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In [[geometry]], '''Legendre's theorem on spherical triangles''', named after [[Adrien-Marie Legendre]], is stated as follows:


: Let ABC be a spherical triangle on the ''unit'' sphere with ''small'' sides ''a'', ''b'', ''c''. Let A'B'C be the planar triangle with the same sides.  Then the angles of the spherical triangle exceed the corresponding angles of the planar triangle by approximately one third of the [[spherical excess]] (the spherical excess is the amount by which the sum of the three angles exceeds {{pi}}).


[[File:Spherical trigonometry legendre.svg|thumb|400px|center]]
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The theorem was very important in simplifying the heavy numerical work in calculating the results of traditional (pre-GPS and pre-computer) geodetic surveys from about 1800 until the middle of the twentieth century.
 
The theorem  was stated by {{harvtxt|Legendre|1787}}  who provided a proof [[#{{harvid|Legendre|1798|}}|(1798)]]  in a supplement to the report of the measurement of the French meridional arc used in the definition of the [[metre]] {{harv|Delambre|1798}}. Legendre does not claim that he was the originator of the theorem despite the attribution to him. {{harvtxt|Tropfke|1903}} maintains that the method was in common use by surveyors at the time and may have been used as early as 1740 by [[Charles Marie de La Condamine|La Condamine]] for the calculation of the [[French Geodesic Mission|Peruvian meridional arc]].
 
[[Spherical trigonometry#Area and spherical excess|Girard's theorem]] states that the spherical excess of a triangle, ''E'', is equal to its area, &Delta;, and therefore Legendre's theorem may be written as
 
::<math>
\begin{align}
A-A'\;\approx\; B-B'\;\approx\; C-C'\;\approx\;\frac13 E\;=\; \frac13\Delta,\qquad a,\;b,\;c\,\ll\, 1.
\end{align}
</math>
 
The excess, or area, of small triangles is very small. For example consider an equilateral spherical triangle with sides of 60&nbsp;km on a spherical Earth of radius 6371&nbsp;km; the side  corresponds to an angular distance of 60/6371=.0094, or approximately 10<sup>−2</sup> radians (subtending an angle of 0.57° at the centre). The area of such a small triangle is well approximated by that of a planar equilateral triangle with the same sides: {{frac|1|2}}''a''<sup>2</sup>sin({{pi}}/3)&nbsp;=&nbsp;0.0000433 radians corresponding to&nbsp;8.9″.
 
When the sides of the triangles exceed 180&nbsp;km, for which the excess is about 80″, the relations between the areas and the differences of the angles must be corrected by terms of fourth order in the sides, amounting to no more than 0.01″:
 
::<math>\begin{align}
\Delta &=\Delta'\left( 1+\frac{a^2+b^2+c^2}{24} \right),\\
A&=A'+\frac{\Delta}{3}  +\frac {\Delta}{180}  \left( -2a^2+b^2+c^2 \right),\\
B&=B'+\frac{\Delta}{3}  +\frac {\Delta}{180}  \left({\quad a^2-2b^2+c^2} \right),\\
C&=C'+\frac{\Delta}{3}  +\frac {\Delta}{180}  \left({\quad a^2+b^2-2c^2} \right).
\end{align}</math>
 
(&Delta;&prime; is the area of the planar triangle.)  This result was proved by {{harvtxt|Buzengeiger|1818}}—an extended proof may be found in {{harvtxt|Osborne|2013}} (Appendix D13).  Other results are surveyed by {{harvtxt|Nádeník|2004}}.
 
The theorem may be extended to the ellipsoid if  ''a'', ''b'', ''c'' are calculated by dividing the true lengths by the square root of the product of the principal radii of curvature (see {{harvtxt|Osborne|2013}} Chapter 5) at the median latitude of the vertices (in place of a spherical radius). {{harvtxt|Gauss|1828|loc=Art. 26&ndash;28}} provided more exact formulae.
 
