Most people first meet conics by “cutting the cone,” but there’s a really nice way to describe them using eccentricity ( e ).

• (0 \le e < 1): ellipse
• (e = 1): parabola
• (e > 1): hyperbola

Here’s a small interactive block — move the slider and watch the curve change.

Conic explorer

e = 0.40 → ellipse

$$r(\theta) = \frac{1}{1 + 0.40\cos\theta}$$

0 ≤ e < 1 → ellipse, e = 1 → parabola, e > 1 → hyperbola.

The polar equation we’re using here is

where is the semi-latus rectum (we fixed it to in the demo).