the polar equation of a conic

If you put the focus of a conic at the center of your coordinate system and measure points by their distance r and angle theta from that focus, all three conics collapse into one strikingly simple formula. This is the polar equation of a conic with the focus at the pole, and it is the natural language of orbits, because a planet's natural reference point is the Sun, sitting at a focus.

The formula is r = l / (1 + e cos theta), where e is the eccentricity and l is the semi-latus rectum (l = b^2/a for an ellipse or hyperbola, l = 2p for a parabola). It falls straight out of the focus-directrix property: r is the focal distance and the directrix distance is what produces the 1 + e cos theta in the denominator. The single number e steers everything — e = 0 gives r constant (a circle); 0 < e < 1 keeps the denominator positive for all theta, so r stays finite and the curve closes (an ellipse); e = 1 lets the denominator hit zero at theta = 180 degrees, so r runs to infinity once (a parabola); e > 1 makes the denominator negative over a range of angles, carving out the two open branches (a hyperbola). Variants like r = l / (1 - e cos theta) or with sin theta just rotate which way the curve opens.

This one equation is why polar coordinates are the home turf of celestial mechanics: it is the exact solution of the Kepler two-body problem, with the Sun at the pole. Plugging in e and l gives a planet's, comet's, or spacecraft's path directly, and the value of e instantly tells you whether the orbit is a closed ellipse (a returning planet), a parabola (a one-time grazing escape), or a hyperbola (a flyby that never comes back).