In geometry, a locus is a set of all points, whose location satisfies or is determined by one or more specified conditions. In other words, the set of the points that satisfy some property is often called the locus of a point satisfying this property. The use of singular in this formulation is a witness that, until the end of 19th century, mathematicians did not consider infinite sets. Instead of viewing lines and curves as sets of points, they viewed them as places where a point may be located or may move.
History and philosophy
Until the beginning of the 20th century, a geometrical shape was not considered as an infinite set of points; rather, it was considered as an entity on which a point may be located or on which it moves. Thus a circle in the Euclidean plane was defined as the locus of a point that is at a given distance of a fixed point, the center of the circle. In modern mathematics, similar concepts are more frequently reformulated by describing shapes as sets; for instance, one says that the circle is the set of points that are at a given distance from the center. In contrast to the set-theoretic view, the old formulation avoids considering infinite collections, as avoiding the actual infinite was an important philosophical position of earlier mathematicians. Once set theory became the universal basis over which the whole mathematics is built, the term of locus became rather old-fashioned. Nevertheless, the word is still widely used, mainly for a concise formulation, for example:
More recently, techniques such as the theory ofschemes, and the use of category theory instead of set theory to give a foundation to mathematics, have returned to notions more like the original definition of a locus as an object in itself rather than as a set of points.
To prove a geometric shape is the correct locus for a given set of conditions, one generally divides the proof into two stages:
Proof that all the points that satisfy the conditions are on the given shape.
Proof that all the points on the given shape satisfy the conditions.
Examples
First example
Find the locus of a point P that has a given ratio of distances k = d1/d2 to two given points. In this example k = 3, A and B are chosen as the fixed points. This equation represents a circle with center and radius. It is the circle of Apollonius defined by these values of k, A, and B.
Second example
A triangle ABC has a fixed side with length c. Determine the locus of the third vertexC such that the medians from A and C are orthogonal. Choose an orthonormalcoordinate system such that A, B. C is the variable third vertex. The center of is M. The median from C has a slopey/x. The median AM has slope 2y/. The locus of the vertex C is a circle with center and radius 3c/4.
Third example
A locus can also be defined by two associated curves depending on one common parameter. If the parameter varies, the intersection points of the associated curves describe the locus. In the figure, the points K and L are fixed points on a given linem. The line k is a variable line through K. The line l through L is perpendicular to k. The angle between k and m is the parameter. k and l are associated lines depending on the common parameter. The variable intersection pointS of k and l describes a circle. This circle is the locus of the intersection point of the two associated lines.
Fourth example
A locus of points need not be one-dimensional. For example, the locus of the inequality is the portion of the plane that is below the line of equation.
