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 are equidistant from each of two parallel lines is a third line parallel to and centered between the two parallel lines (see figure 1b). The locus of points in a plane that are all the same distancer from a single point 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.
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 (see 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.
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.
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.
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