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3.2.analiitiska geometrija
- 2. Taisnes vienādojums plaknē • Taisnes vienādojums koordinātu sistēmas plaknē ir pilnīgi noteikts , ja ir dots kāds punkts M0(x0;y0), caur kuru taisne iet, un vektors s0, kuram taisne ir paralēla.
- 3. . M (x;y) M 0M r r0 . M0 r M0M r kolinears s0 r0 t s0 r0 s0 Vektoriālais vienādojums
- 4. s0 Vienības vektors x y cos cos 1 1 x cos y sin s0 s0 cos ; sin
- 5. . M (x;y) s0 r r0 t s0 cos ; sin y y0 . M0 x x0 y y0 x x0 x x0 t cos y y0 y y0 t sin tg x x0
- 6. Taisnes virziena koeficients y y0 y y0 tg k x x0 x x0 k tg y y0 k x x0 y kx b Ax By C 0
- 7. Taisnes vienādojums asu nogriežņos Ax By C 0 Ax By C x y 1 a b
- 8. Taisnes normālvienādojums Ax By C 0 1 Normējošais reizinātājs 2 2 A B Ax By C 0 x cos y sin p 0
- 9. Taisnes vienādojums caur diviem punktiem . . M2 (x2;y2) M1 (x1;y1) x x1 y y1 0 x2 x1 y2 y1 y y1 x x1 y2 y1 x2 x1 x x1 y2 y1 y y1 x2 x1 0
- 10. Leņķis starp divām taisnēm A1 Taišņu perpendikularitātes A1 x B1 y C1 0 k1 B1 nosacījums A2 1 A2 x B2 y C2 0 k2 k2 B2 k1 Taišņu paralelitātes k 2 k1 nosacījums tg 1 k1k 2 k1 k2
- 11. Divu taišņu krustpunkts A1 x B1 y C1 0 A2 x B2 y C2 0 Jāizmanto Krāmera formulas
- 12. Attālums no punkta līdz taisnei • Aprēķina normējošo reizinātāju. • Iegūst taisnes normālvienādojumu. • Aprēķina attālumu no punkta līdz taisnes – ievieto punkta koordinātas normālvienādojumā. d x cos y sin p
- 13. Riņķa līnija 2 2 Ax 2 Bxy Cy 2 Dx 2 Ey F 0 2 2 2 x y R Kanoniskais vienādojums 2 2 2 x a y b R
- 14. Elipse
- 15. Elipse • Elipse – to punktu ģeometriskā vieta plaknē, kuru attālumu summa līdz diviem dotajiem plaknes punktiem F1 un F2 ir konstanta.
- 16. A1A2 – elipses lielā ass – elipses fokālā ass F1, F2 – elipses fokusi B1B2 – elipses mazā ass 2c – attālums starp fokusiem a – elipses lielā pusass r1, r2 – punkta P fokālie fokusi b – elipses mazā pusass B1 r1 + r2 = 2a A2 F1 F2 A1 x2 y2 2 2 1 a b Ekscentritāte – attiecība a/c a2 c2 b2 a2 b2 x 2 y 2 B2 1 a a2 a 2 c 2
- 17. Hiperbola • Hiperbola – visu to punktu ģeometriskā vieta plaknē, kuru attālumu starpība līdz diviem dotajiem punktiem F1 un F2 pēc absolūtās vērtības ir konstanta.
