# Locus

# Locus

A locus is a set of points that contains all the points, and only the points, that satisfy the condition, or conditions, required to describe a geometric figure.

The word “locus” is Latin for place or location. A locus may also be defined as the path traced by a point in motion, as it moves according to a stated set of conditions, since all the points on the path satisfy the stated conditions. Thus, the phrases “locus of a point” and “locus of points” are often interchangeable.

A locus may be rather simple and appear to be obvious from the stated condition. Examples of loci (plural for locus) include points, lines, and surfaces. The locus of points in a plane that are equidistant from two given points is the straight line that is perpendicular to and passes through the center of the line segment connecting the two points (Figure 1a).

The locus of points in a plane that is equidistant from each of two parallel lines is a third line parallel to and centered between the two parallel lines (Figure 1b). The locus of points in a plane that are all the same distancer from a single point is a circle with radius r. Given the same condition, not confined to a plane but to three-dimensional space, the locus is the surface of a sphere with radius r. However, not every set of conditions leads to an immediately recognizable geometric object.

To find a locus, given a stated set of conditions, first find a number of points that satisfy the conditions. Then, “guess” at the locus by fitting a smooth line, or lines, through the points. Give an accurate description of the guess, then prove that it is correct. To prove that a guess is correct, it is necessary to prove that the points of the locus and the points of the guess coincide. That is, the figure guessed must contain all the points of the locus and no points that are not in the locus. Thus, it is necessary to show that (1) every point of the figure is in the locus and (2) every point in the locus is a point of the figure, or every point not on the figure is not in the locus.

## Compound loci

In some cases, a locus may be defined by more that one distinct set of conditions. In this case the locus is called a compound locus, and corresponds to the intersection of two or more loci. For example, the locus of points that are equidistant from two given points and also equidistant from two given parallel lines (Figure 1c), is a single point. That point lies at the intersection of two lines, one line containing those points equidistant from the two points, and one line containing all those points equidistant from the parallel lines.

## Applications

There are many other interesting loci, for example the cycloid (Figure 2).

The cycloid is the locus of a point on a circle as the circle rolls in a straight line along a flat surface. The cycloid is the path that a falling body takes on a windy day to reach the ground in the shortest possible time. Some interesting loci can be described by using the moving point definition of locus. For example, consider this simple mechanism. (Figure 3.)

It has a pencil at point A, pivots at points B and C and point D is able to slide toward and away from

### KEY TERMS

**Conic section** —A conic section is a figure that results from the intersection of a right circular cone with a plane. The conic sections are the circle, ellipse, parabola, and hyperbola.

**Line** —A line is a collection of points. A line has length, but no width or thickness.

**Plane** —A plane is also a collection of points. It has length and width, but no thickness.

**Point** —In geometric terms a point is a location. It has no size associated with it, no length, width, or thickness.

**Right circular cone** —The surface that results from rotating two intersecting lines in a circle about an axis that is at a right angle to the circle of rotation.

point C. When point D slides back and forth, the pencil moves up and down drawing a line perpendicular to the base (a line through C and D). More complicated devices are capable of tracing figures while simultaneously enlarging or reducing them.

## Resources

### BOOKS

Fuller, Gordon, and Dalton Tarwater. *Analytic Geometry.* 6th ed. Reading, MA: Addison Wesley, 1986.

Larson, Ron. *Calculus With Analytic Geometry.* Boston: Houghton Mifflin College, 2002.

Smith, Stanley A., Charles W. Nelson, Roberta K. Koss, Mervin L. Keedy, and Marvin L. Bittinger. *Addison Wesley Informal Geometry.* Reading, MA: Addison Wesley, 1992.

### OTHER

*Geocities.* “The Cycloid” <http://www.geocities.com/ CapeCanaveral/Lab/3550/cycloid.htm> (accessed December 2, 2006).

*Wolfram MathWorld.* “Cycloid” <http://mathworld.wolfram.com/Cycloid.html> (accessed December 2, 2006).

J. R. Maddocks

# Locus

# Locus

A locus is a set of points that contains all the points, and only the points, that satisfy the condition, or conditions, required to describe a geometric figure. The word *locus* is Latin for place or location. A locus may also be defined as the path traced out by a point in **motion** , as it moves according to a stated set of conditions, since all the points on the path satisfy the stated conditions. Thus, the phrases "locus of a point" and "locus of points" are often interchangeable. A locus may be rather simple and appear to be obvious from the stated condition. Examples of loci (plural for locus) include points, lines, and surfaces. The locus of points in a **plane** that are equidistant from two given points is the straight line that is **perpendicular** to and passes through the center of the line segment connecting the two points (Figure 1a).

