Locus (mathematics)

A set of loci 2cm, 4cm, 6cm and 8cm from l towards P. These curves are half of the Conchoid of Nichomedes.

In geometry, a locus (Latin for "place", plural loci) is a collection of points which share a property. For example a circle may be defined as the locus of points in a plane at a fixed distance from a given point.

A locus may alternatively be described as the path through which a point moves to fulfill a given condition or conditions. So, for example, a circle may also be defined as the locus of a point moving so as to remain at a given distance from a fixed point.

The term occurs in complex dynamics as:

• Bifurcation locus
• Connectedness locus

Proofs involving loci

In general, a proof that a locus is a particular curve has two parts. The first part is to show that every point on the curve satisfies the condition of the locus and second part is to show that every point that satisfies the condition is on the curve. For example, to show that the locus of points equidistant between two given (different) points is their perpendicular bisector, one must show:

• Any point equidistant from the two points lies on the perpendicular bisector,
• Any point on this line is equidistant from the two given points.

See also

• Set (mathematics)

References

• Robert Clarke James, Glenn James: Mathematics Dictionary. Springer 1992, ISBN 9780412990410, p. 255 (restricted online copy at Google Books)
• Alfred North Whitehead: An Introduction to Mathematics. BiblioBazaar LLC 2009 (reprint), ISBN 9781103197842, pp. 121 (restricted online copy at Google Books)
• George Wentworth: Junior High School Mathematics: Book III. BiblioBazaar LLC 2009 (reprint), ISBN 9781103152360, pp. 265 (restricted online copy at Google Books)