"가우스의 놀라운 정리(Theorema Egregium)"의 두 판 사이의 차이

수학노트
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==개요==
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*  학부 미분기하학에서 배우게 되는 중요한 정리 중의 하나
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*  가우스 곡률은 곡면이 얼마나 휘어 있는가를 재는 양
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*  이 가우스 곡률은 그 곡면의 거리와 각도를 재는 것으로 알수 있다는 정리
  
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==가우스 곡률==
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* [[가우스 곡률|가우스곡률]]
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:<math>K = -\frac{1}{2\sqrt{EG}}\left(\frac{\partial}{\partial u}\frac{G_u}{\sqrt{EG}} + \frac{\partial}{\partial v}\frac{E_v}{\sqrt{EG}}\right)</math>
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==지도제작에의 의미==
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*  구면의 아주 작은 부분이라고 할지라도 수학적으로 엄밀하게 거리와 각도가 모두 보존되도록 하는 평면지도를 그릴수 없다는 것을 의미함.
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**  만약 이것이 가능하려면, 구면과 평면의 가우스 곡률이 같아야 함.
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**  그러나 구면의 가우스 곡률은 언제나 양수이고, 평면의 가우스 곡률은 언제나 0 이다.
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*  이것은 지도제작에 언제나 존재하게 되는 딜레마를 의미함.
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*  지도를 제작한다면 원하는 성질을 얻는 대신, 무언가는 희생해야 한다는 것을 뜻함.
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* [[수학과 지도학|지도와 수학]] 항목 참조
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==역사==
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* [[수학사 연표]]
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==관련된 항목들==
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* [[가우스-보네 정리]]
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* [[제3부 지구위의 딱정벌레]]
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==사전형태의 자료==
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* [http://ko.wikipedia.org/wiki/%EA%B0%80%EC%9A%B0%EC%8A%A4%EA%B3%A1%EB%A5%A0 http://ko.wikipedia.org/wiki/가우스곡률]
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* http://en.wikipedia.org/wiki/Theorema_Egregium
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[[분류:미분기하학]]
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== 노트 ==
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===말뭉치===
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# A consequence of the Theorema Egregium is that the Earth cannot be displayed on a map without distortion.<ref name="ref_b7a2f3ff">[https://simple.wikipedia.org/wiki/Theorema_egregium Simple English Wikipedia, the free encyclopedia]</ref>
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# Gauss's Theorema Egregium (Latin for "Remarkable Theorem") is a major result of differential geometry proved by Carl Friedrich Gauss.<ref name="ref_b7a2f3ff" />
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# Gauss's Theorema Egregium (Latin for "Remarkable Theorem") is a major result of differential geometry (proved by Carl Friedrich Gauss in 1827) that concerns the curvature of surfaces.<ref name="ref_5d738f20">[https://en.wikipedia.org/wiki/Theorema_Egregium Theorema Egregium]</ref>
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# As a corollary of Theorema Egregium, a piece of paper cannot be bent onto a sphere without crumpling.<ref name="ref_5d738f20" />
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# It follows from Theorema Egregium that under this bending the Gaussian curvature at any two corresponding points of the catenoid and helicoid is always the same.<ref name="ref_5d738f20" />
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# An application of the Theorema Egregium is seen when a flat object is somewhat folded or bent along a line, creating rigidity in the perpendicular direction.<ref name="ref_5d738f20" />
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# As Lanczos (1970) put it, `In view of his customary reticence, it was an exceptionally jubilant gesture to call one of his theorems “Theorema egregium”.<ref name="ref_dd2d07f1">[https://thatsmaths.com/2018/12/27/gaussian-curvature-the-theorema-egregium/ Gaussian Curvature: the Theorema Egregium]</ref>
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# The Gaussian curvature is still 0 and Theorema Egregium still holds and the pizza toppings are still on the floor.<ref name="ref_7b26a47e">[https://medium.com/cantors-paradise/theorema-egregium-and-pizza-eating-7833cb34d592 Theorema Egregium and Pizza Eating]</ref>
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# Theorema Egregium means Remarkable Theorem.<ref name="ref_c2521653">[https://www.maths.dur.ac.uk/users/pavel.tumarkin/past/spring17/DG/outline_term2_10.pdf Durham university]</ref>
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# Gauss Theorema Egregium allows us to dene the Gauss curvature for any surface S just using the rst fundamental form.<ref name="ref_c2521653" />
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# Gauss (effectively) expressed the theorema egregium by saying that the Gaussian curvature at a point is given by where is the Riemann tensor, and and are an orthonormal basis for the tangent space.<ref name="ref_ca4ea3a1">[https://mathworld.wolfram.com/GausssTheoremaEgregium.html Gauss's Theorema Egregium -- from Wolfram MathWorld]</ref>
