5.2. Eclipses, occultations, risings, settings, and other planetary phenomena
The Swiss Ephemeris also includes functions for many calculations concerning solar and lunar eclipses. You can:
 search for eclipses or occultations, globally or for a given geographical position
 compute global or local circumstances of eclipses or occultations
 compute the geographical position where an eclipse is central
Moreover, you can compute for all planets and asteroids:
 risings and settings (also for stars)
 midheaven and lower heaven transits (also for stars)
 height of a body above the horizon (refracted and true, also for stars)
 phase angle
 phase (illumined fraction of disc)
 elongation (angular distance between a planet and the sun)
 apparent diameter of a planetary disc
 apparent magnitude.

6. Sidereal Time, Ascendant, MC, Houses, Vertex
The Swiss Ephemeris package also includes a function that computes the Ascendant, the MC, the houses, the Vertex, and the Equatorial Ascendant (sometimes called "East Point").

Swiss Ephemeris versions until 1.80 used the IAU 1976 formula for Sidereal time. Since version 2.00 it uses sidereal time based on the IAU2006/2000 precession/nutation model.
As this solution is not good for the whole time range of JPL Ephemeris DE431, we only use it between 1850 and 2050. Outside this period, we use a solution based on the long term precession model Vondrak 2011, nutation IAU2000 and the mean motion of the Earth according to Simon & alii 1994. To make the function contiuous we add some constant values to our longterm function before 1850 and after 2050.
Vondrak/Capitaine/Wallace, "New precession expressions, valid for long time intervals", in A&A 534, A22(2011).
Simon & alii, "Precession formulae and mean elements for the Moon and the Planets", A&A 282 (1994), p. 675/678.

6.1. Astrological House Systems
The following house methods have been implemented so far:

6.1.1. Placidus
This system is named after the Italian monk Placidus de Titis (15901668). The cusps are defined by divisions of semidiurnal and seminocturnal arcs. The 11th cusp is the point on the ecliptic that has completed 2/3 of its semidiurnal arc, the 12th cusp the point that has completed 1/3 of it. The 2nd cusp has completed 2/3 of its seminocturnal arc, and the 3rd cusp 1/3.

6.1.2. Koch/GOH
This system is called after the German astrologer Walter Koch (18951970). Actually it was invented by Fiedrich Zanzinger and Heinz Specht, but it was made known by Walter Koch. In Germanspeaking countries, it is also called the "Geburtsorthäusersystem" (GOHS), i.e. the "Birth place house system". Walter Koch thought that this system was more related to the birth place than other systems, because all house cusps are computed with the "polar height of the birth place", which has the same value as the geographic latitude.
This argumentation shows actually a poor understanding of celestial geometry. With the Koch system, the house cusps are actually defined by horizon lines at different times. To calculate the cusps 11 and 12, one can take the time it took the MC degree to move from the horizon to the culmination, divide this time into three and see what ecliptic degree was on the horizon at the thirds. There is no reason why this procedure should be more related to the birth place than other house methods.

6.1.3. Regiomontanus
Named after the Johannes Müller (called "Regiomontanus", because he stemmed from Königsberg) (14361476).
The equator is divided into 12 equal parts and great circles are drawn through these divisions and the north and south points on the horizon. The intersection points of these circles with the ecliptic are the house cusps.

6.1.4. Campanus
Named after Giovanni di Campani (12331296).
The vertical great circle from east to west is divided into 12 equal parts and great circles are drawn through these divisions and the north and south points on the horizon. The intersection points of these circles with the ecliptic are the house cusps.

6.1.5. Equal Systems 
6.1.5.1. Equal houses from Ascendant
The zodiac is divided into 12 houses of 30 degrees each starting from the Ascendant.

The zodiac is divided into 12 houses of 30 degrees each starting from MC + 90 degrees.

6.1.5.3. Vehlowequal System
Equal houses with the Ascendant positioned in the middle of the first house.

6.1.5.4. Whole Sign houses
The first house starts at the beginning of the zodiac sign in which the ascendant is. Each house covers a complete sign. This method was used in Hellenistic astrology and is still used in Hindu astrology.

6.1.5.5. Whole Sign houses starting at 0° Aries
The first house starts at the beginning of Aries.

6.1.6. Porphyry Houses and Related House Systems 
6.1.5.1. Porphyry Houses
Each quadrants is trisected in three equal parts on the ecliptic.

