Locus

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Sometimes it is useful in mathematics to describe the path that a point traces as it moves in a plane to meet certain conditions. For example, what is the path that a point on the end of the second hand of a clock traces in 60 seconds? This answer, of course, is a circle.

One way to define a circle is to say that a circle is the locus of all the points in a plane that are a given distance from a fixed point, called the center.

A locus in a plane can be thought of as all the possible locations or positions that a point can take as it moves to meet certain stated conditions. What is the locus of, or path traced by, a point in a plane that moves so that it is always three inches from point A? This locus will be a circle, with point A as the center and a radius of three inches.

What is the locus in a plane of all points that are 2 centimeters from a given line? This locus is made up of two lines, each parallel to the given line, one on each side, and at a distance of 2 centimeters from it, as illustrated in the figure below. The two dashed lines form the locus.

What is the locus of all points in a plane that are the same distance from point D and from point E? To answer this, one might draw some example points that are equidistant from D and E, such as the points marked with a star in the left-hand illustration of the figure below.

These example points indicate that the locus of all the points in a plane that are the same distance from D as they are from E is a line that is the perpendicular bisector of the line segment that joins D and E, as shown in the right-hand illustration.

The idea of a locus can be used not just in a plane but also in three-dimensional space. For example, the preceding example, extended into space, becomes the locus of all points that are equidistant from points D and E. This locus will be the entire plane that is perpendicular to the plane containing DE and its perpendicular bisector and that contains the entire perpendicular bisector.

In space, the locus of all points at a given distance from a specific point is a sphere with a center at the point and a radius equal to the given distance. In space, the locus of all points at a given distance from a line segment is a cylinder with a hemisphere at each end.

The idea of locus can also be used to define the conic sections. In a plane, a circle is the locus of all points at a given distance from a specific point; a parabola is the locus of all points such that each point on the curve is the same distance from a specific point as its distance from a specific line; an ellipse is the locus of all points such that, for each point on the curve, the sum of the distances from each of two separate specific points, called the foci, remains the same; and a hyperbola is the locus of all points such that, for each point on the curve, the absolute value of the difference of the distances to each of two separate specific points, called the foci, remains the same.

Conic Sections.

Hogben, Lancelot. Mathematics in the Making. New York: Crescent Books, Inc., 1940.

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