Ellipse

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 $\pi a b$.

Explanations

L1 human

Fix two pins in a board and tie a loose string to both. Pull the string tight with a pencil and move it round: the pencil draws an ellipse. The length of the string is the constant sum $2a$ of the distances to the two pins.