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ellipse


the ellipse as a conic section
Figure 1. The ellipse is one of the family of curves known as conic sections, the others being the circle, parabola, and hyperbola
An ellipse is a conic section and is mathematically defined (1) by passing a plane through a right circular cylinder at an angle between 0 and 90 degrees (see figure 1), or (2) as the locus of a point which moves so that the sum of its distances from two fixed points, known as foci (singular: focus), is constant (see figure 2). If the two foci coincide then the ellipse is a circle. The ellipse is symmetric with respect to both its axes, and is a closed curve.

The line passing through the foci is called the major axis of the ellipse; half this is the semi-major axis, a. The line passing through the center of the ellipse (the midpoint of the foci) at right angles to the major axis is called the minor axis, half of which is the semi-minor axis, b.

An ellipse centered at the origin of an x-y coordinate system with its major axis along the x-axis is defined by the equation

x2/a2 + y2/b2 = 1

The shape of an ellipse is expressed by a number called the eccentricity, e, which is related to a and b by the formula b2 = a2(1 - e2). The eccentricity is a positive number less than 1, or 0 in the case of a circle. The greater the eccentricity, the larger the ratio of a to b, and therefore the more elongated the ellipse. The distance between the foci is 2ae.

The area enclosed by an ellipse is πab. The circumference of an ellipse is 4aE(e), where the function E is the complete elliptical integral of the second kind.


An ellipse, such as that followed by a planet in orbit around the Sun, can be easily drawn by using a pencil, a loop of string, and two nails in a board
An ellipse, such as that followed by a planet in orbit around the Sun, can be easily drawn by using a pencil, a loop of string, and two nails in a board.

Elliptical orbits

The closed path followed by one object that is gravitationally bound to another – for example, by one of the stars in a binary star system or a spacecraft in Earth orbit. That the orbits of the planets are ellipses, not circles, was first established by Johannes Kepler based on the careful observations of Tycho Brahe.


Related categories

   • PLANE CURVES
   • CELESTIAL MECHANICS


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