가우스의 놀라운 정리(Theorema Egregium)

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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

노트

말뭉치

1. A consequence of the Theorema Egregium is that the Earth cannot be displayed on a map without distortion.
2. Gauss's Theorema Egregium (Latin for "Remarkable Theorem") is a major result of differential geometry proved by Carl Friedrich Gauss.
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.
4. As a corollary of Theorema Egregium, a piece of paper cannot be bent onto a sphere without crumpling.
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.
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.
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”.
8. The Gaussian curvature is still 0 and Theorema Egregium still holds and the pizza toppings are still on the floor.
9. Theorema Egregium means Remarkable Theorem.
10. Gauss Theorema Egregium allows us to dene the Gauss curvature for any surface S just using the rst fundamental form.
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.
12. Gauss's theorema egregium states that the Gaussian Curvature of a surface embedded in 3-space may be understood intrinsically to that surface.
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.
14. This leads us to one of the major theorems in differential geometry, Gauss' Theorema Egregium.
15. At this point we introduce a lemma that will be useful in proving the Theorema Egregium.
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.
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.
18. Finally, this essay deals with a remarkable theorem in the theory of surfaces, Gauss Theorema Egregium.
19. The Theorema Egregium ('Remarkable Theorem') is an important theorem of Carl Friedrich Gauss concerning the curvature of surfaces.
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.
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.
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.
23. Theorema egregium ('det märkvärdiga teoremet') är ett matematiskt teorem av Carl Friedrich Gauss som innebär att Gausskrökningen bevaras vid en isometrisk avbildning.
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.
25. however, statements of Gauss' Theorema Egregium can be replaced for statements concerning simple and useful connections between their intrinsic and extrinsic measures.
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.

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Spacy 패턴 목록

• [{'LOWER': 'theorema'}, {'LOWER': 'egregium'}]