Equipollence (geometry)

In Euclidean geometry, equipollence is a binary relation between directed line segments.

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.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 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:

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.