https://en.formulasearchengine.com/index.php?action=history&feed=atom&title=Inverse_bundle 2026-07-02T17:44:46Z Revision history for this page on the wiki MediaWiki 1.43.0-wmf.28 https://en.formulasearchengine.com/index.php?title=Inverse_bundle&diff=24577&oldid=prev 2011-09-19T08:27:26Z

<p>Various citation cleanup + <a href="/index.php?title=WP:AWB&amp;action=edit&amp;redlink=1" class="new" title="WP:AWB (page does not exist)">WP:AWB</a> fixes . Report errors and suggestions at <a href="/index.php?title=User_talk:CitationCleanerBot&amp;action=edit&amp;redlink=1" class="new" title="User talk:CitationCleanerBot (page does not exist)">User talk:CitationCleanerBot</a></p> <p><b>New page</b></p><div>In [[geometry]], &#039;&#039;&#039;equipollence&#039;&#039;&#039; is a certain relationship between ordered pairs of points. A pair (&#039;&#039;a&#039;&#039;,&amp;nbsp;&#039;&#039;b&#039;&#039;) of points and another pair (&#039;&#039;c&#039;&#039;,&amp;nbsp;&#039;&#039;d&#039;&#039;) are equipollent precisely if the distance and direction from &#039;&#039;a&#039;&#039; to&amp;nbsp;&#039;&#039;b&#039;&#039; are respectively the same as the distance and direction from &#039;&#039;c&#039;&#039; to&amp;nbsp;&#039;&#039;d&#039;&#039;.<br /> <br /> == In affine spaces over a field ==<br /> <br /> Let &#039;&#039;K&#039;&#039; be a [[field (mathematics)|field]] (which may be the field &#039;&#039;&#039;R&#039;&#039;&#039; of [[real number]]s). An [[affine space]] &#039;&#039;E&#039;&#039; associated with a &#039;&#039;K&#039;&#039;-vector space &#039;&#039;V&#039;&#039; is a set provided with a mapping &#039;&#039;ƒ&#039;&#039;&amp;nbsp;:&amp;nbsp;&#039;&#039;E&#039;&#039;&amp;nbsp;&amp;times;&amp;nbsp;&#039;&#039;E&#039;&#039;&amp;nbsp;→&amp;nbsp;&#039;&#039;V&#039;&#039;; (&#039;&#039;a&#039;&#039;,&amp;nbsp;&#039;&#039;b&#039;&#039;)&amp;nbsp;→&amp;nbsp;&#039;&#039;ƒ&#039;&#039;(&#039;&#039;a&#039;&#039;,&amp;nbsp;&#039;&#039;b&#039;&#039;) (the vector &#039;&#039;ƒ&#039;&#039;(&#039;&#039;a&#039;&#039;&amp;nbsp;&#039;&#039;b&#039;&#039;) will be denoted &lt;math&gt;\scriptstyle\vec{ab}&lt;/math&gt;) such that:<br /> <br /> 1) for all a in E and all &lt;math&gt;\vec{v}&lt;/math&gt; in V there exist a single b in E such that &lt;math&gt;\overrightarrow{ab} = \vec{v} &lt;/math&gt;<br /> <br /> 2) for all a,b,c in E, &lt;math&gt;\overrightarrow{ab} + \overrightarrow{bc} = \overrightarrow{ac} . &lt;/math&gt; <br /> <br /> Definition:<br /> <br /> Two bipoints (a, b) and (c, d) of ExE are equipollent if <br /> &lt;math&gt;\overrightarrow{ab} = \overrightarrow{cd} , &lt;/math&gt; <br /> <br /> 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.<br /> <br /> The concept of equipollence of bipoints can be also defined axiomatically.<br /> <br /> ==External links==<br /> * [http://sites.google.com/site/contributionsingeometry Axiomatic definition of equipollence]<br /> <br /> [[Category:Affine geometry]]</div>

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