# 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.[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]

## 메타데이터

### Spacy 패턴 목록

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