- 18. A1A2 – hiperbolas reālā ass – fokālā ass F1, F2 – hiperbolas fokusi B1B2 – hiperbolas imaginārā ass 2c – attālums starp fokusiem a – hiperbolas reālā pusass r1, r2 – hiperbolas fokālie rādiusi b – hiperbolas imaginārā pusass r1 - r2 = 2a 2 2 x y 2 2 1 b a b y x a a2 c2 b2 b Asimptotas y x 2 2 x 2 y 2 a a b 1 a2 a 2 c 2 a
- 19. y2 = 4ax y2 = -4ax x2 = 4ay x2 = -4ay
- 20. Parabola y 2 2 px • Parabola – visu to punktu ģeometriskā vieta plaknē, kuriem attālums līdz dotajam punktam (fokusam) ir vienāds ar attālumu līdz dotajai taisnei (direktrisei). p 2a p F ;0 2 p x 2 OF r ON d
Notes de l'éditeur
- Действие фильма разворачивается в городе Александрия Египетская в начале V века нашей эры. Фильм рассказывает о Гипатии Александрийской — первой в истории женщине-учёном, которая была философом, математиком и астрономом. Она была очень умна и известна, к её словам прислушивались правители. Её учениками были многие видные деятели, в частности префект города Орест.В то время в городе начались религиозные столкновения, в результате которых христианами была разграблена и уничтожена одна из богатейших библиотек мира. Бывший раб ГипатииДавус, тайно влюблённый в неё, тоже стал христианином и присоединился к погромщикам. Со временем церковь приобретала всё большую власть, и вскоре епископ Кирилл Александрийский потребовал, чтобы Орест и другие отреклись от Гипатии, а ей самой запретили преподавать и заниматься науками. Она не могла смириться с этим и принять христианство, поэтому была объявлена ведьмой, проституткой и убита.In the 4th century city of Alexandria, in the Roman Empire, Greek philosopher Hypatia (Rachel Weisz), is a teacher at the Platonic school, where future leaders are educated. Hypatia is the daughter of Theon (Michael Lonsdale), the director of the Musaeum of Alexandria. Hypatia, her slave Davus (Max Minghella), and her pupils, Orestes (Oscar Isaac) and Synesius (Rupert Evans), are immersed in the changing political and social landscape. She rejects Orestes's love, because she prefers to stay an independent scientist. Davus assists Hypatia in her classes and is interested in science, and is also secretly in love with her.Meanwhile, social unrest begins challenging the Roman rule of the city as pagans and Christians come into conflict. When the Christians start defiling the statues of the pagan gods, the pagans, including Orestes and Hypatia's father, ambush the Christians to quash their rising influence. However, in the ensuing battle, the pagans unexpectedly find themselves outnumbered by a large Christian mob. Hypatia's father is gravely injured and Hypatia and the pagans take refuge in the Library of the Serapeum. The Christian siege of the library ends when an envoy of the Roman Emperor declares that the pagans are pardoned, however the Christians shall be allowed to enter the library. Hypatia and the pagans flee, trying to save the most important scrolls, before the Christians overtake the library and destroy its contents. Davus, torn between his love for Hypatia and the possibility of gaining his freedom by joining the rising tide of Christianity, and annoyed by Hypatia calling him an idiot for not being present when he is needed most to help saving the scrolls, chooses to join the Christian forces. He returns with a sword and starts sexually assaulting her, but quickly regrets this and offers his sword to her to kill him. However, she removes the slave collar around his neck and tells him he is free now.Several years later, Orestes, now converted to Christianity, is prefect of Alexandria. Hypatia continues to investigate the motions of the Sun, the Moon, the five known "wanderers" (planets) and the stars. Some Christians ridicule Hypatia for thinking that the Earth is a sphere, by arguing that people far from the top would fall off the Earth. When they ask Davus his opinion he avoids conflict by saying that only God knows these things.Hypatia also investigates the heliocentric model of the solar system proposed by Aristarchus of Samos; by having an object dropped from the mast of a moving ship she demonstrates to Orestes that a possible motion of the Earth would not affect the motion, relative to Earth, of a falling object on Earth. However, due to religious objections against heliocentrism, the Christians have now forbidden Hypatia to teach at the school. The Christians and the Jews come into conflict, committing violent acts against each other, with the Christians ultimately wresting power from the only other religious group remaining. Davus, remembering how Hypatia forgave him, suggests that perhaps the Christians should forgive the Jews, as Jesus did, but the Christians respond by rebuking him for comparing himself with Jesus.The leader of the Christians, Cyril (Sami Samir), views Hypatia as having too much influence over Orestes and stages a public ceremony intended to force Orestes to subjugate her. Hypatia's former pupil, Synesius, now the Bishop of Cyrene, comes to her rescue as a religious authority counterweight, but says he cannot help her unless she accepts Christianity; she refuses. Hypatia makes a great discovery, finding that the Earth orbits around the Sun in an elliptic, not circular, orbit with the Sun at one of the foci. Cyril convinces a mob of Christians that Hypatia is a witch and they vow to kill her. Davus tries to run ahead to warn Hypatia, but she is captured by the mob. They strip Hypatia naked and are about to skin her alive until Davus persuades the mob otherwise, and they decide tostone her instead. When everyone goes outside to collect stones, Davus, with Hypatia's consent, secretly suffocates her and tells the mob that she fainted. Since she is already dead, she does not feel anything when the crowd begins to stone her.