==References==
*{{citation|first=Karl Heribert Ignatz
|last=Buzengeiger
|title=Vergleichung zweier kleiner Dreiecke von gleichen Seiten, wovon das eine sphärisch, das andere eben ist
|year=1818
|journal=Zeitschrift für Astronomie und verwandte Wissenschaften (v6)
|pages=264–270
|url=http://books.google.co.uk/books?id=GXUwAAAAYAAJ&pg=PA264#v=onepage&q&f=false
}}
*{{citation
|last=Clarke
|first=Alexander Ross
|year=1880
|title=Geodesy
|url=http://books.google.com/books?id=lfIoAAAAYAAJ
|publisher=Clarendon Press
|postscript=. Republished at [http://www.forgottenbooks.org/info/Geodesy_1000151390.php Forgotten Books].
 
}}
*{{cite book
|ref = {{harvid|Gauss|1828}}
|last = Gauss
|first = C. F.
|year = 1902
|origyear = 1828
|title = General Investigations of Curved Surfaces of 1827 and 1825
|publisher = Princeton Univ. Lib
|url = http://books.google.com/books?id=a1wTJR3kHwUC
|authorlink = Carl Friedrich Gauss
|postscript = .  English translation of [http://books.google.com/books?id=bX0AAAAAMAAJ&pg=PA3 ''Disquisitiones generales circa superficies curvas''] (Dieterich, Göttingen, 1828).
}}
*{{citation|first=Adrien-Marie
|last=Legendre
|authorlink=Adrien-Marie Legendre
|title=Mémoire sur les opérations trigonométriques, dont les résultats dépendent de la figure de la Terre
|year=1787
|page=7
|url=http://gallica.bnf.fr/ark:/12148/bpt6k6395708p
|others=Article &nbsp;VI [http://gallica.bnf.fr/ark:/12148/bpt6k6395708p/f7.zoom]}}
*{{citation|first=Adrien-Marie
|last=Legendre
|title=Méthode pour déterminer la longueur exacte du quart du méridien d’après les observations faites pour la mesure de l’arc compris entre Dunkerque et Barcelone
|year=1798
|pages=12–14 (Note&nbsp;III [http://www.e-rara.ch/zut/content/pageview/580472])
|url=http://www.e-rara.ch/zut/content/pageview/580461
}}
*{{citation|first=Zbynek
|last=Nádeník
|title=Legendre theorem on spherical triangles
|year=2004
|publisher=
|url=http://www.vugtk.cz/odis/sborniky/sb2005/Sbornik_50_let_VUGTK/Part_1-Scientific_Contribution/2-Nadenik.pdf}}
*{{citation|first=Peter
|last=Osborne
|title=The Mercator Projections
|year=2013
|url=http://www.mercator99.webspace.virginmedia.com
}}
*{{citation|first=Johannes
|last=Tropfke
|title= Geschichte der Elementar-Mathematik (Volume 2).
|year=1903
|page=[http://archive.org/stream/geschichtederel01tropgoog#page/n307 295]
|publisher=Verlag von Veit
|url=http://archive.org/stream/geschichtederel01tropgoog#page/n3
}}
 
{{DEFAULTSORT:Spherical Trigonometry}}
[[Category:Spherical trigonometry]]

Latest revision as of 05:37, 3 November 2014


However, dangerous sexual predators like to target kids in these chat rooms also. She spoke briefly about how they work to fund queer artists, especially as the recession has made it difficult for the arts to get money at all. 32) Computer porn: In one of the office segments, you can use some PCs to watch some SFW porn. To this end, Sony started to develop the DVR Blue, a joint project with Pioneer. com offers free full length pornographic and sex movies. Scream had given the horror genre the shot in the arm it needed, and suddenly there was a resurgence. Porn stars are chosen just as any other artists are chosen - to fit the role assigned to them.

