formulasearchengine - User contributions [en]

Parry point (triangle)

2014-12-22T12:26:06Z

117.6.86.37: /* Parry point */

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Pascal's theorem

2014-03-04T03:32:06Z

117.6.129.14: /* A Property of Pascal's Hexagon */

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2014-02-27T14:50:40Z

117.6.86.31:

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Pole and polar

2014-01-27T03:35:15Z

117.6.86.31: /* See also */

{{use mdy dates|date=February 2013}}
The '''Richter magnitude scale''' (often shortened to '''Richter scale''') was developed to assign a single number to quantify the energy that is released during an [[earthquake]].

The scale is a [[Decimal|base-10]] [[logarithmic scale]]. The magnitude is defined as the logarithm of the ratio of the [[amplitude]] of waves measured by a [[seismometer|seismograph]] to an arbitrary small amplitude. An earthquake that measures 5.0 on the Richter scale has a shaking amplitude 10 times larger than one that measures 4.0, and corresponds to a 31.6 times larger release of energy.<ref name=USGS>[http://earthquake.usgs.gov/learn/topics/richter.php The Richter Magnitude Scale]</ref>

Since the mid-20th century, the use of the Richter magnitude scale has largely been supplanted by the [[moment magnitude scale]] (MMS) in many countries. However, the Richter scale is still widely used in [[Russia]] and other [[Commonwealth of Independent States|CIS]] countries. Earthquake measurements under the moment magnitude scale in the [[United States]]—3.5 and up, on the MMS scale—are still usually erroneously referred to as being quoted on the Richter scale by the general public, as well as the media, due to their familiarity with the Richter scale as compared to the MMS.

== Development ==
[[File:CharlesRichter.jpg|thumb|upright|Charles Richter, c. 1970]]
Developed in 1935 by [[Charles Francis Richter]] in partnership with [[Beno Gutenberg]], both from the [[California Institute of Technology]], the scale was firstly intended to be used only in a particular study area in [[California]], and on seismograms recorded on a particular instrument, the Wood-Anderson torsion seismograph. Richter originally reported values to the nearest quarter of a unit, but values were later reported with one decimal place. His motivation for creating the local magnitude scale was to compare the size of different earthquakes.<ref name=USGS/>
Richter, who since childhood had aspirations in astronomy, drew inspiration from the [[apparent magnitude]] scale used to account for the brightness of stars lost due to distance.<ref name=Reitherman>{{cite book|last=Reitherman|first=Robert|title=Earthquakes and Engineers: An International History|year=2012|publisher=ASCE Press|location=Reston, VA|isbn=9780784410714|pages=208–209|url=http://www.asce.org/Product.aspx?id=2147487208&productid=154097877}}</ref> Richter arbitrarily chose a magnitude 0 event to be an earthquake that would show a maximum combined horizontal displacement of 1 µm (0.00004 in) on a seismogram recorded using a Wood-Anderson torsion seismograph {{convert|100|km|mi|abbr=on}} from the earthquake [[epicenter]]. This choice was intended to prevent negative magnitudes from being assigned. The smallest earthquakes that could be recorded and located at the time were around magnitude 3. However, the Richter scale has no lower limit, and sensitive modern seismographs now routinely record quakes with negative magnitudes.

<math>M_\text{L}</math> (local magnitude) was not designed to be applied to data with distances to the [[hypocenter]] of the earthquake greater than 600 km<ref>{{cite web|url=http://earthquake.usgs.gov/aboutus/docs/020204mag_policy.php |title=USGS Earthquake Magnitude Policy |publisher=USGS |date=March 29, 2010}}</ref> (373 mi). For national and local seismological observatories the standard magnitude scale is today still <math>M_\text{L}</math>. Unfortunately this scale saturates{{clarify|date=October 2012}} at around <math>M_\text{L}</math> = 7,<ref name="weather.gov.hk">{{cite web|url=http://www.weather.gov.hk/education/edu02rga/article/ele-EarthquakeMagnetude_e.htm |title=On Earthquake Magnitudes |first=Wang-chun |last=Woo |date=September 2012 |publisher=Hong Kong Observatory |accessdate=18 December 2013}}</ref> because the high frequency waves recorded locally have wavelengths shorter than the rupture lengths{{clarify|date=October 2012}} of large earthquakes.

To express the size of earthquakes around the globe, Gutenberg and Richter later developed a [[surface wave magnitude]] scale, <math>M_\text{s}</math>, and a [[body wave magnitude]] scale <math>M_\text{b}</math>.<ref name="Ellsworth">{{cite journal
| publisher=USGS
| author=William L. Ellsworth
| url=http://www.johnmartin.com/earthquakes/eqsafs/safs_694.htm
| title=SURFACE-WAVE MAGNITUDE (<math>M_\text{s}</math>) AND BODY-WAVE MAGNITUDE (mb)
| year=1991
| accessdate=2008-09-14
}}
</ref> These are types of waves that are recorded at [[teleseism]]ic distances. The two scales were adjusted such that they were consistent with the <math>M_\text{L}</math> scale. This succeeded better with the <math>M_\text{s}</math> scale than with the <math>M_\text{b}</math> scale. Both of these scales saturate when the earthquake is bigger than magnitude 8 and therefore the moment magnitude scale, <math>M_\text{w}</math>, was invented.

These older magnitude scales have been superseded by methods for estimating the [[seismic moment]], creating the [[moment magnitude scale]], although the older scales are still widely used because they can be calculated quickly.

{{quote|I found a paper by Professor [[Kiyoo Wadati|K. Wadati]] of Japan in which he compared large earthquakes by plotting the maximum ground motion against distance to the epicenter. I tried a similar procedure for our stations, but the range between the largest and smallest magnitudes seemed unmanageably large. Dr. [[Beno Gutenberg]] then made the natural suggestion to plot the amplitudes [[logarithmically]]. I was lucky because [[Logarithmic scale|logarithmic plots]] are a device of the [[devil]].|[[Charles Francis Richter]]|[http://earthquake.usgs.gov/learn/topics/people/int_richter.php Charles Richter Interview]}}

==Details==
The Richter scale proper was defined in 1935 for particular circumstances and instruments; the instrument used saturated for strong earthquakes. The scale was replaced by the [[moment magnitude scale]] (MMS); for earthquakes adequately measured by the Richter scale, numerical values are approximately the same. Although values measured for earthquakes now are actually <math>M_w</math> (MMS), they are frequently reported as Richter values, even for earthquakes of magnitude over 8, where the Richter scale becomes meaningless.
Anything above 5 is classified as a risk by the USGS.{{citation needed|date=April 2013}}

The Richter and MMS scales measure the energy released by an earthquake; another scale, the [[Mercalli intensity scale]], classifies earthquakes by their ''effects'', from detectable by instruments but not noticeable to catastrophic. The energy and effects are not necessarily strongly correlated; a shallow earthquake in a populated area with soil of certain types can be far more intense than a much more energetic deep earthquake in an isolated area.

