an ellipse

Tilt a glass of water and the rim of the liquid traces an ellipse; the shadow of a circle, a planet's orbit, the cross-section of a rugby ball seen edge-on — all ellipses. An ellipse is a closed, flattened circle: a smooth oval loop that bulges in one direction and is pinched in the other.

The cleanest definition uses two special inside points, the foci (singular: focus). An ellipse is the set of all points P for which the sum of the two focal distances stays constant: |PF_1| + |PF_2| = 2a, a fixed total. This is the gardener's trick — pin a loop of string at two stakes, pull it taut with a pencil, and the pencil draws an ellipse. With the foci on the x-axis at (-c, 0) and (c, 0), the curve has the standard equation x^2/a^2 + y^2/b^2 = 1, where a is the semi-major axis (half the long way), b is the semi-minor axis (half the short way), and they are tied by b^2 = a^2 - c^2. The center is the midpoint of the foci; the four points where the curve meets its axes are the vertices.

Why it matters: Kepler's first law says every planet orbits the Sun in an ellipse with the Sun at one focus — not at the center, a subtle and important difference. The ellipse's eccentricity e = c/a measures how squashed it is, running from 0 (a perfect circle, the two foci merged) up toward 1 (a long thin sliver). And its whispering-gallery reflection — sound from one focus gathers at the other — turns elliptical rooms into acoustic curiosities and shapes medical lithotripters that focus shock waves on a kidney stone.