Standard Form (Horizontal)
(x-h)²/a² + (y-k)²/b² = 1
Standard Form (Vertical)
(x-h)²/b² + (y-k)²/a² = 1
Foci Distance
c = √(a² - b²)
Eccentricity
e = c/a (0 ≤ e < 1)
a = semi-major axis (longer)
b = semi-minor axis (shorter)
c = distance from center to focus
e = eccentricity (0 = circle, 1 = parabola)
Foci = two special points on major axis
Area = π × a × b
Unit Ellipse
a=2, b=1, center (0,0)
Planetary Orbit
e ≈ 0.017 (Earth's orbit)
Near-Circle
e → 0 as a → b
An ellipse is a closed curve that resembles a stretched or compressed circle. It is defined as the set of all points in a plane where the sum of the distances from two fixed points (called foci) is constant. This elegant definition leads to the familiar oval shape that appears throughout nature, from planetary orbits to the cross-section of a cylinder cut at an angle.
The ellipse has two axes: the major axis (the longest diameter) and the minor axis (the shortest diameter). The semi-major axis (a) and semi-minor axis (b) are half of these respective lengths. The eccentricity (e) measures how "stretched" the ellipse is - a value of 0 gives a perfect circle, while values approaching 1 create increasingly elongated ellipses.
Ellipses have numerous applications in science and engineering. In astronomy, Kepler's first law states that planets orbit the Sun in elliptical paths with the Sun at one focus. This discovery revolutionized our understanding of celestial mechanics and laid the foundation for Newton's theory of gravitation.
In architecture and engineering, elliptical shapes are used in dome construction, bridge arches, and optical systems. The reflective properties of ellipses are particularly useful - sound or light emitted from one focus will reflect to the other focus, a principle used in whispering galleries and medical lithotripsy machines that use shock waves to break up kidney stones.
Ellipse calculations follow standard mathematical formulas. Results depend on correct input values and axis configuration. The perimeter calculation uses Ramanujan's approximation, which is highly accurate but not exact. For critical applications, verify results with additional methods.