There are several scales which have historically been described as the "Richter scale", especially the ''local magnitude'' <math>M_\text{L}</math> and the surface wave <math>M_\text{s}</math> scale. In addition, the ''body wave magnitude'', <math>m_\text{b}</math>, and the ''moment magnitude'', <math>M_\text{w}</math>, abbreviated MMS, have been widely used for decades, and a couple of new techniques to measure magnitude are in the development stage.

All magnitude scales have been designed to give numerically similar results. This goal has been achieved well for <math>M_\text{L}</math>, <math>M_\text{s}</math>, and <math>M_\text{w}</math>.<ref name="Richter 1935">{{cite journal | url=https://www2.bc.edu/~ebel/Richter1935.pdf | title=An instrumental earthquake magnitude scale | author=Richter, C.F. | journal=Bulletin of the Seismological Society of America | year=1935 | volume=25 | issue=1-2 | pages=1–32}}</ref><ref>Richter, C.F., "Elementary Seismology", edn, Vol., W. H. Freeman and Co., San Francisco, 1956.</ref> The <math>m_\text{b}</math> scale gives somewhat different values than the other scales. The reason for so many different ways to measure the same thing is that at different distances, for different [[hypocenter|hypocentral]] depths, and for different earthquake sizes, the amplitudes of different types of elastic waves must be measured.

<math>M_\text{L}</math> is the scale used for the majority of earthquakes reported (tens of thousands) by local and regional seismological observatories. For large earthquakes worldwide, the moment magnitude scale is most common, although <math>M_\text{s}</math> is also reported frequently.

The [[seismic moment]], '''''<math>M_o</math>''''', is proportional to the area of the rupture times the average slip that took place in the earthquake, thus it measures the physical size of the event. <math>M_\text{w}</math> is derived from it empirically as a quantity without units, just a number designed to conform to the <math>M_\text{s}</math> scale.<ref>Hanks, T. C. and H. Kanamori, 1979, "Moment magnitude scale", Journal of Geophysical Research, 84, B5, 2348.</ref> A spectral analysis is required to obtain <math>M_o</math>, whereas the other magnitudes are derived from a simple measurement of the amplitude of a specifically defined wave.

All scales, except <math>M_\text{w}</math>, saturate for large earthquakes, meaning they are based on the amplitudes of waves which have a wavelength shorter than the rupture length of the earthquakes. These short waves (high frequency waves) are too short a yardstick to measure the extent of the event. The resulting effective upper limit of measurement for <math>M_L</math> is about 7<ref name="weather.gov.hk"/> and about 8.5<ref name="weather.gov.hk"/> for <math>M_\text{s}</math>.<ref name="Local magnitude">{{cite web|url=http://earthquake.usgs.gov/hazards/qfaults/glossary.php |title=Richter scale |work=Glossary |publisher=[[United States Geological Survey|USGS]] |date=March 31, 2010 }}</ref>

New techniques to avoid the saturation problem and to measure magnitudes rapidly for very large earthquakes are being developed. One of these is based on the long period P-wave,<ref>Di Giacomo, D., Parolai, S., Saul, J., Grosser, H., Bormann, P., Wang, R. & Zschau, J., 2008. Rapid determination of the enrgy magnitude Me, in European Seismological Commission 31st General Assembly, Hersonissos.</ref> the other is based on a recently discovered channel wave.<ref>Rivera, L. & Kanamori, H., 2008. Rapid source inversion of W phase for tsunami warning, in European Geophysical Union General Assembly, pp. A-06228, Vienna.</ref>

The [[energy]] release of an earthquake,<ref>Marius Vassiliou and Hiroo Kanamori (1982): "The Energy Release in Earthquakes," Bull. Seismol. Soc. Am. 72, 371-387.</ref> which closely correlates to its destructive power, scales with the {{frac|3|2}} power of the shaking amplitude. Thus, a difference in magnitude of 1.0 is equivalent to a factor of 31.6 (<math>=({10^{1.0}})^{(3/2)}</math>) in the energy released; a difference in magnitude of 2.0 is equivalent to a factor of 1000 (<math>=({10^{2.0}})^{(3/2)}</math> ) in the energy released.<ref>{{cite journal|authors=William Spence, Stuart A. Sipkin, and George L. Choy |url=http://earthquake.usgs.gov/learn/topics/measure.php |title=Measuring the Size of an Earthquake |journal=Earthquakes and Volcanoes |volume=21 |number=1 |year=1989}}</ref> The elastic energy radiated is best derived from an integration of the radiated spectrum, but one can base an estimate on <math>m_\text{b}</math> because most energy is carried by the high frequency waves.

==Richter magnitudes==
The Richter magnitude of an earthquake is determined from the [[logarithm]] of the [[amplitude]] of waves recorded by seismographs (adjustments are included to compensate for the variation in the distance between the various seismographs and the [[epicenter]] of the earthquake). The original formula is:<ref>{{cite journal
| publisher=USGS
| last=Ellsworth
| first=William L.
| url=http://www.johnmartin.com/earthquakes/eqsafs/safs_693.htm
| title=The Richter Scale <math>M_\text{L}</math>, from The San Andreas Fault System, California (Professional Paper 1515)
| pages=c6, p177
| year=1991
| accessdate=2008-09-14
}}
</ref>

:<math>M_\mathrm{L} = \log_{10} A - \log_{10} A_\mathrm{0}(\delta) = \log_{10} [A / A_\mathrm{0}(\delta)],\ </math>

where A is the maximum excursion of the Wood-Anderson seismograph, the empirical function A0 depends only on the [[epicentral distance]] of the station, <math>\delta</math>. In practice, readings from all observing stations are averaged after adjustment with station-specific corrections to obtain the <math>M_\text{L}</math> value.

Because of the logarithmic basis of the scale, each whole number increase in magnitude represents a tenfold increase in measured amplitude; in terms of energy, each whole number increase corresponds to an increase of about 31.6 times the amount of energy released, and each increase of 0.2 corresponds to a doubling of the energy released.

Events with magnitudes greater than 4.5 are strong enough to be recorded by a seismograph anywhere in the world, so long as its sensors are not located in the earthquake's [[Seismic shadowing|shadow]].

The following describes the typical effects of earthquakes of various magnitudes near the epicenter. The values are typical only and should be taken with extreme caution, since intensity and thus ground effects depend not only on the magnitude, but also on the distance to the epicenter, the depth of the earthquake's focus beneath the epicenter, the location of the epicenter and geological conditions (certain terrains can amplify seismic signals).

