Conic Sections in Polar

Calculus 2 · Axiom Academy

LESSON Conic Sections in Polar One polar equation, , draws every conic — the eccentricity e alone decides which one. 1. One Equation for Every Conic A conic is the set of points whose distance to a fixed focus is e times the distance to a fixed line, the directrix . Anchor the focus at the pole and that definition collapses into a single polar equation: e = eccentricity · d = focus-to-directrix distance The animation sweeps the angle from 0 to . At each angle it plugs into the formula to get r , drops the point at , and the radius vector pivots around the fixed focus — tracing a full closed curve for e = 0.6 . 2. Eccentricity Decides the Shape Keep the focus and the formula fixed; turn only the dial e . As e climbs from below 1 up past it, the same equation hands you three completely different curves: Closed, bounded. A circle when e=0 . The knife-edge between closed and open. Open curve that races off to infinity. Watch the single curve below as e animates and back — recomputed point-by-point from the formula every frame, focus pinned at the pole. 3. Reading the Geometry off the Formula The two special angles do all the work. Plug them straight into and out fall the curve's key measurements — no separate formulas to memorize. The animation marks those points on the e=0.6 , d=80 ellipse: the vertex nearest the focus at , the far vertex at , and the semi-latus rectum straight up. 4. The Equation That Runs the Solar System

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