[edit]CastRachel Weisz as Hypatia of Alexandria. Weisz was already a fan of Amenábar's work when she received the script, and was very interested in the role.[2] Although she had not heard of Hypatia before, she felt that her history was still relevant to the contemporary world: "Really, nothing has changed. I mean, we have huge technological advances and medical advances, but in terms of people killing each other in the name of God, fundamentalism still abounds. And in certain cultures, women are still second-class citizens, and they’re denied education."[3] Weisz wanted to delve more into Hypatia's sexuality and her desires, but Amenábar disagreed. She also received science lessons to help inform her depiction of the character.[2] At the 2009 Cannes Film Festival, Weisz spoke about her style and approach: "There's no way we could know how people behave in the 4th century. I imagine they were still human beings with the same emotions as we have now. There are cultural customs, I guess, which were different. We approach the acting style to make the people flesh and blood and to make the acting incredibly naturalistic."[4]Max Minghella as Davus, Hypatia's slave. Davus is in love with Hypatia, but it is an unrequited love, and Davus turns towards Christianity instead. The character of Davus was invented as "eyes for the audience" and is not based on any historical account.[5] Minghella grew up in Hampstead, the same town as Weisz, and found it very easy to work with her.[6]Oscar Isaac as governor Orestes. Student of Hypatia, Orestes is an aristocrat, who like Davus, falls in love with his teacher, and has a strong friendship with Hypatia.[5] Isaac was familiar with the history of early Christianity during the period represented in the film, but like Weisz, he had not heard of Hypatia before joining the project.[7]Sami Samir as Saint Cyril of Alexandria, Bishop of AlexandriaManuel Cauchi as bishop Theophilus of Alexandria, uncle of CyrilAshrafBarhom as Ammonius, a parabalani monkMichael Lonsdale as Theon of Alexandria, father of Hypatia and director of the Musaeum of AlexandriaRupert Evans as bishop Synesius of CyreneHomayounErshadi as Aspasius the old slave. He acts as Hypatia's research assistant.[edit]Production[edit]DevelopmentIt's a movie that challenges the audience in terms of reasoning and trying to get into the story. I kept saying the movie is about astronomy and I wanted to express concepts that we study in school—science, mathematics—that don’t show how fascinating the topic is [the way the subjects are taught in modern education]. I wanted to translate [man’s] fascination with the pursuit of knowledge. I wanted to show astronomy and those who study it in the most appealing way. Those are the real heroes of the movie.Alejandro Amenábar[8]After Amenábar completed The Sea Inside (2004), he took a break and traveled to the island of Malta, where he used his free time to explore the night sky. Seeing the Milky Way galaxy, Amenábar began to discuss astronomy with his friends, speculating about extraterrestrial life on other planets. He started to research astronomy and came across Cosmos: A Personal Voyage, by American astronomer Carl Sagan.[9] Amenábar also studied historical figures such as Ptolemy, Copernicus, Johannes Kepler and Galileo,[10] but found himself most interested in the story of Hypatia, a 4th century Greek astronomer whose history, he felt, was still relevant in the 21st century: "We realized that this particular time in the world had a lot of connections with our contemporary reality. Then the project became really, really intriguing, because we realized that we could make a movie about the past while actually making a movie about the present."[11]To prepare for the task of recreating the ancient city of Alexandria without relying on computer generated imagery, Amenábar reviewed older sword-and-sandal films such as The Ten Commandments (1956), Ben-Hur (1959), and Pharaoh (1966).[9] A year before the start of pre-production, designer Guy Hendrix Dyas spent three weeks with Amenábar in Madrid to do some preliminary work on the set designs and the recreation of the ancient city of Alexandria so that previs animations could be generated.The film was produced by Fernando Bovaira, with Telecinco Cinema as the primary producer along with Mod Producciones, Himenoptero, and Sogecable as co-producers.[12][edit]FilmingPrincipal photography began on March 17, 2008, on the island of Malta, and was scheduled to last 15 weeks.[13] Production designer Guy Hendrix Dyas used large sets on location instead of computer generated imagery at Amenábar's direction.[14] The construction of the set employed almost 400 people, and was the largest ever designed on the island. Actor Charles Thake (Hesiquius) suffered minor facial injuries on the set when he collided with extras running during a scene.[15] Filming ended in June.[8][edit]ReleaseAgora premiered at the 2009 Cannes Film Festival,[16] but the film was initially unable to find a domestic distributor due to its large budget and length.[17] The film also had trouble finding a distributor in both the USA and Italy, although it eventually found distributors in both countries.[18][19] The North America premiere was held at the Toronto International Film Festival in September, 2009.[20] Agoraopened in Spain on October 9, 2009, breaking box office records for that country.