{| class="wikitable"
|-
!Magnitude
!Description
![[Mercalli intensity scale|Mercalli intensity]]
!Average earthquake effects
!Average frequency of occurrence (estimated)
|-
|Less than 2.0
|[[Microearthquake|Micro]]
|I
|Microearthquakes, not felt, or felt rarely by sensitive people. Recorded by seismographs.<ref>This is what Richter wrote in his ''Elementary Seismology'' (1958), an opinion copiously reproduced afterwards in Earth's science primers. Recent evidence shows that earthquakes with negative magnitudes (down to −0.7) can also be felt in exceptional cases, especially when the focus is very shallow (a few hundred metres). See: Thouvenot, F.; Bouchon, M. (2008). What is the lowest magnitude threshold at which an earthquake can be felt or heard, or objects thrown into the air?, in Fréchet, J., Meghraoui, M. & Stucchi, M. (eds), ''Modern Approaches in Solid Earth Sciences'' (vol. 2), Historical Seismology: Interdisciplinary Studies of Past and Recent Earthquakes, Springer, Dordrecht, 313–326.</ref>
|Continual/several million per year
|-
|2.0–2.9
|rowspan="2"|Minor
|I to II
|Felt slightly by some people. No damage to buildings.
|Over one million per year
|-
|3.0–3.9
|II to IV
|Often felt by people, but very rarely causes damage. Shaking of indoor objects can be noticeable.
|Over 100,000 per year
|-
|4.0–4.9
|Light
|IV to VI
|Noticeable shaking of indoor objects and rattling noises. Felt by most people in the affected area. Slightly felt outside. Generally causes none to minimal damage. Moderate to significant damage very unlikely. Some objects may fall off shelves or be knocked over.
|10,000 to 15,000 per year
|-
|5.0–5.9
|Moderate
|VI to VIII
|Can cause damage of varying severity to poorly constructed buildings. At most, none to slight damage to all other buildings. Felt by everyone. Casualties range from none to a few.
|1,000 to 1,500 per year
|-
|6.0–6.9
|Strong
|VII to X
|Damage to a moderate number of well built structures in populated areas. Earthquake-resistant structures survive with slight to moderate damage. Poorly-designed structures receive moderate to severe damage. Felt in wider areas; up to hundreds of miles/kilometers from the epicenter. Strong to violent shaking in epicentral area. Death toll ranges from none to 25,000.
|100 to 150 per year
|-
|7.0–7.9
|Major
|rowspan="3"|VIII or greater<ref>{{cite web |url=http://www.city-data.com/city/Anchorage-Alaska.html|title=Anchorage, Alaska (AK) profile: population, maps, real estate, averages, homes, statistics, relocation, travel, jobs, hospitals, schools, crime, moving, houses, news |publisher=City-Data.com |accessdate=2012-10-12 }}</ref>
|Causes damage to most buildings, some to partially or completely collapse or receive severe damage. Well-designed structures are likely to receive damage. Felt across great distances with major damage mostly limited to 250 km from epicenter. Death toll ranges from none to 250,000.
|10 to 20 per year
|-
|8.0–8.9
|rowspan="2"|Great
|Major damage to buildings, structures likely to be destroyed. Will cause moderate to heavy damage to sturdy or earthquake-resistant buildings. Damaging in large areas. Felt in extremely large regions. Death toll ranges from 1,000 to 1 million.
|One per year
|-
|9.0 and greater
|Near or at total destruction - severe damage or collapse to all buildings. Heavy damage and shaking extends to distant locations. Permanent changes in ground topography. Death toll usually over 50,000.
|One per 10 to 50 years
|}

(''Based on U.S. Geological Survey documents.'')<ref>{{cite web|url=http://earthquake.usgs.gov/earthquakes/eqarchives/year/eqstats.php |title=Earthquake Facts and Statistics |publisher=United States Geological Survey|date=29 November 2012 |accessdate=18 December 2013}}</ref>

The intensity and death toll depend on several factors (earthquake depth, epicenter location, population density, to name a few) and can vary widely.

Minor earthquakes occur every day and hour. On the other hand, great earthquakes occur once a year, on average. The largest recorded earthquake was the [[1960 Valdivia earthquake|Great Chilean Earthquake]] of May 22, 1960, which had a magnitude of 9.5 on the [[moment magnitude scale]].<ref>{{cite web|url=http://earthquake.usgs.gov/regional/world/10_largest_world.php |title=Largest Earthquakes in the World Since 1900 |date=30 November 2012 |accessdate=18 December 2013}}</ref> The larger the magnitude, the less frequent the earthquake happens.

===Examples===
The following table lists the approximate [[energy]] equivalents in terms of [[TNT equivalent|TNT]] explosive force – though note that the earthquake energy is released ''underground'' rather than overground.<ref>[http://earthquake.usgs.gov/learn/faq/?faqID=33 FAQs – Measuring Earthquakes]</ref> Most energy from an earthquake is not transmitted to and through the surface; instead, it dissipates into the crust and other subsurface structures. In contrast, a small atomic bomb blast (see [[nuclear weapon yield]]) will not, it will simply cause light shaking of indoor items, since its energy is released above ground.

31.6227 to the ''power of'' 0 ''equals'' 1, 31.6227 to the ''power of'' 1 ''equals'' 31.6227 and 31.6227 to the ''power of'' 2 ''equals'' 1000. Therefore, an 8.0 on the Richter scale releases 31.6227 ''times'' more energy than a 7.0 and a 9.0 on the Richter scale releases 1000 ''times'' more energy than a 7.0. Thus, <math>E \approx 6.3\times 10^4\times 10^{3M/2}\,</math>

{| class="wikitable"
|-
!Approximate Magnitude
!Approximate TNT for Seismic Energy Yield
!Joule equivalent
!Example
|-
| -0.2
|7.5 g
|31.5 kJ
|Energy released by lighting 30 typical matches
|-
|0.0
|15 g
|63 kJ
|-
|0.2
|30 g
|130 kJ
|Large [[hand grenade]]
|-
|0.5
|85 g
|360 kJ
|
|-
|1.0
|480 g
|2.0 MJ
|
|-
|1.2
|1.1 kg
|4.9 MJ
|Single stick of dynamite [DynoMax Pro]
|-
|1.4
|2.2 kg
|9.8 MJ
|Seismic impact of typical small construction blast
|-
|1.5
|2.7 kg
|11 MJ
|
|-
|2.0
|15 kg
|63 MJ
|
|-
|2.1
|21 kg
|89 MJ
|[[West fertilizer plant explosion]]<ref>{{cite web |url=http://earthquake.usgs.gov/earthquakes/eventpage/usb000g9yl#summary |title=2.1 Explosion - 1km NNE of West, Texas (BETA) |date=19 June 2013 |publisher=United States Geological Survey |accessdate=18 December 2013}}</ref>
|-
|2.5
|85 kg
|360 MJ
|
|-
|3.0
|480 kg
|2.0 GJ
|[[Oklahoma City bombing]], 1995
|-
|3.5
|2.7 metric tons
|11 GJ
|[[PEPCON disaster|PEPCON fuel plant explosion, Henderson, Nevada]], 1988
[[Dallas]], [[Texas]] earthquake, September 30, 2012
|-
|3.87
|9.5 metric tons
|40 GJ
|[[Chernobyl disaster|Explosion at Chernobyl nuclear power plant]], 1986
|-
|3.91
|11 metric tons
|46 GJ
|[[GBU-43/B Massive Ordnance Air Blast bomb|Massive Ordnance Air Blast bomb]]
St. Patrick's Day earthquake, [[Auckland]], [[New Zealand]], 2013 <ref>{{cite web |url=http://www.geonet.org.nz/quakes/region/aucklandnorthland/2013p203051 |title=New Zealand Earthquake Report: Magnitude 3.9, Sunday, March 17, 2013 at 4:05:42 pm (NZDT) |publisher=GeoNet |accessdate=18 December 2013}}</ref><ref>{{cite news |url=http://www.nzherald.co.nz/nz/news/article.cfm?c_id=1&objectid=10871821 |title=Quake rattles Auckland |first=Matthew |last=Backhouse |first2=Matthew |last2=Theunissen |date=17 March 2013 |newspaper=The New Zealand Herald |accessdate=18 December 2013}}</ref>
|-
|4.0
|15 metric tons
|63 GJ
|[[Johannesburg]]/[[South Africa]], November 18, 2013
|-
|4.3
|43 metric tons
|180 GJ
|[[2007 Kent earthquake|Kent Earthquake (Britain), 2007]]
Eastern [[Kentucky]] earthquake, November 2012
|-
|5.0
|480 metric tons
|2.0 TJ
|[[2008 Lincolnshire earthquake|Lincolnshire earthquake (UK), 2008]] 
[[Moment magnitude scale|<math>M_\text{w}</math>]] [[2010 Central Canada earthquake|Ontario-Quebec earthquake (Canada), 2010]]<ref>{{cite web |url=http://earthquake.usgs.gov/earthquakes/recenteqsww/Quakes/us2010xwa7.php#details |title=Magnitude 5.0 – Ontario-Quebec border region, Canada |publisher=earthquake.usgs.gov |accessdate=2010-06-23 }}</ref><ref>{{cite web |url=http://news.nationalpost.com/2010/06/23/tremors-felt-in-toronto-ottawa-reports/ |title=Moderate 5.0 earthquake shakes Toronto, Eastern Canada and U.S.|publisher=nationalpost.com |accessdate=2010-06-23 }}</ref>
|-
|5.5
|2.7 kilotons
|11 TJ
|Little Skull Mtn. earthquake (Nevada, USA), 1992 
[[Moment magnitude scale|<math>M_\text{w}</math>]] [[2007 Alum Rock earthquake|Alum Rock earthquake (California), 2007]] 
[[Moment magnitude scale|<math>M_\text{w}</math>]] [[2008 Chino Hills earthquake|Chino Hills earthquake (Southern California), 2008]]
|-
|5.6
|3.8 kilotons
|16 TJ
|[[1989 Newcastle earthquake|Newcastle, Australia, 1989]] 
[[2011 Oklahoma earthquake|Oklahoma, 2011]] 
[[2012 Pernik earthquake|Pernik, Bulgaria, 2012]]
|-
|6.0
|15 kilotons
|63 TJ
|Double Spring Flat earthquake ([[Nevada]], USA), 1994
Approximate magnitude of [[Virginia]]/[[Washington, D.C.]]/[[East Coast of the United States|East Coast]] [[2011 Virginia earthquake|earthquake, 2011]] 
Approximate yield of the [[Little Boy]] Atomic Bomb dropped on [[Hiroshima]] (~16 kt)
|-
|6.3
|43 kilotons
|180 TJ
|[[Moment magnitude scale|<math>M_\text{w}</math>]] [[2008 Dodecanese earthquake|Rhodes earthquake (Greece), 2008]] 
[[1927 earthquake in Palestine|Jericho earthquake (British Palestine), 1927]] 
[[February 2011 Christchurch earthquake|Christchurch earthquake (New Zealand), 2011]]
|-
|6.4
|60 kilotons
|250 TJ
|[[2010 Kaohsiung earthquake|Kaohsiung earthquake (Taiwan), 2010]]
[[2011 Vancouver earthquake|Vancouver earthquake (Canada), 2011]]
|-
|6.5
|85 kilotons
|360 TJ
|[[Surface wave magnitude|<math>M_\text{s}</math>]] [[1967 Caracas earthquake|Caracas earthquake (Venezuela), 1967]] 
[[1980 Irpinia earthquake|Irpinia earthquake (Italy), 1980]] 
[[Moment magnitude scale|<math>M_\text{w}</math>]] [[2010 Eureka earthquake|Eureka earthquake (California, USA), 2010]] 
Zumpango del Rio earthquake (Guerrero, Mexico), 2011<ref>{{cite web|title=Past Earthquakes |url=http://www2.ssn.unam.mx/website/jsp/fuertes.jsp |publisher=Servicio Sismologico Nacional |accessdate=2 March 2013 |language=Spanish}}</ref>
|-
|6.6
|120 kilotons
|500 TJ
|[[Moment magnitude scale|<math>M_\text{w}</math>]] [[1971 San Fernando earthquake|San Fernando earthquake (California, USA), 1971]]
|-
|6.7
|170 kilotons
|710 TJ
|[[Moment magnitude scale|<math>M_\text{w}</math>]] [[1994 Northridge earthquake|Northridge earthquake (California, USA), 1994]]
|-
|6.8
|240 kilotons
|1.0 PJ
|[[Moment magnitude scale|<math>M_\text{w}</math>]] [[2001 Nisqually earthquake|Nisqually earthquake (Anderson Island, WA), 2001]] 
[[Moment magnitude scale|<math>M_\text{w}</math>]] [[Great Hanshin earthquake|Great Hanshin earthquake (Kobe, Japan), 1995]] 
[[2007 Gisborne earthquake|Gisborne earthquake (Gisborne, NZ), 2007]]
|-
|6.9
|340 kilotons
|1.4 PJ
|[[Moment magnitude scale|<math>M_\text{w}</math>]] [[1989 Loma Prieta earthquake|San Francisco Bay Area earthquake (California, USA), 1989]] 
[[Moment magnitude scale|<math>M_\text{w}</math>]] [[2010 Pichilemu earthquake|Pichilemu earthquake (Chile), 2010]] 
[[Moment magnitude scale|<math>M_\text{w}</math>]] [[2011 Sikkim earthquake|Sikkim earthquake (Nepal-India Border), 2011]]
|-
|7.0
|480 kilotons
|2.0 PJ
|[[Moment magnitude scale|<math>M_\text{w}</math>]] [[2009 Java earthquake|Java earthquake (Indonesia), 2009]] 
[[Moment magnitude scale|<math>M_\text{w}</math>]] [[2010 Haiti earthquake|Haiti earthquake, 2010]]
|-
|7.1
|680 kilotons
|2.8 PJ
|[[Moment magnitude scale|<math>M_\text{w}</math>]] [[1908 Messina earthquake|Messina earthquake (Italy), 1908]] 
[[Moment magnitude scale|<math>M_\text{w}</math>]] [[1944 San Juan earthquake|San Juan earthquake (Argentina), 1944]] 
[[Moment magnitude scale|<math>M_\text{w}</math>]] [[2010 Canterbury earthquake|Canterbury earthquake (New Zealand), 2010]]
|-
|7.2
|950 kilotons
|4.0 PJ
|[[1977 Vrancea earthquake|Vrancea earthquake (Romania), 1977]] 
[[Moment magnitude scale|<math>M_\text{w}</math>]] [[1980 Azores Islands earthquake|1980 Azores Islands Earthquake]] 
[[Moment magnitude scale|<math>M_\text{w}</math>]] [[2010 Baja California earthquake|Baja California earthquake (Mexico), 2010]]

