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[어셈블에게 물어봐] 볼록 사각형에 외접하는 타원

피카파이 2022.07.30 20:39 조회 106

페이스북 트위터 카카오톡 카카오스토리 네이버블로그 네이버밴드

제가 함풀문에 문제로도 올린 건데요,

제가 사각형에 외접하는 타원을 그리다 보니까 볼록 사각형을 모두 다 되더라고요.

직관적으로도 세 점을 지나는 타원을 좀 늘렸다 줄였다(?) 하고 옆으로 비틀었다가 그러면 될 거 같은데

증명이 안 되서...

이전글 이것좀...

다음글 "어셈블에게 물어봐" 가이드라인

추천 1 목록보기

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김다인(멘토) Lv.15 2022.07.30 22:05

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피카파이 친구에게,

좌표평면에서 아핀변환은 원을 타원으로 변환시킵니다.

정의 [아핀 변환]. 함수 $\Psi \colon \mathbb R^2 \to \mathbb R^2$ 가 $ad-bc \ne 0$ 인 실수 $a, b, c, d$ 에 대해

$\Psi(x, y) = (ax+by, cx+dy)$

일 때 $\Psi$ 를 아핀 변환이라 합니다.

명제. 아핀 변환의 역함수도 아핀 변환입니다.

볼록 사각형을 좌표평면 위에 적당히 배치하고 아핀 변환의 변수 $a, b, c, d$ 를 연속적으로 변환시킨다면 마주보는 두 각의 합이 연속적으로 작아지거나 커지게 할 수 있습니다. 따라서 중간값 정리에 의해 아핀 변환에 의해 변환된 사각형의 마주보는 두 각의 합이 정확이 $180^\circ$ 인 아핀 변환 $\Psi$ 이 존재하며, 이 때의 원을 역함수 $\Psi^{-1}$ 로 통해 변환시키면 우리가 원하는 타원을 얻게 됩니다.

변수를 연속적으로 변환시키는 과정과 중간값 정리를 적용하는 과정에서는 세세한 부분들에 대한 설명을 생략했습니다. 스스로 증명을 마무리해보세요!

일반적으로 평면 위에 5개의 점을 찍었을 때 임의의 세 점이 한 직선 위에 놓이지 않는다면 이 5개의 점을 지나는 이차곡선이 항상 존재함이 알려져 있습니다. 이차곡선은

$Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$

로 나타나며 포물선, 타원, 쌍곡선으로 분류됩니다.

다인 멘토 드림

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