Eccentricity: Morphing from Circle to Hyperbola

All four conic sections — circle, ellipse, parabola, and hyperbola — are actually the same family of curves, parameterized by a single number: the eccentricity $e$.

The unified definition: a conic is the set of all points $P$ where the ratio of distance to a fixed point (focus) to distance to a fixed line (directrix) equals $e$:

$\dfrac{PF}{PD} = e$

• $e = 0$: Circle (every point equidistant from center; directrix at infinity)
• $0 < e < 1$: Ellipse (points closer to focus than to directrix)
• $e = 1$: Parabola (equal distances — the transition case)
• $e > 1$: Hyperbola (points farther from focus than from directrix)

The polar equation of any conic with focus at the origin:

$r = \dfrac{a(1 - e^{2})}{1 + e\cos\theta}$

This single formula draws all four conic types as $e$ changes.

Try it in the visualization

Drag the eccentricity slider continuously from 0 to 2 and watch the curve morph through all four types. At $e = 0$, it's a perfect circle. As $e$ approaches 1, the ellipse elongates dramatically. At $e = 1$, one end opens to infinity (parabola). Above $e = 1$, both branches of the hyperbola appear. The focus-directrix ratio is shown for a moving point.