|-
|7.5
|2.7 megatons
|11 PJ
| [[Moment magnitude scale|<math>M_\text{w}</math>]] [[2005 Kashmir earthquake|Kashmir earthquake (Pakistan), 2005]] 
[[Moment magnitude scale|<math>M_\text{w}</math>]] [[2007 Antofagasta earthquake|Antofagasta earthquake (Chile), 2007]]
|-
|7.6
|3.8 megatons
|16 PJ
| [[Moment magnitude scale|<math>M_\text{w}</math>]] [[2012 Costa Rica earthquake|Nicoya earthquake (Costa Rica), 2012]] 
[[Moment magnitude scale|<math>M_\text{w}</math>]] [[2012 San Juan Cacahuatepec earthquake|Oaxaca earthquake (Mexico), 2012]] 
[[Moment magnitude scale|<math>M_\text{w}</math>]] [[2001 Gujarat earthquake|Gujarat earthquake (India), 2001]] 
[[Moment magnitude scale|<math>M_\text{w}</math>]] [[1999 İzmit earthquake|İzmit earthquake (Turkey), 1999]] 
[[Moment magnitude scale|<math>M_\text{w}</math>]] [[1999 Jiji earthquake|Jiji earthquake (Taiwan), 1999]]
|-
|7.7
|5.4 megatons
|22 PJ
|[[Moment magnitude scale|<math>M_\text{w}</math>]] [[October 2010 Sumatra earthquake and tsunami|Sumatra earthquake (Indonesia), 2010]] 
[[Moment magnitude scale|<math>M_\text{w}</math>]] [[2012 Haida Gwaii earthquake|Haida Gwaii earthquake (Canada), 2012]]
|-
|7.8
|7.6 megatons
|32 PJ
|[[Moment magnitude scale|<math>M_\text{w}</math>]] [[1976 Tangshan earthquake|Tangshan earthquake (China), 1976]] 
[[Surface wave magnitude|<math>M_\text{s}</math>]] [[1931 Hawke's Bay earthquake|Hawke's Bay earthquake (New Zealand), 1931]] 
[[Surface wave magnitude|<math>M_\text{s}</math>]] [[1990 Luzon earthquake|Luzon earthquake (Philippines), 1990]] 
|-
|7.9
|10-15 megatons
|42-63 PJ
|[[Tunguska event]] [[1802 Vrancea earthquake]] 
[[Moment magnitude scale|<math>M_\text{w}</math>]] [[1923 Great Kanto earthquake|Great Kanto earthquake (Japan), 1923]]
|-
|8.0
|15 megatons
|63 PJ
| [[Surface wave magnitude|<math>M_\text{s}</math>]] [[1891 Mino-Owari earthquake|Mino-Owari earthquake (Japan), 1891]] 
[[1894 San Juan earthquake|San Juan earthquake (Argentina), 1894]] 
[[1906 San Francisco earthquake|San Francisco earthquake (California, USA), 1906]] 
[[Surface wave magnitude|<math>M_\text{s}</math>]] [[1949 Queen Charlotte Islands earthquake|Queen Charlotte Islands earthquake (B.C., Canada), 1949]] 
[[Moment magnitude scale|<math>M_\text{w}</math>]] [[2007 Peru earthquake|Chincha Alta earthquake (Peru), 2007]] 
[[Surface wave magnitude|<math>M_\text{s}</math>]] [[2008 Sichuan earthquake|Sichuan earthquake (China), 2008]] [[1905 Kangra earthquake|Kangra earthquake, 1905]] 
|-
|8.1
|21 megatons
|89 PJ
|[[1985 Mexico City earthquake|México City earthquake (Mexico), 1985]] 
Guam earthquake, August 8, 1993<ref>{{cite web |url=http://www.eeri.org/site/reconnaissance-activities/64-guam/182-m81southendofisland |title=M8.1 South End of Island August 8, 1993.|publisher=eeri.org |accessdate=2011-03-11}}</ref>
|-
|8.35
|50 megatons
|210 PJ
|[[Tsar Bomba]] - Largest thermonuclear weapon ever tested
|-
|8.5
|85 megatons
|360 PJ
|[[Moment magnitude scale|<math>M_\text{w}</math>]] [[September 2007 Sumatra earthquakes|Sumatra earthquake (Indonesia), 2007]]
|-
|8.6
| 120 megatons
| 500 PJ
|[[Moment magnitude scale|<math>M_\text{w}</math>]] [[April 2012 Banda Aceh earthquakes|Sumatra earthquake (Indonesia), 2012]]
|-
|8.7
|170 megatons
|710 PJ
|[[Moment magnitude scale|<math>M_\text{w}</math>]] [[2005 Sumatra earthquake|Sumatra earthquake (Indonesia), 2005]]
|-
|8.75
|200 megatons
|840 PJ
|[[Krakatoa]] 1883
|-
|8.8
|240 megatons
|1.0 EJ
|[[Moment magnitude scale|<math>M_\text{w}</math>]] [[2010 Chile earthquake|Chile earthquake, 2010]],
|-
|9.0
|480 megatons
|2.0 EJ
| [[Moment magnitude scale|<math>M_\text{w}</math>]] [[1755 Lisbon earthquake|Lisbon earthquake (Portugal), All Saints Day, 1755]] [[Moment magnitude scale|<math>M_\text{w}</math>]] [[2011 Tōhoku earthquake and tsunami|The Great Japan earthquake, March 2011]]
|-
|9.15
|800 megatons
|3.3 EJ
|[[Toba catastrophe theory#The Eruption|Toba eruption]] 75,000 years ago; among the largest known volcanic events.<ref>Petraglia, M.; R. Korisettar, N. Boivin, C. Clarkson,4 P. Ditchfield,5 S. Jones,6 J. Koshy,7 M.M. Lahr,8 C. Oppenheimer,9 D. Pyle,10 R. Roberts,11 J.-C. Schwenninger,12 L. Arnold,13 K. White. (6 July 2007). [http://toba.arch.ox.ac.uk/pub_files/Petraglia2007Science.pdf "Middle Paleolithic Assemblages from the Indian Subcontinent Before and After the Toba Super-eruption"]. ''Science'' 317 (5834): 114–116. {{doi|10.1126/science.1141564}}. PMID 17615356.</ref>
|-
|9.2
|950 megatons
|4.0 EJ
|[[Moment magnitude scale|<math>M_\text{w}</math>]] [[1964 Alaska earthquake|Anchorage earthquake (Alaska, USA), 1964]] [[Moment magnitude scale|<math>M_\text{w}</math>]] [[2004 Indian Ocean earthquake|Sumatra-Andaman earthquake and tsunami (Indonesia), 2004]]
|-
|9.5
|2.7 gigatons
|11 EJ
|[[Moment magnitude scale|<math>M_\text{w}</math>]] [[1960 Valdivia earthquake|Valdivia earthquake (Chile), 1960]]
|-
|10.0
|15 gigatons
|63 EJ
|Never recorded, equivalent to an earthquake rupturing a very large, lengthy fault, or an extremely rare/impossible mega-earthquake, shown in science fiction {{Clarify|date=April 2013}}
|-
|12.55
|100 teratons
|420 ZJ
|[[Yucatán Peninsula]] impact (creating [[Chicxulub crater]]) 65 [[mya (unit)|Ma]] ago (108 megatons; over 4x1029 ergs = [[1 E21 J|400 ZJ]]).<ref>{{Cite journal | doi = 10.1130/0091-7613(1998)026<0331:TCTBCC>2.3.CO;2 | issn = 0091-7613 | year = 1998 | volume = 26 | pages = 331–334 | title = The Cretaceous-Tertiary boundary cocktail: Chicxulub impact triggers margin collapse and extensive sediment gravity flows |author = Bralower, Timothy J. | coauthors= Charles K. Paull; R. Mark Leckie| journal = Geology |url=http://www.geosc.psu.edu/people/faculty/personalpages/tbralower/Braloweretal1998.pdf |accessdate=2009-09-03 | bibcode=1998Geo....26..331B}}</ref><ref>{{Cite journal | doi = 10.1130/0091-7613(2000)28<319:IMWATK>2.0.CO;2 | issn = 0091-7613 | year = 2000 | volume = 28 | pages = 319–322 | title = Impact-induced mass wasting at the K-T boundary: Blake Nose, western North Atlantic | author = Klaus, Adam | journal = Geology|accessdate=2009-09-03 | last2 = Norris | first2 = Richard D. | last3 = Kroon | first3 = Dick | last4 = Smit | first4 = Jan | bibcode=2000Geo....28..319K}}</ref><ref>{{Cite journal | doi = 10.1130/0091-7613(2002)030<0687:CLACST>2.0.CO;2 | issn = 0091-7613 | year = 2002 | volume = 30 | pages = 687–690 | title = Coastal landsliding and catastrophic sedimentation triggered by Cretaceous-Tertiary bolide impact: A Pacific margin example? | author = Busby, Cathy J. | coauthors= Grant Yip; Lars Blikra; Paul Renne |journal = Geology|accessdate=2009-09-03 | bibcode=2002Geo....30..687B}}</ref><ref>{{Cite journal | doi = 10.1130/0091-7613(2003)031<0557:UESFTL>2.0.CO;2 | issn = 0091-7613 | year = 2003 | volume = 31 | pages = 557–560 | title = Uniquely extensive seismite from the latest Triassic of the United Kingdom: Evidence for bolide impact? | author = Simms, Michael J. | journal = Geology|accessdate=2009-09-03 | bibcode=2003Geo....31..557S}}</ref><ref>{{cite web |author= Simkin, Tom |coauthors= Robert I. Tilling; Peter R. Vogt; Stephen H. Kirby; Paul Kimberly; David B. Stewart |title= This dynamic planet. World map of volcanoes, earthquakes, impact craters, and plate tectonics. Inset VI. Impacting extraterrestrials scar planetary surfaces|url=http://mineralsciences.si.edu/tdpmap/pdfs/impact.pdf |year=2006 |publisher= U.S. Geological Survey |accessdate=2009-09-03}}</ref>
|-
|22.88 or 32
|310 yottatons
|1.3×1039 J
|Approximate magnitude of the [[Starquake (astrophysics)|starquake]] on the [[magnetar]] [[SGR 1806-20]], registered on December 27, 2004.{{Clarify|date=April 2013}}
|}
* Quakes using the more modern magnitude scales will denote their abbreviations: [[Moment magnitude scale|<math>M_\text{w}</math>]] and [[Surface wave magnitude|<math>M_\text{s}</math>]]. Those that have no denoted prefix are [[Moment magnitude scale|<math>M_\text{L}</math>]]. Please be advised that the magnitude "number" (example 7.0) displayed for those quakes on this table may represent a significantly greater or lesser release in energy than by the correctly given magnitude (example [[Moment magnitude scale|<math>M_\text{w}</math>]]).

