Equipollence (geometry)

In Euclidean geometry, equipollence is a homogeneous 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. 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 (possibly degenerate) parallelogram:

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'':
:$AB \bumpeq CD .$

The following passages, translated by Michael J. Crowe, show the anticipation that Bellavitis had of vector concepts:
:Equipollences continue to hold when one substitutes for the lines in them, other lines which are respectively equipollent to them, however they may be situated in space. From this it can be understood how any number and any kind of lines may be ''summed'', and that in whatever order these lines are taken, the same equipollent-sum will be obtained...

:In equipollences, just as in equations, a line may be transferred from one side to the other, provided that the sign is changed...
Thus oppositely directed segments are negatives of each other: $AB + BA \bumpeq 0 .$

:The equipollence $AB \bumpeq n.CD ,$ where ''n'' stands for a positive number, indicates that ''AB'' is both parallel to and has the same direction as ''CD'', and that their lengths have the relation expressed by ''AB'' = ''n.CD''.

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:
:To appreciate Hamilton's method, let us first recall the much simpler case of the Abelian group of translations in Euclidean three-dimensional space. Each translation is representable as a vector in space, only the direction and magnitude being significant, and the location irrelevant. The composition of two translations is given by the head-to-tail parallelogram rule of vector addition; and taking the inverse amounts to reversing direction. In Hamilton's theory of turns, we have a generalization of such a picture from the Abelian translation group to the non-Abelian SU(2). Instead of vectors in space, we deal with directed great circle arcs, of length < π on a unit sphere S² in a Euclidean three-dimensional space. Two such arcs are deemed equivalent if by sliding one along its great circle it can be made to coincide with the other.
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
:$\exp(a r) = \cos a + r \sin a ,$ where ''a'' is arc length and ''r'' determines the plane of the great circle by perpendicularity.

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 $\overrightarrow$ in ''V'', and for any ''a'' in ''A'' and ''v'' in ''V'' there is ''b'' in ''A'' such that $\overrightarrow = v ,$ and for any three points in ''A'' there is the vector equation
:$\overrightarrow + \overrightarrow = \overrightarrow .$

Evidently this development depends on previous introduction to abstract vector spaces, in contrast to the introduction of vectors via equivalence classes of directed segments. Mikhail Postnikov (1982
Lectures in Geometry Semester I Analytic Geometry
pages 45 and 46, via Internet Archive

References

{{Reflist
* Giusto Bellavitis (1835) "Saggio di applicazioni di un nuovo metodo di Geometria Analitica (Calcolo delle equipollenze)", ''Annali delle Scienze del Regno Lombardo-Veneto, Padova'' 5: 244–59.
* Giusto Bellavitis (1854
Sposizione del Metodo della Equipollenze
link from Google Books.
:* Charles-Ange Laisant (1874): French translation with additions of Bellavitis (1854
Exposition de la méthode des equipollences
link from Google Books.
* Giusto Bellavitis (1858
Calcolo dei Quaternioni di W.R. Hamilton e sua Relazione col Metodo delle Equipollenze
link from HathiTrust.
* Charles-Ange Laisant (1887
Theorie et Applications des Equipollence
Gauthier-Villars, link from University of Michigan Historical Math Collection.
* Lena L. Severance (1930
The Theory of Equipollences; Method of Analytical Geometry of Sig. Bellavitis
link from HathiTrust.

Vectors (mathematics and physics)
History of mathematics
Binary relations
Equivalence (mathematics)