AI Tutor
Shine a flashlight straight at a wall, and you get a perfect circle. Tilt your wrist slightly, and that circle stretches into an ellipse, opens into a parabola, and finally flares into a hyperbola.
How can one single beam of light create every fundamental orbital shape in the universe?

Slicing the Double Cone

A double-napped cone consists of two identical infinite cones touching tip-to-tip at a single point called the apex.
The angle between the central axis and the cone's outer surface is the semi-vertical angle Ξ±\alpha, while the cutting plane tilts at an angle Ξ²\beta relative to that same central axis.
When the cutting plane does not pass through the apex, the tilt angle determines the resulting curve:
  • Ξ²=90∘\beta = 90^\circ: Circle (a special symmetric closed loop).
  • Ξ±<Ξ²<90∘\alpha < \beta < 90^\circ: Ellipse (a tilted closed oval).
  • Ξ²=Ξ±\beta = \alpha: Parabola (an open curve cutting one nappe).
  • 0βˆ˜β‰€Ξ²<Ξ±0^\circ \le \beta < \alpha: Hyperbola (two open symmetric branches cutting both nappes).
If the plane passes directly through the apex, you get degenerate conics: a single point, a single straight line, or two intersecting lines.
These 3D slices look completely distinct, but can we describe every single one of them with just one universal rule?

The Unified Focus-Directrix Definition

Around 300 CE, the Greek mathematician Pappus of Alexandria discovered that all conics share a single 2D geometric rule based on a fixed point and a fixed line.
A conic is the locus of all points PP where the distance to a fixed point called the focus (FF) divided by the perpendicular distance to a fixed line called the directrix (LL) is a constant value called the eccentricity (ee).