6.1.5.2. Sripati Houses
This is a traditional Indian house system. In a first step, Porphyry houses are calculated. The cusps of each new house will be the midpoint between the last and the current. So house 1 will be equal to:
H1' = (H1  H12) / 2 + H12.
H2' = (H2  H1) / 2 + H1;
And so on.

6.1.5.3. Pullen SD (Sinusoidal Delta, also known as “NeoPorphyry”)
This house system was invented in 1994 by Walter Pullen, the author of the astrology software Astrolog. Like the Porphyry house system, this house system is based on the idea that the division of the houses must be relative to the ecliptic sections of the quadrants only, not relative to the equator or diurnal arcs. For this reason, Pullen originally called it “NeoPorphyry”. However, the sizes of the houses of a quadrant are not equal. Pullen describes it as follows:
“Like Porphyry except instead of simply trisecting quadrants, fit the house widths to a sine wave such that the 2nd/5th/8th/11th houses are expanded or compressed more based on the relative size of their quadrants.”
In practice, an ideal house size of 30° each is assumed, then half of the deviation of the quadrant from 90° is added to the middle house of the quadrant. The remaining half is bisected again into quarters, and a quarter is added to each of the remaining houses. Pullen himself puts it as follows:
"Sinusoidal Delta" (formerly "NeoPorphyry") Houses.
Asc 12th 11th MC 9th 8th 7th
      
+++++++ ^ ^ ^ ^ ^ ^
angle angle angle angle angle angle
x+n x x+n x+3n x+4n x+3n
In January 2016, in a discussion in the Swiss Ephemeris Yahoo Group, Alois Treindl criticised that Pullen’s code only worked as long as the quadrants were greater than 30°, whereas negative house sizes resulted for the middle house of quadrants smaller than 30°. It was agreed upon that in such cases the size of the house had to be set to 0.
https://groups.yahoo.com/neo/groups/swisseph/conversations/topics/5579
https://groups.yahoo.com/neo/groups/swisseph/conversations/topics/5606

6.1.5.4. Pullen SR (Sinusoidal Ratio)
On 24 Jan. 2016, during the abovementioned discussion in the Swiss Ephemeris Yahoo Group, Walter Pullen proposed a better solution of a sinusoidal division of the quadrants, which does not suffer from the same problem. He wrote:
“It's possible to do other than simply add sine wave offsets to houses (the "Sinusoidal Delta" house system above). Instead, let's proportion or ratio the entire house sizes themselves to each other based on the sine wave constants, or multiply instead of add. That results in using a "sinusoidal ratio" instead of a "sinusoidal delta", so this alternate method could be called "Sinusoidal Ratio houses". As before, let "x" be the smallest house in the compressed quadrant. There's a ratio "r" multiplied by it to get the slightly larger 10th and 12th houses. The value "r" starts out at 1.0 for 90 degree quadrants, and gradually increases as quadrant sizes differ. Houses in the large quadrant have "r" multiplied to "x" 3 times (or 4 times for the largest quadrant). That results in the (0r, 1r, 3r, 4r) distribution from the sine wave above. This is summarized in the chart below:”
"Sinusoidal Ratio" Houses.
Asc 12th 11th MC 9th 8th 7^{th}
      
+++++++
^ ^ ^ ^ ^ ^
angle angle angle angle angle angle
rx x rx (r^3)x (r^4)x (r^3)x
“The unique values for "r" and "x" can be computed based on the quadrant size "q", given the equations rx + x + rx = q, xr^3 + xr^4 + xr^3 = 180q.”
https://groups.yahoo.com/neo/groups/swisseph/conversations/topics/5579

6.1.7. Axial Rotation Systems 
6.1.7.1. Meridian System
The equator is partitioned into 12 equal parts starting from the ARMC. Then the ecliptic points are computed that have these divisions as their right ascension. Note: The ascendant is different from the 1^{st} house cusp.

6.1.7.2. Carter’s poliequatorial houses 
The equator is partitioned into 12 equal parts starting from the right ascension of the ascendant. Then the ecliptic points are computed that have these divisions as their right ascension. Note: The MC is different from the 10^{th} house cusp.

The prefix “poli“ might stand for “polar”. (Speculation by DK.)

Carter’s own words:

“...the houses are demarcated by circles passing through the celestial poles and dividing the equator into twelve equal arcs, the cusp of the 1st house passing through the ascendant. This system, therefore, agrees with the natural rotation of the heavens and also produces, as the Ptolemaic (equal) does not, distinctive cusps for each house....”

