# 타원

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## 개요

- 원뿔의 단면에서 얻어지는 원뿔곡선의 하나
- 이차곡선의 하나이다
- 타원위의 점들은 어떤 두 점(초점)에서의 거리의 합이 일정하다

## 타원의 방정식

- 타원은 이차곡선 \(ax^2+bxy+cy^2+dx+ey+f=0\)의 판별식이 \(\Delta=b^2-4ac<0\)인 경우
- 타원의 방정식의 표준형
- \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\)
- \(a=b\) 이면 원이다. \(a>b\) 이면 가로( 축)로 납작한 타원, \(a<b\) 이면 세로로 길쭉한 타원이 된다.
- 두 축 중 긴 것을 장축, 짧은 것을 단축이라 한다.

- 평행이동, 회전변환에 의해서도 변형해도 여전히 타원이 얻어짐.

\(\frac{1}{4} \left(\frac{\sqrt{3} x}{2}+\frac{y}{2}\right)^2+\left(-\frac{x}{2}+\frac{\sqrt{3} y}{2}\right)^2=1\)

## 타원 둘레의 길이

- 타원 둘레의 길이 항목 참조

## 타원내부의 면적

- 다음과 같이 주어진 타원 내부의 면적은 \(\pi a b\) 이다\[\frac{x^2}{a^2}+\frac{y^2}{b^2}\leq 1\]
- 타원의 넓이 항목 참조

## 배우기 전에 알고 있어야 하는 것들

- 다항식
- 일차식과 이차식

- 원의 방정식

## 중요한 개념 및 정리

- 타원 \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1 (0<b<a)\)을 고려하자.
- 초점 \(f=\sqrt{a^2-b^2}\)라 두면, \((\pm f,0)\)
- 이심률 (eccentricity)
- 타원이 원에서 멀어지는 것을 재는 양 . 이심률은 \(e=\frac{f}{a}=\frac{\sqrt{a^2-b^2}}{a}=\sqrt{1-\frac{b^2}{a^2}}\)로 주어진다

## 재미있는 문제

- 타원 \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) 에 외접하는 사각형의 최소 넓이는
- 빛의 반사성 : 한 초점에서 나온 빛은 타원 벽에서 반사되어 다른 초점으로 들어간다.
- 매개변수표현
- 타원과 포물선 가 직교하기 위해서는 를 만족하면 된다.

## 관련된 항목들

### 관련된 개념 및 나중에 더 배우게 되는 것들

### 관련있는 다른 과목

- 물리
- 행성운동
- 지구는 태양의 주위를, 태양을 하나의 초점으로 하는 타원궤도로 돌고 있음.

- 미술
- 원근법
- 원을 바르게 그리려면, 타원으로 그려야 함.

### 관련된 대학교 수학

## 블로그

- 미적분과 인문계(3) : 타원 - 자연, 예술, 인간 (피타고라스의 창)