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# Gauss's theorema egregium states that the Gaussian Curvature of a surface embedded in 3-space may be understood intrinsically to that surface.<ref name="ref_08ffd5eb">[https://archive.lib.msu.edu/crcmath/math/math/g/g075.htm Gauss's Theorema Egregium]</ref>
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# Gauß (effectively) expressed the theorema egregium by saying that the Gaussian Curvature at a point is given by where is the Riemann Tensor, and and are an orthonormal basis for the Tangent Space.<ref name="ref_08ffd5eb" />
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# This leads us to one of the major theorems in differential geometry, Gauss' Theorema Egregium.<ref name="ref_1ae5ded2">[https://www.math.brown.edu/tbanchof/balt/ma106/dtext88.html @include course]</ref>
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# At this point we introduce a lemma that will be useful in proving the Theorema Egregium.<ref name="ref_1ae5ded2" />
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# Gauss's Theorema Egregium (Latin: "Remarkable Theorem") is a foundational result in differential geometry proved by Carl Friedrich Gauss that concerns the curvature of surfaces.<ref name="ref_9d3c2fe4">[https://enacademic.com/dic.nsf/enwiki/160565 Theorema Egregium]</ref>
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# It follows from Theorema Egregium that the Gaussian curvature at the two points of the catenoid and helicoid corresponding to each other under this bending is the same.<ref name="ref_9d3c2fe4" />
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# Finally, this essay deals with a remarkable theorem in the theory of surfaces, Gauss Theorema Egregium.<ref name="ref_58adece4">[http://www.diva-portal.org/smash/record.jsf?pid=diva2:1104562 Gauss Theorema Egregium]</ref>
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# The Theorema Egregium ('Remarkable Theorem') is an important theorem of Carl Friedrich Gauss concerning the curvature of surfaces.<ref name="ref_8f070fe5">[http://50.116.16.126/wiki/index.php/Theorema_Egregium Theorema Egregium]</ref>
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# A somewhat whimsical application of the Theorema Egregium is seen in a common pizza-eating strategy: A slice of pizza can be seen as a surface with constant Gaussian curvature 0.<ref name="ref_8f070fe5" />
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# We shall deduce the Theorema Egregium from two results which relate the first and second fundamental forms of a surface, and which have other important consequences.<ref name="ref_7d976830">[https://www.semanticscholar.org/paper/Gauss%E2%80%99-Theorema-Egregium-Pressley/d371c196d546250e08bd1895284ec0d6d9e34885 Gauss’ Theorema Egregium]</ref>
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# Gauss's Theorema Egregium (Latin for "Remarkable Theorem") is a major result of differential geometry proved by Carl Friedrich Gauss that concerns the curvature of surfaces.<ref name="ref_06a6a958">[https://en.glosbe.com/en/sv/Theorema%20Egregium Theorema Egregium in Swedish - English-Swedish Dictionary]</ref>
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# Theorema egregium ('det märkvärdiga teoremet') är ett matematiskt teorem av Carl Friedrich Gauss som innebär att Gausskrökningen bevaras vid en isometrisk avbildning.<ref name="ref_06a6a958" />
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# For smooth surfaces, Gauss Theorema Egregium says that the Gaussian curvature can be calculated by using distances on S alone; it is independent of the embedding S R3.<ref name="ref_41a8df35">[https://www.mtholyoke.edu/courses/adurfee/ICM-2010/ICM-2010-talk.pdf Polyhedral differential geometry]</ref>
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# however, statements of Gauss' Theorema Egregium can be replaced for statements concerning simple and useful connections between their intrinsic and extrinsic measures.<ref name="ref_c18f8b71">[https://pp.bme.hu/ci/article/view/3849 'GAUSS' THEOREMA EGREGIUM FOR TRIANGULATED SURFACES]</ref>
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# This is the most wonderful thing I learned this week (though I don’t truly understand it), from the Wikipedia article on Gauss’s Theorema Egregium (Remarkable Theorem) about the curvature of surfaces.<ref name="ref_e6290c1f">[https://www.miskatonic.org/2016/12/09/gauss-and-pizza/ Miskatonic University Press]</ref>
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===소스===
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<references />
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== 메타데이터 ==
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===위키데이터===
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* ID :  [https://www.wikidata.org/wiki/Q1048874 Q1048874]
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===Spacy 패턴 목록===
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* [{'LOWER': 'theorema'}, {'LOWER': 'egregium'}]

