In Euclidean geometry, equipollence is a binary relation between directed line segments. Two segments are said to be equipollent when they have the same length and direction. Two equipollent segments are parallel but not necessarily colinear nor overlapping, and vice versa. For example, a segment AB, from point A to point B, has the opposite direction to segment BA; thus AB and BA are equipollent.

Parallelogram property

A property of Euclidean spaces is the parallelogram property of vectors: If two segments are equipollent, then they form two sides of a parallelogram: "If a given vector holds between a and b, c and d, then the vector which holds between a and c is the same as that which holds between b and d."

History

The concept of equipollent line segments was advanced by Giusto Bellavitis in 1835. Subsequently, the term vector was adopted for a class of equipollent line segments. Bellavitis's use of the idea of a relation to compare different but similar objects has become a common mathematical technique, particularly in the use of equivalence relations. Bellavitis used a special notation for the equipollence of segments AB and CD: The following passages, translated by Michael J. Crowe, show the anticipation that Bellavitis had of vector concepts: Thus oppositely directed segments are negatives of each other: The segment from A to B is a bound vector, while the class of segments equipollent to it is a free vector, in the parlance of Euclidean vectors.

Spherical geometry

Geometric equipollence is also used on the sphere: On a great circle of a sphere, two directed circular arcs are equipollent when they agree in direction and arc length. An equivalence class of such arcs is associated with a quaternion versor

Abstraction

Properties of the equivalence classes of equipollent segments can be abstracted to define affine space: If A is a set of points and V is a vector space, then (A, V) is an affine space provided that for any two points a,b in A there is a vector in V, and for any a in A and v in V there is b in A such that and for any three points in A there is the vector equation Evidently this development depends on previous introduction to abstract vector spaces, in contrast to the introduction of vectors via equivalence classes of directed segments.