Conic Sections and Their Polar Representations

Parabolas, ellipses, and hyperbolas are the three conic sections — curves formed by cutting a cone with a plane. Their Cartesian equations are the familiar $x^2/a^2 + y^2/b^2 = 1$ (ellipse) and $y = ax^2$ (parabola), but the deepest statement about conics is their unified polar form: every conic section has polar equation $r = \frac{ed}{1 - e\cos\theta}$ where $e > 0$ is the eccentricity and $d$ is the distance from the focus to the directrix. When $e < 1$ the conic is an ellipse; when $e = 1$ it is a parabola; when $e > 1$ it is a hyperbola. In orbital mechanics, this equation describes the path of every planetary orbit and comet trajectory. In finance, the hyperbola $\text{Var}/(\text{return}^2) = \text{const}$ is the Markowitz efficient frontier — the same conic section geometry governing risk–return trade-offs.

Learning objectives. By the end of this note you should be able to: