An ellipse is a closed curve on a plane surrounding two focal points. For every point on the ellipse, the sum of the distances to the two foci is constant. It is a generalization of a circle (which is an ellipse where both foci coincide at the center). Ellipses are conic sections formed when a plane intersects a cone at an angle.

The major axis is the longest diameter of the ellipse, passing through both foci and the center. Its half-length is the semi-major axis (a). The minor axis is the shortest diameter, perpendicular to the major axis at the center. Its half-length is the semi-minor axis (b). In the standard equation (x−h)²/a² + (y−k)²/b² = 1, a is always ≥ b.

Eccentricity (e) measures how "stretched" an ellipse is. It is calculated as e = c/a, where c = √(a² − b²) is the focal distance (half the distance between foci). Eccentricity ranges from 0 (perfect circle, a = b) to nearly 1 (very elongated). Earth's orbital eccentricity is approximately 0.017 — nearly circular!

Unlike a circle, the ellipse perimeter has no exact closed-form formula using elementary functions. This tool uses Ramanujan's approximation: P ≈ π(a+b)(1 + 3h/(10+√(4−3h))), where h = (a−b)²/(a+b)². This approximation is remarkably accurate, with error less than 0.001% for most ellipses. The exact perimeter requires elliptic integrals.

The foci (plural of focus) are two special points on the major axis, equidistant from the center. Their defining property: for any point P on the ellipse, distance(P, F₁) + distance(P, F₂) = 2a (constant). This property makes ellipses useful in optics, acoustics, and orbital mechanics. A whispering gallery is an elliptical room where sound from one focus concentrates at the other.

Ellipses appear everywhere in science and engineering: Planetary orbits (Kepler's first law), GPS satellite trajectories, lithotripsy (kidney stone treatment using elliptical reflectors), whispering galleries, ellipse-based cryptography, gear design, and architectural acoustics. Understanding ellipse geometry is fundamental in physics, astronomy, and engineering design.