Equipollence (geometry)

In Euclidean geometry, equipollence is a binary relation between directed line segments. A line segment "AB" from point "A" to point "B" has the opposite direction to line segment "BA". Two directed line segments are equipollent when they have the same length and direction.
History.
Overview.
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: formula_2
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.
Examples.
Among the historical applications of equipollences by Bellavitis and others, the conjugate diameters of ellipses as well as hyperbolas shall be discussed:
a) Conjugate diameter of ellipses.
Bellavitis (1854) defined the equipollence OM of an ellipse and the respective tangent MT as
where OA and OB are conjugate semi-diameters of the ellipse, both of which he related to two other conjugated semi-diameters OC and OD by the following relation and its inverse:
producing the invariant
Substituting the inverse into (1a), he showed that OM retains its form
b) Conjugate diameter of hyperbolas.
In the French translation of Bellavitis' 1854-paper, Charles-Ange Laisant (1874) added a chapter in which he adapted the above analysis to the hyperbola. The equipollence OM and its tangent MT of a hyperbola is defined by
Here, OA and OB are conjugate semi-diameters of a hyperbola with OB being imaginary, both of which he related to two other conjugated semi-diameters OC and OD by the following transformation and its inverse:
producing the invariant relation
Substituting into (1b), he showed that OM retains its form
From a modern perspective, Laisant's transformation between two pairs of conjugate semi-diameters can be interpreted as Lorentz boosts in terms of hyperbolic rotations, as well as their visual demonstration in terms of Minkowski diagrams.
Extension.
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