At times Man would open up and bring me closer to a world he shared with no one, not even his own siblings. In addition, there are many websites where one can download all of them from. We had the sex toy revolution first - no more giant cucumbers but a gold-plated mini vibrator perfect for clit stimulation - and now the second wave is the porn revolution: porn that is made by and for women, that focuses on female pleasure and features male sex objects. Because of the insights provided by the portable Penetrator Wi - Fi pen testing appliance, you will be able to implement solutions to patch the vulnerabilities of your network. When you loved this short article and you want to receive much more information concerning hdporn generously visit our page. You'll love them through and through even though they're just here for the show. In another meeting with Man, he implied that I might be a US Marshal trying to capture his brothers and wanted to know if I was wired. In addition to keeping good security software in place, it doesn't hurt to run regular scans of the entire system.

t remember meeting a man or woman who struggled with porn and didn. The church is coming under fire because it meets weekly at Grassfield High School in Chesapeake where school officials have expressed their concerns to the pastor. The best step for this is treatment and it is something you need to do for you and those you love most in the world. His mom was a book editor who was asked to serve the government's OSS at the time of World War because she fluently spoke French and German. Lee started his career at age 18 when he portrayed gay roles in contemporary Asian adult films. Another way to seduce her is to add some romance to your day. They are in fact looking for a job in order to earn some money.

' Then I realized there were more than three prizes and I felt disappointed that I had put my ticket into that bowl, but it was too late to take it back. , The Weakest Link, and The Million Pound Drop Live have all seen record audience numbers when their American adaptations have aired on television networks over there, but the United States has a reputation of taking British formats or programmes and creating an adaptation that is doomed to fail from the start. The best part about all these enticing scenes is that you do not have to pay a single penny to watch them, and hence it is great for many people because they just watch free porn anywhere they like without having to pay any money. While this is not actually recognized in the “Diagnostic and Statistical Manual of Mental Disorders”, more and more mental health professionals are starting to recognize this as a serious addiction that can lead to many negative life changing emotions and behaviors. 16: 'In a controversial interview, author John Grisham called for lighter sentences for some of the people convicted of sexual offenses. Second to his record was another fellow top porn star named Peter North, who appeared in 1,700 porn flicks. There are some which fleece hardworking models out of their money or set a minimum so their money remains locked inside for quite some time.

There are plenty of different types of these types of porn videos available on the web that you can down load. Seeking diversion in fun and games is a sign of the times, and noble or not, the astute marketer can realize that there is a lot of money to be made catering to the praetorian. When you are all alone and you don't have anything to do, that is when you are most likely to watch porn. Remember this golden rule: Be prepared for the worst, so when the worst happens you will be doing just fine. The Ability to Block or Limit Social Networking Sites. Not sure, what he meant I pushed on, but his big southern smile always brought the conversation to another topic. That said, taking out enemies is still a blast, and Heroes' Paradise does a great job of allowing you to let loose in some pretty cool environments against hordes of enemies.

All of it is to produce the mood of the scene and it does so rather well. Until the general public becomes more informed about the reality of how pornography affects the human brain it will continue to be looked at as a moral weakness or a form of mere entertainment rather than a true chemical addiction. A phase preoccupied with credit binging, fiat money and spectacle. It presents a host of erotic programming to suit each and every taste. Hacking is preventable because it uses known methods to exploit the vulnerabilities of Wi - Fi networks. Generally situations are unpredictable and ending a relationship often is the only choice for a person which is true significantly in case of marital relationships. Iggy Azalea's alleged porn video could be released any day now because the footage is now in the hands of a major company in the porn industry who is said to have offered Azalea a substantial amount of cash for permission to release the footage for commercialization, IBT said in a Sep.

If it's Such a Big Problem Then Why Do We Never Hear About it. This is a 30-day program that caters to many different individuals. Lust, Joy and Span have previously won awards in different categories. Downloading harmful links and software is becoming easier and easier. Some years ago, Unsolved Mysteries did a segment about the Anglin brothers; I had the good fortune of meeting the Director of that segment who was now one of the producers I was working with. So you are using Ubuntu and you can't access some important files eh. He craves attention from real, live, people, flesh and blood.