Charles Carter (1947, 2nd ed. 1978) Essays on the Foundations of Astrology. Theosophical Publishing House, London. p. 158159.

http://www.exeterastrologygroup.org.uk/2014/12/charlescartersforgottenhousesystem.html

6.1.8. The Morinus System
The equator is divided into 12 equal parts starting from the ARMC. The resulting 12 points on the equator are transformed into ecliptic coordinates. Note: The Ascendant is different from the 1^{st} cusp, and the MC is different from the 10^{th} cusp.

6.1.9. Horizontal system
The house cusps are defined by division of the horizon into 12 directions. The first house cusp is not identical with the Ascendant but is located precisely in the east.

6.1.10. The PolichPage (“topocentric”) system
This system was introduced in 1961 by Wendel Polich and A.P. Nelson Page. Its construction is rather abstract: The tangens of the polar height of the 11^{th} house is the tangens of the geo. latitude divided by 3. (2/3 of it are taken for the 12^{th} house cusp.) The philosophical reasons for this algorithm are obscure. Nor is this house system more “topocentric” (i.e. birthplacerelated) than any other house system. (c.f. the misunderstanding with the “birth place system”.) The “topocentric” house cusps are close to Placidus house cusps except for high geographical latitudes. It also works for latitudes beyond the polar circles, wherefore some consider it to be an improvement of the Placidus system. However, the striking philosophical idea behind Placidus, i.e. the division of diurnal and nocturnal arcs of points of the zodiac, is completely abandoned.

6.1.11. Alcabitus system
A method of house division which first appears with the Hellenistic astrologer Rhetorius (500 A.D.) but is named after Alcabitius, an Arabic astrologer, who lived in the 10th century A.D. This is the system used in a few remaining examples of ancient Greek horoscopes.
The MC and ASC are the 10th and 1st house cusps. The remaining cusps are determined by the trisection of the semidiurnal and seminocturnal arcs of the ascendant, measured on the celestial equator. The houses are formed by great circles that pass through these trisection points and the celestial north and south poles.

6.1.12. Gauquelin sectors
This is the “house” system used by the Gauquelins and their epigones and critics in statistical investigations in Astrology. Basically, it is identical with the Placidus house system, i.e. diurnal and nocturnal arcs of ecliptic points or planets are subdivided. There are a couple of differences, though.

Up to 36 “sectors” (or house cusps) are used instead of 12 houses.

The sectors are counted in clockwise direction.

There are socalled plus (+) and minus (–) zones. The plus zones are the sectors that turned out to be significant in statistical investigations, e.g. many top sportsmen turned out to have their Mars in a plus zone. The plus sectors are the sectors 36 – 3, 9 – 12, 19 – 21, 28 – 30.

More sophisticated algorithms are used to calculate the exact house position of a planet (see chapters 6.4 and 6.5 on house positions and Gauquelin sector positions of planets).

