Polar Equations of Conics

Definition

In polar coordinates, a conic section is defined by its eccentricity $e$ and the distance $p$ from the focus (placed at the pole) to the directrix. This representation provides a unified equation for all conics based on the focus-directrix property.

Why It Matters

In space, everything is a conic section. The polar form is the only equation that handles all trajectories—from a stable orbit to a deep-space escape—in a single line of math. If you don’t use this unified view, you have to switch formulas every time a satellite’s speed changes. It is the essential “Navigation Logic” for anything moving under the influence of a central mass.

Core Concepts

• Focus at the Pole: By convention, one focus of the conic is placed at the origin $(0,0)$.
• Unified Equation:
  • Vertical Directrix: $r = \frac{ep}{1 \pm e \cos \theta}$.
  • Horizontal Directrix: $r = \frac{ep}{1 \pm e \sin \theta}$.
  • How to read: “The radius r is equal to the product of eccentricity e and the focal parameter p, all divided by the quantity one plus or minus e times the cosine of theta, or similarly using the sine function.”
  • Meaning: Unified polar conic form—$e$ is eccentricity, $p$ is focus-to-directrix distance; cosine/sine picks directrix orientation.
• Classification by Eccentricity ($e$):
  • Parabola: $e = 1$.
  • Ellipse: $0 < e < 1$.
  • Hyperbola: $e > 1$.
  • How to read: “An eccentricity e equal to one describes a parabola; an eccentricity e between zero and one describes an ellipse; and an eccentricity e strictly greater than one describes a hyperbola.”
  • Meaning: Eccentricity alone classifies the conic in polar form.
• Directrix Orientation:
  • $r = \frac{ep}{1 + e \cos \theta} \implies$ Directrix is $x = p$.
  • $r = \frac{ep}{1 - e \cos \theta} \implies$ Directrix is $x = -p$.
  • $r = \frac{ep}{1 + e \sin \theta} \implies$ Directrix is $y = p$.
  • $r = \frac{ep}{1 - e \sin \theta} \implies$ Directrix is $y = -p$.
  • How to read: “The polar equation with one plus e times the cosine of theta in the denominator corresponds to a directrix at x equals p; with one minus e times the cosine of theta it corresponds to x equals negative p; with one plus e times the sine of theta it corresponds to y equals p; and with one minus e times the sine of theta it corresponds to y equals negative p.”
  • Meaning: Sign in the denominator determines which side of the focus the directrix sits—cosine for vertical, sine for horizontal.

Connected Concepts

• Polar Coordinates
• Polar Coordinate Conversions
• Gallery of Polar Curves
• Graphs of Polar Equations
• Systems of Autonomous Equations
• Map-Territory Relation