Equipollence (geometry)

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:

${\displaystyle AB\bumpeq CD.}$

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: ${\displaystyle AB+BA\bumpeq 0.}$

The equipollence ${\displaystyle AB\bumpeq n.CD,}$ 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.

Among the historical applications of equipollences by Bellavitis and others, the conjugate diameters of ellipses as well as hyperbolas shall be discussed:

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

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

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 (see History of Lorentz transformations#Laisant).