Ellipses also arise as images of a circle or a sphere under parallel projection, and some cases of perspective projection. Indeed, circles are special cases of ellipses. An ellipse is also the closed and bounded case of an implicit curve of degree 2, and of a rational curve of degree 2. It is also the simplest Lissajous figure, formed when the horizontal and vertical motions are sinusoids with the same frequency.
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Ellipses also arise as images of a circle or a sphere under parallel projection, and some cases of perspective projection. Indeed, circles are special cases of ellipses. An ellipse is also the closed and bounded case of an implicit curve of degree 2, and of a rational curve of degree 2. It is also the simplest Lissajous figure, formed when the horizontal and vertical motions are sinusoids with the same frequency.
Elements of an ellipse
An ellipse is a smooth closed curve which is symmetric about its center. The distance between antipodal points on the ellipse, or pairs of points whose midpoint is at the center of the ellipse, is maximum and minimum along two perpendicular directions, the major axis or transverse diameter, and the minor axis or conjugate diameter.
The semimajor axis (denoted by a in the figure) and the semiminor axis (denoted by b in the figure) are one half of the major and minor diameters, respectively. These are sometimes called (especially in technical fields) the major and minor semi-axes,, the major and minor semiaxes, or major radius and minor radius. When a and b are equal, the foci coincide with the center, and the ellipse becomes a circle with radius a=b.
There are two special points F1 and F2 on the ellipse's major axis, on either side of the center, such that the sum of the distances from any point of the ellipse to those two points is constant and equal to the major diameter (2 a). Each of these two points is called a focus of the ellipse.
The eccentricity of an ellipse, usually denoted by ε or e, is the ratio of the distance between the foci to the length of the major axis. The eccentricity is necessarily between 0 and 1; it is zero if and only if a=b, in which case the ellipse is a circle. The distance ae from a focal point to the centre is called the linear eccentricity of the ellipse.
The pins-and-string method
An ellipse can be drawn using two pins, a length of string, and a pencil:
- Push the pins into the paper at two points, which will become the ellipse's foci. Tie the string into a loose loop around the two pins. Pull the loop taut with the pencil's tip, so as to form a triangle. Move the pencil around, while keeping the string taut, and its tip will trace out an ellipse.
