measures how far a body has traveled from the nearer meridian (MC or IC), in equatorial degrees:
MD = angular distance from nearest meridian along the equatorMD ranges from 0° (on the meridian) to the relevant semi-arc (on the horizon). The ratio MD/SA is the fundamental input to the DMO calculation.
When |δ| > 90° − |φ|, the body is circumpolar — its cone never intersects the horizon. The system handles this by treating the full 360° diurnal circle as a single extended arc.
A planet with positive declination at a northern latitude has DSA > 90° (longer visible arc). At the equator (φ = 0°), all semi-arcs are exactly 90° and the cone geometry is symmetric.Oblique Ascension (OA) and Oblique Descension (OD) are the equatorial degrees that rise or set at the horizon simultaneously with a given celestial body, computed under that body's own topocentric pole. They are the fundamental values for all equatorial aspect computations.
A planet above the horizon uses its OA (the equatorial degree rising when the planet rises). A planet below uses its OD (the equatorial degree setting when the planet sets). The choice depends on which semi-arc the planet currently occupies.
Two planets whose OA (or OD) values differ by an exact harmonic angle (0°, 60°, 90°, 120°, 180°) are in equatorial aspect. Unlike ecliptical aspects (which compare zodiacal longitudes), equatorial aspects reflect actual geometric relationships on the celestial sphere as seen from the observer's specific latitude.
Because OA/OD incorporate both the planet's position and its topocentric pole, two charts cast for different latitudes will show different equatorial aspects for the same planets. This is astronomically correct and astrologically meaningful.
The distinction between OA and OD is critical: OA measures the horizon interaction at the rising point; OD at the setting point. Confusing them produces incorrect aspect values.The speculum is the tabular display of every planet's complete positional data in the topocentric framework. It is the primary data output of the system, from which all aspects and directions are derived.
The speculum makes explicit how each planet relates to the local frame. A planet with DMO near 0° is angular (near MC or IC); near 90° it is on the horizon. The pole value shows the effective latitude used for that planet's OA/OD computation. Cross-referencing DMO and pole with the aspect table reveals the geometric basis of every contact.
The speculum is computed simultaneously under four RAMC values (Radix, Temporal, Local, Full), producing four parallel data sets for the same planets. This is the foundation of the transposition method.The Distance from Meridian in Oblique Ascension (DMO) is the core measurement of the topocentric system. It expresses every planet's mundane position as a single number in the range 0°–90°, regardless of declination or observer latitude.
where MD is the meridian distance and SA is the relevant semi-arc (DSA if above horizon, NSA if below).
Two planets with the same DMO are mundanely parallel — they occupy the same proportional position in their respective quadrants, even if their ecliptic longitudes, RAs, and declinations differ completely. DMO creates a universal positional measure that transcends each planet's individual diurnal arc geometry.
Sun at RA = 83.60°, δ = +23.32°; observer at φ = 40.72°N, RAMC = 9.51°:
HA = (9.51 − 83.60 + 360) mod 360 = 285.91° → Q1 (above, east) AD = arcsin(tan(23.32°) × tan(40.72°)) = 21.77° DSA = 90° + 21.77° = 111.77° MD = 360° − 285.91° = 74.09° DMO = (74.09 / 111.77) × 90 = 59.66°The Sun is at DMO 59.66° in Q1 — deep in House 11, about two-thirds from MC to ASC.
The Polich & Page system recognizes multiple classes of aspect, each computed in a different coordinate space. Together they form a multi-layered picture of planetary relationships.
Compare OA values (both above), OD values (both below), or cross-compare OA/OD (different hemispheres):
Equatorial aspect: |OAA − OAB| = harmonic angle (0°, 60°, 90°, 120°, 180°)Compare DMO values directly. Since DMO already normalizes for declination and latitude, these aspects are universal:
Mundane aspect: |DMOA − DMOB| = harmonic fraction of 90°Two planets in adjacent quadrants whose DMO values sum to a harmonic angle — the topocentric equivalent of antiscia:
Symmetric aspect: DMOA + DMOB = harmonic angle (adjacent quadrants)Equal DMO values regardless of quadrant — same proportional position in their respective semi-arcs.
