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

In mathematics, a locus (Latin for "place", plural loci) is a collection of points which share a property. The term locus is typically applied to a condition which defines a continuous figure or figures—that is, a curve. For example, in two-dimensional space a line is the locus of points equidistant from two fixed points or from two lines (parallel or non parallel). The locus may alternatively be described as the path through which a particle moves to fulfil preset conditions. So imagined, the locus of a point P(x,y) such that P is always three units from the origin is the circle x² + y² = 9.

[edit] Examples

The epitrochoid is an example of a locus generated by a point on a disk rolling around a circle.

The conic sections may be defined in terms of loci:

• A circle is the locus of points where the distance from a certain point, called the center or focus of the circle, is equidistant to all points on the locus; the distance between the center and the locus is the radius.
• An ellipse is the locus of points such that the sum of the distances from any point to the two foci on the ellipse's major axis is constant.
• A hyperbola is the locus of points such that the difference between the distance between a point and one focus and the same point and the other focus is constant.
• A parabola is the locus of points such that the distance from a point to the focus and from the same point to the directrix is always 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 is known as an algebraic variety, the properties of which are studied in the branch of mathematics called algebraic geometry.

In complex dynamics:

• Bifurcation locus
• Connectedness locus.

Further examples of complex geometric shapes are generated by a point on a disk which is made to roll on a flat or curved surface.

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