The locus of points in a plane that is equidistant from each of two **parallel** lines is a third line parallel to and centered between the two parallel lines (Figure 1b). The locus of points in a plane that are all the same **distance** *r* from a single point is a **circle** with radius *r*. Given the

same condition, not confined to a plane but to three-dimensional space, the locus is the surface of a **sphere** with radius *r*. However, not every set of conditions leads to an immediately recognizable geometric object.

To find a locus, given a stated set of conditions, first find a number of points that satisfy the conditions. Then, "guess" at the locus by fitting a smooth line, or lines, through the points. Give an accurate description of the guess, then prove that it is correct. To prove that a guess is correct, it is necessary to prove that the points of the locus and the points of the guess coincide. That is, the figure guessed must contain all the points of the locus and no points that are not in the locus. Thus, it is necessary to show that (1) every point of the figure is in the locus and (2) every point in the locus is a point of the figure, or every point not on the figure is not in the locus.

## Compound loci

In some cases, a locus may be defined by more that one distinct set of conditions. In this case the locus is called a compound locus, and corresponds to the intersection of two or more loci. For example, the locus of points that are equidistant from two given points and also equidistant from two given parallel lines (Figure 1c), is a single point. That point lies at the intersection of two lines, one line containing those points equidistant from the two points, and one line containing all those points equidistant from the parallel lines.

## Applications

There are many other interesting loci, for example the cycloid.

The cycloid is the locus of a point on a circle as the circle rolls in a straight line along a flat surface. The cycloid is the path that a falling body takes on a windy day in order to reach the ground in the shortest possible **time** . Some interesting loci can be described by using the moving point definition of locus. For example, consider this simple mechanism. (Figure 3.)

It has a pencil at point A, pivots at points B and C and point D is able to slide toward and away from point C. When point D slides back and forth, the pencil moves up and down drawing a line perpendicular to the base (a line through C and D). More complicated devices are capable of tracing figures while simultaneously enlarging or reducing them.

## Resources

### books

Fuller, Gordon, and Dalton Tarwater. *Analytic Geometry.* 6th ed. Reading, MA: Addison Wesley, 1986.

Gowar, Norman. *An Invitation to Mathematics.* New York: Oxford University Press, 1979.

Larson, Ron. *Calculus With Analytic Geometry.* Boston: Houghton Mifflin College, 2002.

Smith, Stanley A., Charles W. Nelson, Roberta K. Koss, Mervin L. Keedy, and Marvin L. Bittinger. *Addison Wesley Informal Geometry.* Reading, MA: Addison Wesley, 1992.

J. R. Maddocks

## KEY TERMS

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .**Conic section**—A conic section is a figure that results from the intersection of a right circular cone with a plane. The conic sections are the circle, ellipse, parabola, and hyperbola.

**Line**—A line is a collection of points. A line has length, but no width or thickness.

**Plane**—A plane is also a collection of points. It has length and width, but no thickness.

**Point**—In geometric terms a point is a location. It has no size associated with it, no length, width, or thickness.

**Right circular cone**—The surface that results from rotating two intersecting lines in a circle about an axis that is at a right angle to the circle of rotation.

# Locus

# Locus

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.

see also Conic Sections.

*Lucia McKay*

## Bibliography

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

# locus

lo·cus / ˈlōkəs/ •
n. (pl. lo·ci / ˈlōˌsī; -ˌsē; -ˌkē; -ˌkī/ ) 1. technical a particular position, point, or place: *it is impossible to specify the exact locus in the brain of these neural events.* ∎ the effective or perceived location of something abstract: *the real locus of power is the informal council.* ∎ Genetics the position of a gene or mutation on a chromosome.2. Math. a curve or other figure formed by all the points satisfying a particular equation of the relation between coordinates, or by a point, line, or surface moving according to mathematically defined conditions.

# locus

**locus (pl. loci)** A specific place on a chromosome where a gene is located. In diploids, loci pair during meiosis and, unless there have been translocations, inversions, etc., the homologous chromosomes contain identical sets of loci in the same linear order. At each locus is 1 gene; if that gene can take several forms (alleles), only 1 of these will be present at a given locus.

# locus

**locus (pl. loci)** The specific place on a chromosome where a gene is located. In diploids, loci pair during meiosis and unless there have been translocations, inversions, etc., the homologous chromosomes contain identical sets of loci in the same linear order. At each locus is one gene; if that gene can take several forms (alleles), only one of these will be present at a given locus.

# locus

**locus (pl. loci)** The position of a gene on a chromosome or within a nucleic acid molecule. The alleles of a gene occupy the same locus on homologous chromosomes.

# locus

**locus** In geometry, the path traced by a specified point when it moves to satisfy certain conditions. For example, a circle is the locus of a point in a plane moving in such a way that its distance from a fixed point (the centre) is constant.

# locus

**locus ( loh-kŭs) n.**

**1.**(in anatomy) a region or site.

**2.**(in genetics) the region of a chromosome occupied by a particular gene.

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