6.1.13. Krusinski/Pisa/Goelzer system
This house system was first published in 1994/1995 by three different authors independently of each other:
 by Bogdan Krusinski (Poland)
 by Milan Pisa (Czech Republic) under the name “Amphora house system”.
 by Georg Goelzer (Switzerland) under the name “IchKreisHäusersystem” (“ICircle house system”)..
Krusinski defines the house system as “… based on the great circle passing through ascendant and zenith. This circle is divided into 12 equal parts (1st cusp is ascendant, 10th cusp is zenith), then the resulting points are projected onto the ecliptic through meridian circles. The house cusps in space are halfcircles perpendicular to the equator and running from the north to the south celestial pole through the resulting cusp points on the house circle. The points where they cross the ecliptic mark the ecliptic house cusps.” (Krusinski, email to Dieter Koch)
It may seem hard to believe that three persons could have discovered the same house system at almost the same time. But apparently this is what happened. Some more details are given here, in order to refute wrong accusations from the Czech side against Krusinski of having “stolen” the house system.
Out of the documents that Milan Pisa sent to Dieter Koch, the following are to be mentioned: Private correspondence from 1994 and 1995 on the house system between Pisa and German astrologers Böer and SchubertWeller, two typewritten (apparently unpublished) treatises in German on the house system dated from 1994. A printed booklet of 16 pages in Czech from 1997 on the theory of the house system (“Algoritmy noveho systemu astrologickych domu”). House tables computed by Michael Cifka for the geographical latitude of Prague, copyrighted from 1996. The house system was included in the Czech astrology software Astrolog v. 3.2 (APAS) in 1998. Pisa’s first publication on the house system happened in spring 1997 in “Konstelace“ No. 22, the periodical of the Czech Astrological Society.
Bogdan Krusinski first published the house system at an astrological congress in Poland, the “"XIV Szkola Astrologii Humanistycznej" held by Dr Leszek Weres, which took place between 15.08.1995 and 28.08.1995 in Pogorzelica at cost of the Baltic Sea.” Since then Krusinski has distributed printed house tables for the Polish geographical latitudes 4955° and floppy disks with house tables for latitudes 090°. In 1996, he offered his program code to Astrodienst for implementation of this house system into Astrodienst’s then astronomical software Placalc. (At that time, however, Astrodienst was not interested in it.) In May 1997, Krusinski put the data on the web at http://www.ci.uw.edu.pl/~bogdan/astrol.htm (now at http://www.astrologia.pl/main/domy.html) In February 2006 he sent SwissEphemeriscompatible program code in C for this house system to Astrodienst. This code was included into Swiss Ephemeris Version 1.70 and released on 8 March 2006.
Georg Goelzer describes the same house system in his book “Der IchKosmos”, which appeared in July 1995 at Dornach, Switzerland (Goetheanum). The book has a second volume with house tables according to the house method under discussion. The house tables were created by Ulrich Leyde. Goelzer also uses a house calculation programme which has the time stamp “9 April 1993” and produces the same house cusps.
By March 2006, when the house system was included in the Swiss Ephemeris under the name of “Krusinski houses”, neither Krusinski nor Astrodienst knew about the works of Pisa and Goelzer. Goelzer heard of his codiscoverers only in 2012 and then contacted Astrodienst.
Conclusion: It seems that the house system was first “discovered” and published by Goelzer, but that Pisa and Krusinski also “discovered” it independently. We do not consider this a great miracle because the number of possible house constructions is quite limited.

6.1.14. APC house system
This house system was introduced by the Dutch astrologer L. Knegt and is used by the Dutch Werkgemeenschap van Astrologen (WvA, also known as “Ram school”).
The parallel of declination that goes through the ascendant is divided in six equal parts both above and below the horizon. Position circles through the north and the south point on the horizon are drawn through he division points. The house cusps are where the position circles intersect the ecliptic.
Note, the house cusps 11, 12, 2, and 3 are not exactly opposite the cusps 5, 6, 8, and 9.

6.1.15. Sunshine house system
This house system was invented by Bob Makransky and published in 1988 in his book Primary Directions. A Primer of Calculation (free download: http://www.dearbrutus.com/buyprimarydirections.html).
The diurnal and nocturnal arcs of the Sun are trisected, and great circles are drawn through these trisection points and the north and the south point on the horizon. The intersection points of these great circles with the ecliptic are the house cusps. Note that the cusps 11, 12, 2, and 3 are not in exact opposition to the cusps 5, 6, 8, and 9.
For the polar region and during times where the Sun does not rise or set, the diurnal and nocturnal arc are assumed to be either 180° or 0°. If the diurnal arc is 0°, the house cusps 8 – 12 coincide with the meridian. If the nocturnal arc is 0°, the cusps 2 – 6 coincide with the meridian. As with the closely related Regiomontanus system, an MC below the horizon and IC above the horizon are exchanged.

The Vertex is the point of the ecliptic that is located precisely in western direction. The Antivertex is the opposition point and indicates the precise east in the horoscope. It is identical to the first house cusp in the horizon house system.
There is a lot of confusion about this, because there is also another point which is called the "East Point" but is usually not located in the east. In celestial geometry, the expression "East Point" means the point on the horizon which is in precise eastern direction. The equator goes through this point as well, at a right ascension which is equal to ARMC + 90 degrees. On the other hand, what some astrologers call the "East Point" is the point on the ecliptic whose right ascension is equal to ARMC + 90 (i.e. the right ascension of the horizontal East Point). This point can deviate from eastern direction by 23.45 degrees, the amount of the ecliptic obliquity. For this reason, the term "East Point" is not very wellchosen for this ecliptic point, and some astrologers (M. Munkasey) prefer to call it the Equatorial Ascendant. This, because it is identical to the Ascendant at a geographical latitude 0.
The Equatorial Ascendant is identical to the first house cusp of the axial rotation system.
Note: If a projection of the horizontal East Point on the ecliptic is wanted, it might seem more reasonable to use a projection rectangular to the ecliptic, not rectangular to the equator as is done by the users of the "East Point". The planets, as well, are not projected on the ecliptic in a right angle to the ecliptic.
The Swiss Ephemeris supports three more points connected with the house and angle calculation. They are part of Michael Munkasey's system of the 8 Personal Sensitive Points (PSP). The PSP include the Ascendant, the MC, the Vertex, the Equatorial Ascendant, the Aries Point, the Lunar Node, and the "CoAscendant" and the "Polar Ascendant".
The term "CoAscendant" seems to have been invented twice by two different people, and it can mean two different things. The one "CoAscendant" was invented by Walter Koch (?). To calculate it, one has to take the ARIC as an ARMC and compute the corresponding Ascendant for the birth place. The "CoAscendant" is then the opposition to this point.
The second "CoAscendant" stems from Michael Munkasey. It is the Ascendant computed for the natal ARMC and a latitude which has the value 90°  birth_latitude.
The "Polar Ascendant" finally was introduced by Michael Munkasey. It is the opposition point of Walter Koch's version of the "CoAscendant". However, the "Polar Ascendant" is not the same as an Ascendant computed for the birth time and one of the geographic poles of the earth. At the geographic poles, the Ascendant is always 0 Aries or 0 Libra. This is not the case for Munkasey's "Polar Ascendant".

