Locus (mathematics)

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A set of loci 2cm, 4cm, 6cm and 8cm from [l] towards [P]. These curves are half of the Conchoid of Nichomedes. (Ink Drawing by [Susan Murray])

In mathematics, a locus (Latin for "place", plural loci) is a collection of points which share a property. The term 'locus' is usually used of a condition which defines a continuous figure or figures, that is, a curve. For example, a line is the locus of points equidistant from two fixed points.

The conic sections may be defined in terms of loci:

A circle is the locus of points from which the distance to the center is a given value, the radius.
An ellipse is the locus of points, the sum of the distances from which to the foci is a given value.
A hyperbola is the locus of points, the difference of the distances from which to the foci is a given value.
A parabola is the locus of points, the distances from which to the focus and to the directrix are equal.

Very complex geometric shapes may be described as the locus of zeros of a function or polynomial. Thus, for example, the quadric surfaces are defined as the loci of zeros of the quadratic polynomials. More generally, the locus of zeros of a set of polynomials are called an algebraic variety, the properties of which are studied in the branch of mathematics known as algebraic geometry.

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