## 노트

- You see here, we're really, if we're on this point on the ellipse, we're really close to the origin.
^{[1]} - And the way I drew this, we have kind of a short and fat ellipse you can also have kind of a tall and skinny ellipse.
^{[1]} - But in the short and fat ellipse, the direction that you're short in that's called your minor axis.
^{[1]} - If b was larger than a, I would have a tall and skinny ellipse.
^{[1]} - ; Description Draws an ellipse (oval) to the screen.
^{[2]} - A straight line drawn through the foci and extended to the curve in either direction is the major diameter (or major axis) of the ellipse.
^{[3]} - This section focuses on the four variations of the standard form of the equation for the ellipse.
^{[4]} - We can draw an ellipse using a piece of cardboard, two thumbtacks, a pencil, and string.
^{[4]} - Place the thumbtacks in the cardboard to form the foci of the ellipse.
^{[4]} - The derivation of the standard form of the equation of an ellipse relies on this relationship and the distance formula.
^{[4]} - Eccentricity is a number that describe the degree of roundness of the ellipse.
^{[5]} - is the line segment passing the foci and intersects with the ellipse.
^{[5]} - is the line segment perpendicular to the major axis, passing the center of foci, and intersects with the ellipse.
^{[5]} - The formula for ellipse can be derived in many ways.
^{[5]} - It is an ellipse! and draw a curve.
^{[6]} - It goes from one side of the ellipse, through the center, to the other side, at the widest part of the ellipse.
^{[6]} - Try bringing the two focus points together (so the ellipse is a circle) ...
^{[6]} - An ellipse is the set of all points \((x,y)\) in a plane such that the sum of their distances from two fixed points is a constant.
^{[7]} - If \((a,0)\) is a vertex of the ellipse, the distance from \((−c,0)\) to \((a,0)\) is \(a−(−c)=a+c\).
^{[7]} - It follows that \(d_1+d_2=2a\) for any point on the ellipse.
^{[7]} - What is the standard form equation of the ellipse that has vertices \((\pm 8,0)\) and foci \((\pm 5,0)\)?
^{[7]} - Populations of the ellipse are declining across the state, Inoue said.
^{[8]} - The storm will move into the right entrance region of the jet streak, shown by the large red ellipse.
^{[8]} - In other words, the orbit can be elliptical, but the ellipse can have any orientation in space.
^{[8]} - Supermoons occur because the moon orbits the Earth in the shape of an ellipse.
^{[8]} - In "primitive" geometrical terms, an ellipse is the figure you can draw in the sand by the following process: Push two sticks into the sand.
^{[9]} - The resulting shape drawn in the sand is an ellipse.
^{[9]} - Each of the two sticks you first pushed into the sand is a " focus " of the ellipse; the two together are called "foci" (FOH-siy).
^{[9]} - The points where the major axis touches the ellipse are the " vertices " of the ellipse.
^{[9]} - An ellipse is the set of all points P in a plane such that the sum of the distances from P to two fixed points is a given constant.
^{[10]} - The center of the ellipse is the midpoint of the line segment joining its foci.
^{[10]} - The major axis of the ellipse is the chord that passes through its foci and has its endpoints on the ellipse.
^{[10]} - The graph of an ellipse can be translated so that its center is at the point ( h , k ) .
^{[10]} - The ellipse is one of the four classic conic sections created by slicing a cone with a plane.
^{[11]} - The shape of the ellipse is described by its eccentricity.
^{[11]} - The larger the semi-major axis relative to the semi-minor axis, the more eccentric the ellipse is said to be.
^{[11]} - The equation of the ellipse can also be written in terms of the polar coordinates (r, f).
^{[11]} - The ellipse was first studied by Menaechmus Euclid wrote about the ellipse and it was given its present name by Apollonius .
^{[12]} - There is no exact formula for the length of an ellipse in elementary functions and this led to the study of elliptic functions.
^{[12]} - The evolute of the ellipse with equation given above is the Lamé curve.
^{[12]} - The ellipse is the set of all points R in the plane such that PR + QR is a fixed constant.
^{[13]} - An ellipse can be constructed using a piece of string.
^{[13]} - Then with a pencil pull the string so that the string is tight and move the string around to form the ellipse.
^{[13]} - This number tells us how squished the ellipse is.
^{[13]} - In other words, the caustic by refraction of the ellipse for rays parallel to the axis reduces to the two foci.
^{[14]} - Conversely, the ellipse is the boundary of any convex set with oblique axes of symmetry in every direction.
^{[14]} - An ellipse is the locus of all those points in a plane such that the sum of their distances from two fixed points in the plane, is constant.
^{[15]} - The shape of the ellipse is in an oval shape and the area of an ellipse is defined by its major axis and minor axis.
^{[15]} - The ellipse is one of the conic sections, that is produced, when a plane cuts the cone at an angle with the base.
^{[15]} - In geometry, an ellipse is a two-dimensional shape, that is defined along its axes.
^{[15]} - An ellipse is the set of points in a plane such that the sum of the distances from two fixed points in that plane stays constant.
^{[16]} - The midpoint of the segment joining the foci is called the center of the ellipse.
^{[16]} - When an ellipse is written in standard form, the major axis direction is determined by noting which variable has the larger denominator.
^{[16]} - Graph the following ellipse.
^{[16]} - We know that an ellipse is characterized by its squished circle or oval shape.
^{[17]} - An ellipse eccentricity measures how imperfectly round or squished an ellipse is.
^{[17]} - As the foci of an ellipse are moved towards the center, the shape of the ellipse becomes closer to that of the circle.
^{[17]} - If the foci of the ellipse are at the center, i.e. c = 0, then the value of eccentricity will become 0.
^{[17]} - The set of all points in a plane, the sum of whose distances from two fixed points in the plane is constant is an ellipse.
^{[18]} - These two fixed points are the foci of the ellipse (Fig. 1).
^{[18]} - We know that both points P and Q are on the ellipse.
^{[18]} - Hence, the ellipse becomes a circle.
^{[18]} - An ellipse is a circle that has been stretched in one direction, to give it the shape of an oval.
^{[19]} - But not every oval is an ellipse, as shown in Figure 1, below.
^{[19]} - There is a specific kind of stretching that turns a circle into an ellipse, as we shall see on the next page.
^{[19]} - Figure 2 hints at the nature of the type of stretching that creates an ellipse.
^{[19]} - The major axis is the segment that contains both foci and has its endpoints on the ellipse.
^{[20]} - An ellipse looks like a circle that has been squashed into an oval.
^{[21]} - An ellipse is defined by two points, each called a focus.
^{[21]} - If you take any point on the ellipse, the sum of the distances to the focus points is constant.
^{[21]} - In the figure above, drag the point on the ellipse around and see that while the distances to the focus points vary, their sum is constant.
^{[21]} - The term ellipse has been coined by Apollonius of Perga, with a connotation of being "left out".
^{[22]} - There are many ways to define an ellipse.
^{[22]} - We cite several common definitions, prove that all are equivalent, and, based on these, derive additional properties of ellipse.
^{[22]} - The ellipse touches the sides at the points (± a 1 ± a 2 cos δ) and (± a 1 cos δ, ± a 2 ).
^{[23]} - We distinguish two cases of polarization, according to the sense in which the end point of the electric vector describes the ellipse.
^{[23]} - F2 are called the foci of the ellipse (singular: focus).
^{[24]} - F2 is called the major axis of the ellipse, and the axis perpendicular to the major axis is the minor axis.
^{[24]} - A tunnel opening is shaped like a half ellipse.
^{[24]} - Find the equation of the ellipse assuming it is centered at the origin.
^{[24]} - is the line segment through the center of an ellipse defined by two points on the ellipse where the distance between them is at a minimum.
^{[25]} - If the major axis of an ellipse is parallel to the x-axis in a rectangular coordinate plane, we say that the ellipse is horizontal.
^{[25]} - If the major axis is parallel to the y-axis, we say that the ellipse is vertical.
^{[25]} - However, the ellipse has many real-world applications and further research on this rich subject is encouraged.
^{[25]} - As such, it generalizes a circle, which is the special type of ellipse in which the two focal points are the same.
^{[26]} - The midpoint C {\displaystyle C} of the line segment joining the foci is called the center of the ellipse.
^{[26]} - An arbitrary line g {\displaystyle g} intersects an ellipse at 0, 1, or 2 points, respectively called an exterior line, tangent and secant.
^{[26]} - \displaystyle w} which is different from P {\displaystyle P} cannot be on the ellipse.
^{[26]} - Click on the blue point on the ellipse and drag it to change the figure.
^{[27]} - The eccentricity of an ellipse is a measure of how much it is changed from a circle.
^{[27]} - The ellipse was first studied by Menaechmus, investigated by Euclid, and named by Apollonius.
^{[28]} - The focus and conic section directrix of an ellipse were considered by Pappus.
^{[28]} - In 1602, Kepler believed that the orbit of Mars was oval; he later discovered that it was an ellipse with the Sun at one focus.
^{[28]} - Let an ellipse lie along the x-axis and find the equation of the figure (1) where and are at and .
^{[28]}