==Magnitude empirical formulae==

These formulae are an alternative method to calculate Richter magnitude instead of using Richter correlation tables based on Richter standard seismic event (<math>M_\mathrm{L}</math>=0, A=0.001mm, D=100 km).

The Lillie empirical formula:

:<math>M_\mathrm{L} = \log_{10}A - 2.48+ 2.76\log_{10}\Delta</math>
Where:
* A is the amplitude (maximum ground displacement) of the P-wave, in micrometers, measured at 0.8 Hz.
* <math>\Delta</math> is the epicentral distance, in km.

For distance less than 200 km:
:<math>M_\mathrm{L} = \log_{10} A + 1.6\log_{10} D - 0.15</math>

For distance between 200 km and 600 km:
:<math>M_\mathrm{L} = \log_{10} A + 3.0\log_{10} D - 3.38</math>

where A is [[seismograph]] signal amplitude in mm, D distance in km.

The Bisztricsany (1958) empirical formula for epicentral distances between 4˚ to 160˚:

:<math>M_\mathrm{L} = 2.92 + 2.25 \log_{10} (\tau) - 0.001 \Delta^{\circ} </math>
Where:
* <math>M_\mathrm{L}</math> is magnitude (mainly in the range of 5 to 8)
* <math>\tau</math> is the duration of the surface wave in seconds
* <math>\Delta</math> is the epicentral distance in degrees.

