# Distance

## Path length

Distance has two different meanings. It is a number used to characterize the shortest length between two geometric figures, and it is the total length of a path. In the first case, the distance between two points is the simplest instance.

The absolute distance between two points, sometimes called the displacement, can only be a positive number. It can never be a negative number, and can only be zero when the two points are identical. Only one straight line exists between any two points P1 and P2. The length of this line is the shortest distance between P1 and P2.

In the case of parallel lines, the distance between the two lines is the length of a perpendicular segment connecting them. If two figures such as line segments, triangles, circles, cubes, etc. do not intersect, then the distance between them is the shortest distance between any pair of points, one of which lies on one figure, one of which lies on the other.

To determine the distance between two points, we must first consider a coordinate system. An xy coordinate system consists of a horizontal axis (x) and vertical axis (y). Both axes are infinite for positive and negative values. The crossing point of the lines is the origin (O), at which both x and y values are zero.

We define the coordinates of point P 1 as (x1, y1),and point P2 by (x 2, y2). The distance, the length of the connecting straight line (P1, P2) which is the shortest distance between the two points, is given by the equation d = RADIC(x MINUS x 2 1 2) + (y 2 1 - y2) in many types of physics and engineering problems, for example, in tracking the trajectory of an atomic particle, or in determining the lateral motion of a suspension bridge in the presence of high winds.

The other meaning of distance is the length of a path. This is easily understood if the path consists entirely of line segments, such the perimeter of a pentagon. The distance is the sum of the lengths of the line segments that make up the perimeter. For curves that are not line segments, a continuous path can usually be approximated by a sequence of line segments. Using shorter line segments produces a better approximation. The limiting case, when the lengths of the line segments go to zero, is the distance. A common example would be the circumference of a circle, which is a distance.

Kristin Lewotsky

## KEY TERMS

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Absolute distance

—The displacement between two points, independent of orientation.

Hessian normal form

—A definition of a line such that the line cuts the plane into positive and negative regions; this form attaches sign (+/-) to distance so that a distance measurement indicates displacement and orientation of the measured object from the line.

Oriented distance

—A distance that indicates direction by sign.

Path length

—The length of a path, generally determine by calculus.