Talk:Equipollence (geometry)

Historical

The term equipollence is now historical. The following unreferenced text was removed today:

In affine spaces over a field

Let K be a field (which may be the field R of real numbers). An affine space E associated with a K-vector space V is a set provided with a mapping ƒ : E × E → V; (a, b) → ƒ(a, b) (the vector ƒ(a b) will be denoted $\scriptstyle {\vec {ab}}$) such that:

1) for all a in E and all ${\vec {v}}$ in V there exist a single b in E such that ${\overrightarrow {ab}}={\vec {v}}$

2) for all a,b,c in E, ${\overrightarrow {ab}}+{\overrightarrow {bc}}={\overrightarrow {ac}}.$

Definition:

Two bipoints (a, b) and (c, d) of ExE are equipollent if ${\overrightarrow {ab}}={\overrightarrow {cd}},$

when K=R (or K is a field of characteristic different from 2) then (a, b) and (c, d) are equipollent if and only if (a,d) and (b,c) have the same midpoint.

The concept of equipollence of bipoints can be also defined axiomatically.

A reference is needed to use the term in the context of affine geometry.Rgdboer (talk) 22:41, 25 July 2015 (UTC)

Vector

The article states:

Subsequently the term vector was adopted for a class of equipollent line segments.

What exactly ist meant by "subsequently"? --Digamma (talk) 18:56, 21 February 2016 (UTC)

By 1858 Bellavitis saw that Hamilton had absorbed equipollence into the notion of a vector. Reference to Bellavitis' on Hamilton was posted today. Note also that in 1887 Charles-Ange Laisant put the two notions together. — Rgdboer (talk) 04:13, 23 February 2016 (UTC)

Examples

I've added two sections on applications of equipollences, in particular on conjugate diameters: One by Bellavitis on ellipses, and one by Laisant on hyperbolas. --D.H (talk) 13:05, 4 April 2020 (UTC)

Edited version moved here for improvement (diameters are not equipollent):

Equipollence and conjugate diameters In 1854 Bellavitis showed that conjugate diameters in an ellipse are equipollent.(no) In 1874 Laisant extended this observation to conjugate diameters of an hyperbola:

Of an ellipse

Bellavitis (1854)<ref Bellavitis (1854), §§ 145ff /ref> defined the equipollence OM of an ellipse and the respective tangent MT as

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

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:

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

producing the invariant

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

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

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

Of a hyperbola

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<ref Laisant (1874), pp. 133ff /ref>

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

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:

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

producing the invariant relation

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

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

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

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.

The tangent to a one diameter may be equipollent to conjugate diameter (?) Rgdboer (talk) 04:08, 3 November 2021 (UTC)

disable tags Rgdboer (talk) 04:12, 3 November 2021 (UTC)

Equipollences that occur here are clear when looking on the figure in Conjugate diameters. IMO, this example does not belong to this article, and, in any case, is of very low encyclopedic value. So, I suggest to add somewhere in the article the sentence In the 19th century, "Equipollence" has been used with another meaning in the context of Conjugate diameters of a conic section.<ref Laisant (1874), pp. 133ff /ref> It would also be fine to forgive completely this obscure old work. D.Lazard (talk) 09:40, 3 November 2021 (UTC)