### 소스

- ↑
^{1.0}^{1.1}^{1.2}^{1.3}Intro to ellipses (video) - ↑ ellipse() \ Language (API) \ Processing 3+
- ↑ Ellipse | mathematics
- ↑
^{4.0}^{4.1}^{4.2}^{4.3}Equations of Ellipses - ↑
^{5.0}^{5.1}^{5.2}^{5.3}Ellipse - ↑
^{6.0}^{6.1}^{6.2}Ellipse - ↑
^{7.0}^{7.1}^{7.2}^{7.3}12.2: The Ellipse - ↑
^{8.0}^{8.1}^{8.2}^{8.3}Definition of Ellipse by Merriam-Webster - ↑
^{9.0}^{9.1}^{9.2}^{9.3}Conics: Ellipses: Introduction - ↑
^{10.0}^{10.1}^{10.2}^{10.3}Ellipses - ↑
^{11.0}^{11.1}^{11.2}^{11.3}Ellipse - ↑
^{12.0}^{12.1}^{12.2}Ellipse - ↑
^{13.0}^{13.1}^{13.2}^{13.3}The Ellipse - ↑
^{14.0}^{14.1}Ellipse - ↑
^{15.0}^{15.1}^{15.2}^{15.3}Ellipse (Definition, Equation, Properties, Eccentricity, Formulas) - ↑
^{16.0}^{16.1}^{16.2}^{16.3}Ellipse - ↑
^{17.0}^{17.1}^{17.2}^{17.3}Superprof - ↑
^{18.0}^{18.1}^{18.2}^{18.3}Ellipse: Definition, Equations, Derivations, Observations, Q&A - ↑
^{19.0}^{19.1}^{19.2}^{19.3}The Most Marvelous Theorem in Mathematics - ↑ Equation of an Ellipse in Standard Form and how it relates to the graph of the Ellipse.
- ↑
^{21.0}^{21.1}^{21.2}^{21.3}math word definition- Math Open Reference - ↑
^{22.0}^{22.1}^{22.2}What Is Ellipse? - ↑
^{23.0}^{23.1}Ellipse - an overview - ↑
^{24.0}^{24.1}^{24.2}^{24.3}Brilliant Math & Science Wiki - ↑
^{25.0}^{25.1}^{25.2}^{25.3}Ellipses - ↑
^{26.0}^{26.1}^{26.2}^{26.3}Wikipedia - ↑
^{27.0}^{27.1}Ellipse: A closed curve with an equation in the form (x-h)^2/a+-(y-k)^2/b=1. - ↑
^{28.0}^{28.1}^{28.2}^{28.3}Ellipse -- from Wolfram MathWorld