6.3. House cusps beyond the polar circle
Beyond the polar circle, we proceed as follows:
We make sure that the ascendant is always in the eastern hemisphere.
Placidus and Koch house cusps sometimes can, sometimes cannot be computed beyond the polar circles. Even the MC doesn't exist always, if one defines it in the Placidus manner. Our function therefore automatically switches to Porphyry houses (each quadrant is divided into three equal parts) and returns a warning.
Beyond the polar circles, the MC is sometimes below the horizon. The geometrical definition of the Campanus and Regiomontanus systems requires in such cases that the MC and the IC are swapped. The whole house system is then oriented in clockwise direction.
There are similar problems with the Vertex and the horizon house system for birth places in the tropics. The Vertex is defined as the point on the ecliptic that is located in precise western direction. The ecliptic east point is the opposition point and is called the Antivertex. Our program code makes sure that the Vertex (and the cusps 11, 12, 1, 2, 3 of the horizon house system) is always located in the western hemisphere. Note that for birthplaces on the equator the Vertex is always 0 Aries or 0 Libra.
Of course, there are no problems in the calculation of the Equatorial Ascendant for any place on earth.

6.3.1. Implementation in other calculation modules:
a) PLACALC
Placalc is the predecessor of Swiss Ephemeris; it is a calculation module created by Astrodienst in 1988 and distributed as C source code. Beyond the polar circles, Placalc‘s house calculation did switch to Porphyry houses for all unequal house systems. Swiss Ephemeris still does so with the Placidus and Koch method, which are not defined in such cases. However, the computation of the MC and Ascendant was replaced by a different model in some cases: Swiss Ephemeris gives priority to the Ascendant, choosing it always as the eastern rising point of the ecliptic and accepting an MC below the horizon, whereas Placalc gave priority to the MC. The MC was always chosen as the intersection of the meridian with the ecliptic above the horizon. To keep the quadrants in the correct order, i.e. have an Ascendant in the left side of the chart, the Ascendant was switched by 180 degrees if necessary.
In the discussions between Alois Treindl and Dieter Koch during the development of the Swiss Ephemeris it was recognized that this model is more unnatural than the new model implemented in Swiss Ephemeris.
Placalc also made no difference between Placidus/Koch on one hand and Regiomontanus/Campanus on the other as Swiss Ephemeris does. In Swiss Ephemeris, the geometrical definition of Regiomontanus/Campanus is strictly followed, even for the price of getting the houses in ”wrong” order. (see above, chapter 4.1.)
b) ASTROLOG program as written by Walter Pullen
While the freeware program Astrolog contains the planetary routines of Placalc, it uses its own house calculation module by Walter Pullen. Various releases of Astrolog contain different approaches to this problem.
c) ASTROLOG on our website
ASTROLOG is also used on Astrodienst’s website for displaying free charts. This version of Astrolog used on our website however is different from the Astrolog program as distributed on the Internet. Our webserver version of Astrolog contains calls to Swiss Ephemeris for planetary positions. For Ascendant, MC and houses it still uses Walter Pullen's code. This will change in due time because we intend to replace ASTROLOG on the website with our own charting software.
d) other astrology programs
Because most astrology programs still use the Placalc module, they follow the Placalc method for houses inside the polar circles. They give priority to keep the MC above the horizon and switch the Ascendant by 180 degrees if necessary to keep the quadrants in order.

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