2021년 2월 23일 (화) 18:00 기준 최신판

개요

  • 학부 미분기하학에서 배우게 되는 중요한 정리 중의 하나
  • 가우스 곡률은 곡면이 얼마나 휘어 있는가를 재는 양
  • 이 가우스 곡률은 그 곡면의 거리와 각도를 재는 것으로 알수 있다는 정리



가우스 곡률

\[K = -\frac{1}{2\sqrt{EG}}\left(\frac{\partial}{\partial u}\frac{G_u}{\sqrt{EG}} + \frac{\partial}{\partial v}\frac{E_v}{\sqrt{EG}}\right)\]


지도제작에의 의미

  • 구면의 아주 작은 부분이라고 할지라도 수학적으로 엄밀하게 거리와 각도가 모두 보존되도록 하는 평면지도를 그릴수 없다는 것을 의미함.
    • 만약 이것이 가능하려면, 구면과 평면의 가우스 곡률이 같아야 함.
    • 그러나 구면의 가우스 곡률은 언제나 양수이고, 평면의 가우스 곡률은 언제나 0 이다.
  • 이것은 지도제작에 언제나 존재하게 되는 딜레마를 의미함.
  • 지도를 제작한다면 원하는 성질을 얻는 대신, 무언가는 희생해야 한다는 것을 뜻함.
  • 지도와 수학 항목 참조