[21] A limited release in the U.S. began on May 28, 2010, opening on two screens at the Paris Theatre and the Sunshine Cinema in New York. The film opened on the West Coast of the U.S. on June 4, playing only two screens: at The Landmark theatre in Los Angeles and at Regal'sWestpark 8 in Irvine.[22][edit]Home mediaIn March 2010, Agora was released in Region 2-locked DVD and Blu-Ray formats.[23] A Region 1-locked DVD was released October 2010.[24][edit]Reception[edit]Critical responseA visually imposing, high-minded epic that ambitiously puts one of the pivotal moments in Western history onscreen for the first time.Todd McCarthy, Variety[25]The film holds a 53% 'Rotten' rating on Rotten Tomatoes based on 85 reviews.[26] It also scored more than 7.5 stars out of 10 on IMDb(The Internet Movie Database) within the first three months of its release, and as of June 2011 the rating lies at 7.2.[27]British writer and film critic Peter Bradshaw of The Guardian praised Alejandro Amenábar and his film, describing Agora as "an ambitious, cerebral and complex movie...Unlike most toga movies, it doesn't rely on CGI spectacle, but real drama and ideas." Bradshaw also applauded Rachel Weisz's role as Hypatia, calling it "an outstanding performance".[28] Critic Roger Ebert liked the film and gave it 3 stars out of four. He said: "I went to see "Agora" expecting an epic with swords, sandals and sex. I found swords and sandals, some unexpected opinions about sex, and a great deal more. " [29]The Religious Anti-Defamation Observatory (ObservatorioAntidifamaciónReligiosa), a Spanish Catholic group, claimed that the film was responsible for "promoting hatred of Christians and reinforcing false clichés about the Catholic Church."[18] Michael Ordoña of the Los Angeles Times acknowledges that the film has been criticized for "perceived slights against Christians" but that "its lack of condemnation of specific dogma makes the film's target seem to be fundamentalism in general".[30][edit]Box officeAgora was Spain's highest grossing film of 2009, earning over $10.3 million within four days of its release on October 9.[26] The film grossed over $32.3 million (€ 21.4 million) by December 1 and grossed about $35 million by February 1, 2010. Based on North American theatre tracking data from Rentrak Theatrical, indieWIRE reported that Agora "scored the highest per-theater-average of any film in the marketplace" during the Memorial Day holiday weekend from May 28 through May 31, just after its U.S. limited release.[31] Despite this very high per-theater average, Agora was never widely released in the USA. According to Box Office Mojo, its widest release in the USA was just 17 theaters.[32]As of January 10, 2011, Agora's worldwide box office earnings are approximately $39 million.[33] DVD and Blu-Ray sales numbers for Agora are not publicly available.[edit]AccoladesAgora was nominated for 13 Goya Awards, winning 7.[14] The film won the Alfred P. Sloan Foundation Feature Film Prize ($25,000.00 USD) in 2009 at the Hamptons International Film Festival.[34][edit]Historical accuracyBefore its release, the distribution company insisted on screening the film at the Vatican. No objections were reported and Vatican officials assisted in some of the religious depictions. According to Amenábar, "There’s one scene in which Cyril reads from St. Paul and [the Vatican] tried to look for the softest version. In the English version, [it is] taken from the King James version of the Bible."[8]The line is excerpted from 1 Timothy 2:8 - 2:12.[35]Antonio Mampaso, a Spanish astrophysicist and one of Agora's scientific advisors, stated in an interview that "we know that Hypatia lived in Alexandria in the IV and V centuries CE, until her death in 415. Only three primary sources mention Hypatia of Alexandria, apart from other secondary ones". He added that none of Hypatia's work has survived but it is thought, from secondary sources, that her main fields of study and work were geometry and astronomy. Mampaso claimed that Hypatia invented the hydrometer, an instrument still in use today, and that probably her father Theon of Alexandria, together with Hypatia, invented the astrolabe.[36] However, it is generally accepted that the astrolabe had already been invented a couple of centuries earlier, and that the instrument was known to the Greeks before the Christian era.[37][38] Similarly, the hydrometer was invented before Hypatia, and already known in her time. Synesius sent Hypatia a letter describing a hydrometer, and requesting her to have one constructed for him.[39][40][41]Some reviewers have heavily criticized Agora for historical innacuracies, heavy artistic licenses and perceived anti-Christian bias in the movie. Robert Barron, an American Catholic priest, writes in an article: "Hypatia was indeed a philosopher and she was indeed killed by a mob in 415, but practically everything else about the story that Gibbon and Sagan and Amenábar tell is false".[42] Irene A. Artemi, a Dr. of Theology of the Athens University, states that "the movie - albeit seemingly not turning against the Christian religion - is in fact portraying the Christians as fundamentalist, obscurantist, ignorant and fanatic".[43] Similarly, the atheist Armarium Magnum blog said: "Over and over again, elements are added to the story that are not in the source material: the destruction of the library, the stoning of the Jews in the theatre, Cyril condemning Hypatia's teaching because she is a woman, the heliocentric "breakthrough" and Hypatia's supposed irreligiousity