Ellipse
An ellipse is a geometric shape classified as one of the four conic sections, alongside hyperbolas, parabolas, and circles. It is formed by the intersection of a double-cone and a plane at an angle, resulting in a closed curve. An ellipse can be defined in several ways, including as the locus of points for which the sum of the distances to two fixed points (called foci) is constant. The ellipse has a major axis and a minor axis, with its eccentricity (a measure of how much it deviates from being circular) being defined by the distance between the foci relative to the length of the major axis.
Ellipses are significant in various natural phenomena, such as the orbits of celestial objects; for example, planets and satellites follow elliptical paths around larger bodies due to gravitational forces. The reflective property of ellipses enables applications in architecture and acoustics, where sound or light emanating from one focus will converge at the other. When an ellipse is rotated around either its major or minor axis, it forms a spheroid, impacting how shapes like the Earth are modeled in geodesy. Overall, ellipses are fundamental in both mathematical theory and practical applications across diverse fields.
Subject Terms
Ellipse
An ellipse is one of the four conic sections: hyperbola, parabola, ellipse, and circle. They are called conic because they are the result of cutting a double-cone with a plane. Demonstrating Kepler’s laws, an object moving around the sun follows a path which is one of the conic sections. If the object is moving slower than the sun’s escape velocity, then its path will be an ellipse (or a circle, which is the limiting form of an ellipse) with the sun at one focus. Similarly, a satellite moving around the earth, or a charged object moving around another charged object, will follow a conic, and, if moving too slowly to escape, will follow an ellipse.
Overview
There are five definitions of an ellipse. All five are equivalent:
Definition. The intersection of a double-cone and a plane which intersects the axis of the double-cone at an angle between 90° and the angle the cone makes with its axis. (A 90° plane produces a circle and a plane parallel to the side of the cone produces a parabola. Both are limiting cases of the ellipse).
Definition. Given two points (each is called a focus, plural foci), the locus of all points for which the sum of their distances to each of the two foci is the same.
Definition. The curve in the xy coordinate plane defined by
.
Definition. The curve defined by the parametric equations x = a cos t and y = b sin t.
Definition. The locus of all points whose distance from a fixed point is ε times its distance from a fixed line, with 0 < ε < 1.
Parts of an Ellipse
In the above definitions, ε is called the eccentricity. The length of the "major axis" is 2a, and 2b is the length of the "minor axis." If f is the distance from one focus to the origin, the eccentricity is the ratio of the distance between the two foci to the length of the major axis, or ε = 2f/2a. If the two foci are zero distance apart, ε = 0 and the ellipse becomes a circle. See Figure 1.
Uses of the Ellipse
An ellipse has the property that if light (or sound) starting at one focus meets the ellipse, it will be reflected so that it will cross the other focus. Echo chambers, such as in the US Capitol or the Mormon Tabernacle, use this property. A person standing at one focus can clearly hear a whisper at the other focus.
If an ellipse is rotated around its major or minor axis, the result is a three-dimensional shape known as a "spheroid." If rotated around the major axis the resulting figure is called a "prolate spheroid" and resembles an American football. If rotated around the minor axis the resulting figure is an "oblate spheroid" and resembles a lentil or a pumpkin. The earth, due to its rotation, is very close to a slightly oblate spheroid. (The 1984 World Geodetic System, used in GPS, models the earth as an oblate spheroid measuring 6,378.137 km at the equator and 6,356.752 km at the poles.) Note that each of these spheroids has one cross-section which is a circle.
A solid with three perpendicular cross-sections, each of which is an ellipse, is an "ellipsoid."
Bibliography
Aarts, J. M. Plane and Solid Geometry. New York: Springer, 2009.
McKellar, Danica. Girls Get Curves. New York: Penguin, 2012.
Posamentier, Alfred S, and Robert L. Bannister. Geometry, Its Elements and Structure. Mineola, NY: Dover, 2014.
Sautoy, Marcus. Symmetry: A Journey into the Patterns of Nature. New York: Harper, 2008.
Stankowski, James F., ed. Geometry and Trigonometry. New York: Rosen, 2015.