역사




관련된 항목들



사전형태의 자료

노트

말뭉치

  1. A consequence of the Theorema Egregium is that the Earth cannot be displayed on a map without distortion.[1]
  2. Gauss's Theorema Egregium (Latin for "Remarkable Theorem") is a major result of differential geometry proved by Carl Friedrich Gauss.[1]
  3. Gauss's Theorema Egregium (Latin for "Remarkable Theorem") is a major result of differential geometry (proved by Carl Friedrich Gauss in 1827) that concerns the curvature of surfaces.[2]
  4. As a corollary of Theorema Egregium, a piece of paper cannot be bent onto a sphere without crumpling.[2]
  5. It follows from Theorema Egregium that under this bending the Gaussian curvature at any two corresponding points of the catenoid and helicoid is always the same.[2]
  6. An application of the Theorema Egregium is seen when a flat object is somewhat folded or bent along a line, creating rigidity in the perpendicular direction.[2]
  7. As Lanczos (1970) put it, `In view of his customary reticence, it was an exceptionally jubilant gesture to call one of his theorems “Theorema egregium”.[3]
  8. The Gaussian curvature is still 0 and Theorema Egregium still holds and the pizza toppings are still on the floor.[4]
  9. Theorema Egregium means Remarkable Theorem.[5]
  10. Gauss Theorema Egregium allows us to dene the Gauss curvature for any surface S just using the rst fundamental form.[5]
  11. Gauss (effectively) expressed the theorema egregium by saying that the Gaussian curvature at a point is given by where is the Riemann tensor, and and are an orthonormal basis for the tangent space.[6]
  12. Gauss's theorema egregium states that the Gaussian Curvature of a surface embedded in 3-space may be understood intrinsically to that surface.[7]
  13. Gauß (effectively) expressed the theorema egregium by saying that the Gaussian Curvature at a point is given by where is the Riemann Tensor, and and are an orthonormal basis for the Tangent Space.[7]
  14. This leads us to one of the major theorems in differential geometry, Gauss' Theorema Egregium.[8]
  15. At this point we introduce a lemma that will be useful in proving the Theorema Egregium.[8]
  16. Gauss's Theorema Egregium (Latin: "Remarkable Theorem") is a foundational result in differential geometry proved by Carl Friedrich Gauss that concerns the curvature of surfaces.[9]
  17. It follows from Theorema Egregium that the Gaussian curvature at the two points of the catenoid and helicoid corresponding to each other under this bending is the same.[9]
  18. Finally, this essay deals with a remarkable theorem in the theory of surfaces, Gauss Theorema Egregium.[10]
  19. The Theorema Egregium ('Remarkable Theorem') is an important theorem of Carl Friedrich Gauss concerning the curvature of surfaces.[11]
  20. A somewhat whimsical application of the Theorema Egregium is seen in a common pizza-eating strategy: A slice of pizza can be seen as a surface with constant Gaussian curvature 0.[11]
  21. We shall deduce the Theorema Egregium from two results which relate the first and second fundamental forms of a surface, and which have other important consequences.[12]
  22. Gauss's Theorema Egregium (Latin for "Remarkable Theorem") is a major result of differential geometry proved by Carl Friedrich Gauss that concerns the curvature of surfaces.[13]
  23. Theorema egregium ('det märkvärdiga teoremet') är ett matematiskt teorem av Carl Friedrich Gauss som innebär att Gausskrökningen bevaras vid en isometrisk avbildning.[13]
  24. For smooth surfaces, Gauss Theorema Egregium says that the Gaussian curvature can be calculated by using distances on S alone; it is independent of the embedding S R3.[14]
  25. however, statements of Gauss' Theorema Egregium can be replaced for statements concerning simple and useful connections between their intrinsic and extrinsic measures.[15]
  26. This is the most wonderful thing I learned this week (though I don’t truly understand it), from the Wikipedia article on Gauss’s Theorema Egregium (Remarkable Theorem) about the curvature of surfaces.[16]

소스

  1. ↑ 1.0 1.1 Simple English Wikipedia, the free encyclopedia
  2. ↑ 2.0 2.1 2.2 2.3 Theorema Egregium
  3. ↑ Gaussian Curvature: the Theorema Egregium
  4. ↑ Theorema Egregium and Pizza Eating
  5. ↑ 5.0 5.1 Durham university
  6. ↑ Gauss's Theorema Egregium -- from Wolfram MathWorld
  7. ↑ 7.0 7.1 Gauss's Theorema Egregium
  8. ↑ 8.0 8.1 @include course
  9. ↑ 9.0 9.1 Theorema Egregium
  10. ↑ Gauss Theorema Egregium
  11. ↑ 11.0 11.1 Theorema Egregium
  12. ↑ Gauss’ Theorema Egregium
  13. ↑ 13.0 13.1 Theorema Egregium in Swedish - English-Swedish Dictionary
  14. ↑ Polyhedral differential geometry
  15. ↑ 'GAUSS' THEOREMA EGREGIUM FOR TRIANGULATED SURFACES
  16. ↑ Miskatonic University Press

메타데이터

위키데이터

Spacy 패턴 목록

  • [{'LOWER': 'theorema'}, {'LOWER': 'egregium'}]
원본 주소 "https://wiki.mathnt.net/index.php?title=가우스의_놀라운_정리(Theorema_Egregium)&oldid=52673"