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:

\( AB \bumpeq CD . \)

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.[1]

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)[2] defined the equipollence OM of an ellipse and the respective tangent MT as

(1a) \( {\displaystyle {\begin{matrix}&\mathrm {OM} \bumpeq x\mathrm {OA} +y\mathrm {OB} \\&\mathrm {MT} \bumpeq -y\mathrm {OA} +x\mathrm {OB} \\&\left[x^{2}+y^{2}=1;\ x=\cos t,\ y=\sin t\right]\\\Rightarrow &\mathrm {OM} \bumpeq \cos t\cdot \mathrm {OA} +\sin t\cdot \mathrm {OB} \end{matrix}}} \)

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:

\( {\displaystyle {\begin{matrix}{\begin{aligned}\mathrm {OC} &\bumpeq c\mathrm {OA} +d\mathrm {OB} &\qquad &&\mathrm {OA} &\bumpeq c\mathrm {OC} -d\mathrm {OD} \\\mathrm {OD} &\bumpeq -d\mathrm {OA} +c\mathrm {OB} &&&\mathrm {OB} &\bumpeq d\mathrm {OC} +c\mathrm {OD} \end{aligned}}\\\left[c^{2}+d^{2}=1\right]\end{matrix}}} \)

producing the invariant

\( {\displaystyle (\mathrm {OC} )^{2}+(\mathrm {OD} )^{2}\bumpeq (\mathrm {OA} )^{2}+(\mathrm {OB} )^{2}}. \)

Substituting the inverse into (1a), he showed that OM retains its form

\( {\displaystyle {\begin{matrix}\mathrm {OM} \bumpeq (cx+dy)\mathrm {OC} +(cy-dx)\mathrm {OD} \\\left[(cx+dy)^{2}+(cy-dx)^{2}=1\right]\end{matrix}}} \)

b) Conjugate diameter of hyperbolas
Further information: History of Lorentz transformations § Lorentz transformation via hyperbolic functions

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[3]

(1b) \( {\displaystyle {\begin{matrix}&\mathrm {OM} \bumpeq x\mathrm {OA} +y\mathrm {OB} \\&\mathrm {MT} \bumpeq y\mathrm {OA} +x\mathrm {OB} \\&\left[x^{2}-y^{2}=1;\ x=\cosh t,\ y=\sinh t\right]\\\Rightarrow &\mathrm {OM} \bumpeq \cosh t\cdot \mathrm {OA} +\sinh t\cdot \mathrm {OB} \end{matrix}}} \)

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:

\( {\displaystyle {\begin{matrix}{\begin{aligned}\mathrm {OC} &\bumpeq c\mathrm {OA} +d\mathrm {OB} &\qquad &&\mathrm {OA} &\bumpeq c\mathrm {OC} -d\mathrm {OD} \\\mathrm {OD} &\bumpeq d\mathrm {OA} +c\mathrm {OB} &&&\mathrm {OB} &\bumpeq -d\mathrm {OC} +c\mathrm {OD} \end{aligned}}\\\left[c^{2}-d^{2}=1\right]\end{matrix}}} \)

producing the invariant relation

\( {\displaystyle (\mathrm {OC} )^{2}-(\mathrm {OD} )^{2}\bumpeq (\mathrm {OA} )^{2}-(\mathrm {OB} )^{2}}. \)

Substituting into (1b), he showed that OM retains its form

\( {\displaystyle {\begin{matrix}\mathrm {OM} \bumpeq (cx-dy)\mathrm {OC} +(cy-dx)\mathrm {OD} \\\left[(cx-dy)^{2}-(cy-dx)^{2}=1\right]\end{matrix}}} \)

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:

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.[4]

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

\( {\displaystyle \exp(ar)=\cos a+r\sin a,} \) where a is arc length and r determines the plane of the great circle by perpendicularity.

References

Michael J. Crowe (1967) A History of Vector Analysis, "Giusto Bellavitis and His Calculus of Equipollences", pp 52–4, University of Notre Dame Press
Bellavitis (1854), §§ 145ff
Laisant (1874), pp. 133ff

N. Mukunda, Rajiah Simon and George Sudarshan (1989) "The theory of screws: a new geometric representation for the group SU(1,1), Journal of Mathematical Physics 30(5): 1000–1006 MR0992568