Ellipse

Builds on Euclidean plane Euclidean distance

Definition

Let $F_1$ and $F_2$ be points of the Euclidean plane $\mathbb{R}^2$ , and let $d$ be the Euclidean distance. Let $a$ be a real number with $2a > d(F_1, F_2)$ . The ellipse with the foci $F_1$ , $F_2$ and the semi-major axis $a$ is the set $\{\, P \in \mathbb{R}^2 \;:\; d(P, F_1) + d(P, F_2) = 2a \,\}.$

Remarks

Let $c = \frac{1}{2} d(F_1, F_2)$ and $b = \sqrt{a^2 - c^2}$ . For $F_1 = (-c, 0)$ and $F_2 = (c, 0)$ , the ellipse is the set of the points with $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ . If $F_1 = F_2$ , the ellipse is the circle with the centre $F_1$ and the radius $a$ . The eccentricity of the ellipse is $e = c / a$ , with $0 \leq e < 1$ . The ellipse bounds a region of area $\pi a b$ .

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Euclidean distance concept
Euclidean plane object
Ellipse definition

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Joukowsky transform

Tag	`0FF6CAQT2Y`
URI	`m2://0FF6CAQT2Y`
Kind	definition
Writing	W1 stated
MSC	51N20
Defines	Ellipse
Uses	Euclidean plane, Euclidean distance

Ellipse

Definition

Remarks

Reading path

Where this leads

Further reading

What this page needs for W2

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