The Tsumura empirical formula:

:<math>M_\mathrm{L} = -2.53 + 2.85 \log_{10} (F-P) + 0.0014 \Delta^{\circ} </math>
Where:
* <math>M_\mathrm{L}</math> is the magnitude (mainly in the range of 3 to 5).
* <math>F-P</math> is the total duration of oscillation in seconds.
* <math>\Delta</math> is the epicentral distance in kilometers.

The Tsuboi, University of Tokio, empirical formula:

:<math>M_\mathrm{L} = \log_{10}A + 1.73\log_{10}\Delta - 0.83 </math>
Where:
* <math>M_\mathrm{L}</math> is the magnitude.
* <math>A</math> is the amplitude in um.
* <math>\Delta</math> is the epicentral distance in kilometers.

==See also==
{{portal|Earthquakes}}
{{div col|colwidth=30em}}
* [[1935 in science]]
* [[Japan Meteorological Agency seismic intensity scale]]{{spaced ndash}} does the same thing as the Mercalli Scale, but in different numbers
* [[Largest earthquakes by magnitude]]
* [[Mercalli intensity scale]] - Measures the intensity of an earthquake
* [[Moment magnitude scale]]
* [[Order of magnitude]]
* [[Rohn Emergency Scale]] for measuring the magnitude (intensity) of any emergency
* [[Seismic scale]]
* [[Seismite]]
* [[Timeline of United States inventions (1890–1945)]]
{{div col end}}

==References==
{{Reflist|30em}}

==External links==
* [http://www.iris.edu/seismon/ IRIS Real-time Seismic Monitor of the Earth]
* [http://earthquake.usgs.gov/learning/topics/mag_vs_int.php USGS: magnitude and intensity comparison]
* [http://earthquake.usgs.gov/aboutus/docs/020204mag_policy.php USGS: Earthquake Magnitude Policy]
* [http://neic.usgs.gov/neis/eqlists/eqstats.html USGS: 2000–2006 Earthquakes worldwide]
* [http://neic.usgs.gov/neis/eqlists/info_1990s.html USGS: 1990–1999 Earthquakes worldwide]
* [http://www.alaskarails.org/historical/earthquake/earthquake-richter.html Alaska Railroad Earthquake] with a table of yield-to-magnitude relations.
* [http://www.alabamaquake.com/energy.html Earthquake Energy Calculator] with seismic energy approximated in everyday equivalent measures.
{{Seismic scales}}

{{DEFAULTSORT:Richter Magnitude Scale}}
[[Category:1935 in science]]
[[Category:1935 introductions]]
[[Category:American inventions]]
[[Category:German inventions]]
[[Category:California Institute of Technology]]
[[Category:Seismic scales]]

Parry point (triangle)

2014-01-23T05:41:49Z

117.6.86.31:

In mathematics, in particular in [[functional analysis]], the '''Rademacher system''', named after [[Hans Rademacher]], is an [[Complete orthogonal system # Incomplete orthogonal sets|incomplete orthogonal system]] of functions on the [[unit interval]] of the following form:

: <math>\{ t \mapsto r_{n}(t)=\sgn ( \sin 2^{n+1} \pi t ) ; t \in [0,1], n \in \N \}.</math>

The Rademacher system is stochastically-independent, and is closely related to the [[Walsh transform|Walsh]] system. Specifically, the [[Walsh transform|Walsh system]] can be constructed as a product of Rademacher functions.

==References==
* {{Cite web| url=http://eom.springer.de/O/o070380.htm |title=Orthogonal system |work=Encyclopaedia of Mathematics }}
* {{Cite web| last=Heil | first=Christopher E. | title=A basis theory primer | url=http://www.math.gatech.edu/~heil/papers/bases.pdf | format=PDF | date=1997 }}

[[Category:Functional analysis]]

{{Mathanalysis-stub}}

Nine-point circle

2014-01-23T05:37:47Z

117.6.86.31: /* See also */

[[File:Orthosystem.png|thumb|ight|250px|'''Orthocentric system'''. Notice any point is the orthocenter of the triangle formed by the other three.]]
In [[geometry]], an '''orthocentric system''' is a [[Set (mathematics)|set]] of four [[point (geometry)|points]] on a [[plane (mathematics)|plane]], where one of which is the [[Altitude (triangle)|orthocenter]] of the [[triangle]] formed by the other three.

If four points form an orthocentric system, then ''each'' of the four points is the orthocenter of the other three. These four possible triangles will all have the same [[nine-point circle]]. Consequently these four possible triangles must all have [[circumcircle]]s with the same [[circumradius]].

==The common nine-point circle==

[[File:Nine point circle for orthocentric system.PNG|thumb|ight|250px|'''Common nine-point circle'''. Where O, O4 and A4 are the nine-point center, circumcenter orthocenter, respectively of the triangle formed from the other three orthocentric points A1, A2 and A3.]]

The center of this common nine-point circle lies at the [[centroid]] of the four orthocentric points. The radius of the common nine-point circle is the distance from the nine-point center to the midpoint of any of the six connectors that join any pair of orthocentric points through which the common nine-point circle passes. The nine-point circle also passes through the three orthogonal intersections at the feet of the altitudes of the four possible triangle.

This common nine-point center lies at the midpoint of the connector that joins any orthocentric point to the circumcenter of the triangle formed from the other three orthocentric points.

==The common orthic triangle, its incenter and excenters==

If the six connectors that join any pair of orthocentric points are extended to six lines that intersect each other, they generate seven intersection points. Four of these points are the original orthocentric points and the additional three points are the [[orthogonal]] intersections at the feet of the [[altitude]]s. The joining of these three orthogonal points into a triangle generates an [[orthic triangle]] that is common to all the four possible triangles formed from the four orthocentric points taken three at a time.

Note that the [[incenter]] of this common orthic triangle must be one of the original four orthocentric points. Furthermore, the three remaining points become the [[excenter]]s of this common orthic triangle. The orthocentric point that becomes the incenter of the orthic triangle is that orthocentric point closest to the common nine-point center. This relationship between the orthic triangle and the original four orthocentric points leads directly to the fact that the incenter and excenters of a reference triangle form an orthocentric system.

It is normal to choose the orthocentric point that is the incenter of the orthic triangle as H the orthocenter of the outer three orthocentric points that are chosen as a reference triangle ABC. In this normalized configuration the point H will always lie within the triangle ABC and all the angles of triangle ABC will be acute. The four possible triangles referred above are then triangles ABC, ABH, ACH and BCH. The six connectors referred above are AB, AC, BC, AH, BH and CH. The seven intersections referred above are A, B, C, H (the original orthocentric points) and HA, HB, HC (the feet of the altitudes of triangle ABC and the vertices of the orthic triangle).

