An ellipse is a set of points such that the sum of the distances from two fixed points (called foci) is constant. It appears as an elongated circle and arises frequently in orbital mechanics, optics, and geometry.
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
Where:
\[ a^2 = b^2 + c^2 \quad \text{(where } c \text{ is the distance from center to focus)} \]
From this relation, we derive many other properties.
\[ \varepsilon = \frac{c}{a} = \frac{\sqrt{a^2 - b^2}}{a}, \quad \varepsilon < 1 \]
The eccentricity measures how "stretched" the ellipse is. If \( \varepsilon = 0 \), the ellipse is a circle.
For any point \( M(x, y) \) on the ellipse, the sum of distances to the foci \( F \) and \( F_1 \) is:
\[ FM + F_1M = 2a \]
\[ r = a \pm \varepsilon x \]
\[ A = \pi a b \]
