Ellipse

ellipse, closed plane curve consisting of all points for which the sum of the distances between a point on the curve and two fixed points (foci) is the same. It is the conic section formed by a plane cutting all the elements of the cone in the same nappe. The center of an ellipse is the point halfway between its foci. The major axis is the chord that passes through the foci. The minor axis is the chord that passes through the center perpendicular to the major axis. The latus rectum is the chord through either focus perpendicular to the major axis. The vertices are the two points of intersection of the major axis with the curve. The eccentricity of an ellipse, a ratio of two lengths, is a measure of its flatness; it is the distance from the center to either focus divided by the distance from the center to either vertex. The circle may be considered an ellipse of eccentricity zero, i.e., one in which the center and the two foci all coincide.

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ellipse

Closed curve, one of the conic sections of analytic geometry, consisting of all points whose distances from each of two fixed points (foci) add up to the same value. The midpoint between the foci is the center. One property of an ellipse is that the reflection off its boundary of a line from one focus will pass through the other. In an elliptical room, a person whispering at one focus is easily heard by someone at the other. An oval may or may not fit the definition of an ellipse.

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ellipse

a closed conic section shaped like a flattened circle and formed by an inclined plane that does not cut the base of the cone. Standard equation x²/a² + y²/b² = 1, where 2a and 2b are the lengths of the major and minor axes. Area: πab

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ellipse [ə′lips]

(mathematics)

The locus of all points in the plane at which the sum of the distances from a fixed pair of points, the foci, is a given constant.

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Ellipse 

the intersection of a plane and one of the nappes of a circular conical surface (Figure 1).

Figure 1

An ellipse may also be defined as the locus of points M in the plane for which the sum of the distances from two fixed points F₁ and F₂ in the plane is constant; F₁ and F₂ are known as the foci of the ellipse. If a system of coordinates xOy is chosen as shown in Figure 2, then the equation of the ellipse assumes the form

Here, OF₁ = OF₂ = c, 2a = F₁M + F₂M, and .

Figure 2

An ellipse is a quadratic curve. It is symmetric with respect to its axes AB and CD, and its center O is its center of symmetry. The line segments AB = 2a and CD = 2b are called the major and minor axes of the ellipse, respectively. The number e = cla < 1 is known as the eccentricity of the ellipse; if a = b, then e = 0 and the ellipse is a circle. The lines whose equations are x = –a/e and x = a/e are called the directrices of the ellipse; the ratio of the distance of a point of the ellipse from the nearest focus and the distance of the point from the nearest directrix is constant and equal to the eccentricity. The points A and B where the ellipse intersects its major axis are called its vertices. (See alsoCONIC SECTION.)