==The orthocentric system and its orthic axes==

The orthic axis associated with a normalized orthocentric system A, B, C and H, where ABC is the reference triangle, is a line that passes through three intersection points formed when each side of the orthic triangle meets each side of the reference triangle. Now consider the three other possible triangles, ABH, ACH and BCH. They each have their own orthic axis.

==Euler lines and homothetic orthocentric systems==

[[File:Orthocentric system and their circumcenters.PNG|thumb|ight|250px|'''Orthocentric system'''. Where O1, O2, O3 and O4 are the circumcenters of the four possible triangles formed from the orthocentric points A1, A2, A3 and A4.]]

Let [[vector (geometry)|vectors]] ''a'', ''b'', ''c'' and ''h'' determine the position of each of the four orthocentric points and let ''n'' = (''a'' + ''b'' + ''c'' + ''h'') / 4 be the position vector of N, the common nine-point center. Join each of the four orthocentric points to their common nine-point center and extend them into four lines. These four lines now represent the Euler lines of the four possible triangles where the extended line ''HN'' is the Euler line of triangle ''ABC'' and the extended line ''AN'' is the [[Euler line]] of triangle ''BCH'' etc. If a point ''P'' is chosen on the Euler line ''HN'' of the reference triangle ''ABC'' with a position vector ''p'' such that ''p'' = ''n'' + α (''h'' − ''n'') where α is a pure constant independent of the positioning of the four orthocentric points and three more points ''P''''A'', ''P''''B'', ''P''''C'' such that ''pa'' = ''n'' + α (''a'' − ''n'') etc., then ''P'', ''P''''A'', ''P''''B'', ''P''''C'' form an orthocentric system. This generated othocentric system is always [[homothetic]] to the original system of four points with the common nine-point center as the homothetic center and α the ratio of [[similitude]].

When ''P'' is chosen as the centroid ''G'', then α = −1/3. When ''P'' is chosen as the [[circumcenter]] O, then α = −1 and the generated orthocentric system is [[congruence (geometry)|congruent]] to the original system as well as being a reflection of it about the nine-point center. In this configuration ''P''''A'', ''P''''B'', ''P''''C'' form a [[Johnson circles|Johnson triangle]] of the original reference triangle ''ABC''. Consequently the [[circumcircle]]s of the four triangles ''ABC'', ''ABH'', ''ACH'', ''BCH'' are all equal and form a set of [[Johnson circles]] as shown in the diagram adjacent.

==Further properties==

The four Euler lines of an orthocentric system are orthogonal to the four orthic axes of an orthocentric system.

The six connectors that join any pair of the original four orthocentric points will produce pairs of connectors that are orthogonal to each other such that they satisfy the distance equations

:<math> AB^2 + CH^2 = AC^2 + BH^2 = BC^2 + AH^2 = 4R^2 </math>

where ''R'' is the common circumradius of the four possible triangles. These equations together with the [[law of sines]] result in the identity

:<math> \frac{BC}{\sin{A}} = \frac{AC}{\sin{B}} = \frac{AB}{\sin{C}} = \frac{HA}{|\cos{A}|} = \frac{HB}{|\cos{B}|} = \frac{HC}{|\cos{C}|} = 2R. </math>

[[Feuerbach's theorem]] states that the nine-point circle is tangent to the incircle and the three excircles of a reference triangle. Because the nine-point circle is common to all four possible triangles in an orthocentric system it is tangent to 16 circles comprising the incircles and excircles of the four possible triangles.

Any conic that passes through the four orthocentric points can only be a rectangular [[hyperbola]].
This is a result of Feuerbach's conic theorem that states that for all circumconics of a reference triangle that also passes through its orthocenter, the [[locus (mathematics)|locus]] of the center of such circumconics form the nine-point circle and that the circumconics can only be rectangular hyperbolas.

Note that the locus of the perspectors of this family of rectangular hyperbolas will always lie on the four orthic axes. So if a rectangular hyperbola is drawn through four orthocentric points it will have one fixed center on the common nine-point circle but it will have four perspectors one on each of the orthic axes of the four possible triangles. Note also that the one point on the nine-point circle that is the center of this rectangular hyperbola will have four different definitions dependent on which of the four possible triangles is used as the reference triangle.

The well documented rectangular hyperbolas that pass through four orthocentric points are the Feuerbach, Jeřábek and Kiepert circumhyperbolas of the reference triangle ABC in a normalized system with H as the orthocenter.

The four possible triangles have a set of four [[Circumconic and inconic|inconics]] known as the orthic inconics that share certain properties. The contacts of these inconics with the four possible triangles occur at the vertices of their common orthic triangle. In a normalized orthocentric system the orthic inconic that is tangent to the sides of the triangle ABC is an inellipse and the orthic inconics of the other three possible triangles are hyperbolas. These four orthic inconics also share the same [[Brianchon theorem|Brianchon]] point, H, the orthocentric point closest to the common nine-point center. The centers of these orthic inconics are the [[symmedian point]]s, K of the four possible triangles.

There are many documented cubics that pass through a reference triangle and its orthocenter. The circumcubic known as the orthocubic - K006 is interesting in that it passes through three orthocentric systems as well as the three vertices of the orthic triangle (but not the orthocenter of the orthic triangle). The three orthocentric systems are the incenter and excenters, the reference triangle and its orthocenter and finally the orthocenter of the reference triangle together with the three other intersection points that this cubic has with the circumcircle of the reference triangle.

==References==
* {{mathworld|urlname=OrthocentricSystem|title=Orthocentric System}}
* {{mathworld|urlname=Orthocenter|title=Orthocenter}}
* {{mathworld|urlname=FeuerbachsTheorem|title=Feuerbach's Theorem}}
* {{mathworld|urlname=FeuerbachsConicTheorem|title=Feuerbach's Conic Theorem}}
* {{mathworld|urlname=FeuerbachHyperbola|title=Feuerbach Hyperbola}}
* {{mathworld|urlname=JerabekHyperbola|title=Jerabek Hyperbola}}
* {{mathworld|urlname=KiepertHyperbola|title=Kiepert Hyperbola}}
* {{mathworld|urlname=OrthicInconic|title=Orthic Inconic}}
* {{mathworld|urlname=OrthicAxis|title=Orthic Axis}}
* {{mathworld|urlname=Perspector|title=Perspector}}
* Bernard Gibert [http://perso.orange.fr/bernard.gibert/Exemples/k006.html Circumcubic K006]
* Clark Kimberling, "[http://faculty.evansville.edu/ck6/encyclopedia/ETC.html Encyclopedia of triangle centers]". ''(Lists some 5000 interesting points associated with any triangle.)''

[[Category:Triangle geometry]]
[[Category:Quadrilaterals]]

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2012-01-17T03:34:06Z

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