Standard zodiacal longitude comparison, included for completeness and cross-reference with the equatorial system.
where h is the harmonic number. A soft power law that tapers from 25 arcminutes at H1 to approximately 6 arcminutes at H10. The master orb slider scales all values proportionally.
The dodekatemorion (12th-part) is a fractal subdivision technique from Hellenistic astrology, adapted here to the topocentric mundane framework. Each 30° house is subdivided into twelve 2.5° segments that cycle through the zodiacal signs in diurnal order.
In the topocentric frame, dodekatemoria are computed in DMO space rather than ecliptic longitude. A planet at DMO 15° falls in the second decan of its house, and the specific dodekatemorion tells you which sign-quality (in diurnal order) colors that position. This creates a mundane micro-structure invisible to ecliptic-only analysis.
Toggle the DEC display to reveal the subdivision rings on the chart. They appear progressively as you zoom in, maintaining readability at each scale.
The dodekatemoria system follows the ancient method: the first 2.5° of each house corresponds to the sign on the cusp, the next 2.5° to the following sign, and so on through all twelve signs within each 30° house.Primary directions model the rotation of the celestial sphere after birth. They measure how far the RAMC must advance (or retreat) before one body (the promissor) reaches the mundane position of another (the significator). This equatorial arc is then converted to years of life.
The promissor is projected to the significator's pole (not its own pole), creating a true geometric meeting:
1. posS = RAS ± ADS(poleS) 2. aspect_point = posS ± aspect_angle 3. posP@S = RAP ± ADP(poleS) 4. Direction arc = aspect_point − posP@SThe transposition method is Polich & Page's technique for synthesizing primary directions with transits. Instead of listing isolated direction hits, the entire natal chart is rotated by shifting its RAMC, producing a new speculum for any moment or location.
At any moment, four RAMC values are simultaneously meaningful:
RAMCradix = natal RAMC (fixed at birth) RAMCtemporal = RAMCradix + solar_arc RAMClocal = current sidereal time at natal location RAMCfull = current sidereal time at current locationThe natal chart rotated forward by the solar arc (~1° RAMC per year of life). This is a continuous primary direction applied to the whole chart at once, producing directed house cusps and planet positions.
The current RAMC at the birthplace. This generates the transit speculum at the natal location — how transiting planets interact with the natal framework at the place of birth.
The current RAMC at the current location. The most complete picture: both temporal and geographic shifts applied simultaneously. Aspects between the radix and full specula reveal relocated directed-transit contacts.
The temporal RAMC can advance (direct) or retreat (converse). Both yield valid and distinct aspect patterns. Direct models the natural solar arc progression; converse reverses it.
The Trutina Hermetis (Scale of Hermes) is an ancient rectification technique linking the natal chart to a conception epoch. The topocentric system provides the geometric framework for testing epoch candidates.
The last New or Full Moon before birth. In Hellenistic doctrine, this syzygy establishes the releaser (apheta/hyleg) and defines the starting point for life-span calculations.
The rule states that the Moon's position at conception corresponds to a natal angle, and vice versa:
This creates testable epoch candidates separated by ~10 lunar months from birth.
The application generates candidate conception charts and tests them against known life events using primary directions in both the natal and epoch frames. Convergence between the two charts (natal directions and epoch directions pointing to the same events) provides evidence for the correct birth time.
Rectification is necessarily iterative. The Epoch tab generates candidates; the Rectify tab tests them against events using direction arcs. Convergence across multiple events narrows the birth time.The topocentric house system was developed by Wendel Polich and A.P. Nelson Page in Buenos Aires, Argentina, and published in 1964.
Polich and Page were part of a broader movement in Argentine astrology that emphasized mathematical rigor and empirical testing. Their system was designed to satisfy three criteria:
The two systems are often compared because their cusps are numerically similar at most latitudes. However, the theoretical foundations differ entirely:
They agree at the four angles and diverge at intermediate cusps, with divergence increasing at higher latitudes.
With the Swiss Ephemeris and modern computation, the topocentric system achieves sub-arcsecond precision in milliseconds. This application implements the complete Polich-Page framework: six direction systems, four arc-to-time keys, four RAMC transpositions, five aspect types, and the full progression hierarchy.
Based on the work of Wendel Polich & A.P. Nelson Page (Buenos Aires, 1964)
Built by Cameron Cassidy