http://debianws.lexgopc.com/wiki143/index.php?action=history&feed=atom&title=Concentric_objectsConcentric objects - Revision history2026-05-06T01:06:25ZRevision history for this page on the wikiMediaWiki 1.43.1http://debianws.lexgopc.com/wiki143/index.php?title=Concentric_objects&diff=7238132&oldid=previmported>David Eppstein: cs22024-08-19T21:44:29Z<p>cs2</p>
<p><b>New page</b></p><div>{{Short description|Geometric objects with a common centre}}<br />
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{{Other uses|Concentric (disambiguation)}}<br />
[[File:WA 80 cm archery target.svg|thumb|An [[shooting target|archery target]], featuring evenly spaced '''concentric'''&nbsp;circles that surround&nbsp;a&nbsp;"[[Bullseye (target)|bullseye]]".]]<br />
[[File:Kepler-solar-system-2.png|thumb|Kepler's cosmological model formed by concentric spheres and regular polyhedra]]<br />
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In [[geometry]], two or more [[mathematical object|objects]] are said to be '''''concentric''''' when they share the same [[center (geometry)|center]]. Any pair of (possibly unalike) objects with well-defined centers can be concentric, including [[circles]], [[sphere]]s, [[regular polygon]]s, [[regular polyhedron|regular polyhedra]], parallelograms, cones, conic sections, and quadrics.<ref><br />
Circles: {{citation|title=Elementary Geometry for College Students|first1=Daniel C.|last1=Alexander|first2=Geralyn M.|last2=Koeberlein|year=2009|publisher=Cengage Learning|isbn=9781111788599|page=279|url=https://books.google.com/books?id=cRIFAAAAQBAJ&pg=PA279}}<br />
<p>Spheres: {{harvp|Apostol|2013}}</p><br />
<p>Regular polygons: {{citation|title=A Course of Pure Mathematics|first=Godfrey Harold|last=Hardy|author-link=G. H. Hardy|publisher=The University Press|year=1908|page=107|url=https://books.google.com/books?id=tUY7AQAAIAAJ&pg=PA107}}</p><br />
<p>Regular polyhedra: {{citation|title=Comprehensive Coordination Chemistry: Theory & background|first=Robert D.|last=Gillard|isbn=9780080262321|pages=[https://archive.org/details/comprehensivecoo0001unse/page/137 137, 139]|publisher=Pergamon Press|year=1987|url=https://archive.org/details/comprehensivecoo0001unse/page/137}}.</p><br />
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Geometric objects are '''''[[coaxial]]''''' if they share the same [[Coordinate axis|axis]] (line of symmetry). Geometric objects with a well-defined axis include circles (any line through the center), spheres, [[cylinder (geometry)|cylinders]],<ref>{{citation|title=Fluid Mechanics|first1=Joseph|last1=Spurk|first2=Nuri|last2=Aksel|publisher=Springer|year=2008|isbn=9783540735366|page=174|url=https://books.google.com/books?id=7_FrhazRTgsC&pg=PA174}}.</ref> conic sections, and surfaces of revolution.<br />
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Concentric objects are often part of the broad category of ''[[whorl]]ed patterns'', which also includes ''[[spiral]]s'' (a curve which emanates from a point, moving farther away as it revolves around the point).<br />
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==Geometric properties==<br />
In the [[Euclidean plane]], two circles that are concentric necessarily have different radii from each other.<ref name="srm">{{citation|title=Surveyor Reference Manual|first1=George M.|last1=Cole|first2=Andrew L.|last2=Harbin|publisher=www.ppi2pass.com|year=2009|isbn=9781591261742|url=https://books.google.com/books?id=bFun4fjl4P0C&pg=SA2-PA6|at=§2, p.&nbsp;6}}.</ref><br />
However, circles in three-dimensional space may be concentric, and have the same radius as each other, but nevertheless be different circles. For example, two different [[Meridian (astronomy)|meridians]] of a terrestrial [[globe]] are concentric with each other and with the [[globe]] of the earth (approximated as a sphere). More generally, every two [[great circle]]s on a sphere are concentric with each other and with the sphere.<ref>{{citation|title=The American universal geography;: or, A view of the present state of all the kingdoms, states, and colonies in the known world, Volume 1|first=Jedidiah|last=Morse|edition=6th|publisher=Thomas & Andrews|year=1812|page=19|url=https://books.google.com/books?id=J_5AAAAAcAAJ&pg=PA19}}.</ref><br />
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By [[Euler's theorem in geometry]] on the distance between the [[circumcenter]] and [[incenter]] of a triangle, two concentric circles (with that distance being zero) are the [[circumcircle]] and [[incircle]] of a triangle [[if and only if]] the radius of one is twice the radius of the other, in which case the triangle is [[equilateral triangle|equilateral]].<ref>{{citation |author=Dragutin Svrtan and Darko Veljan |title=Non-Euclidean versions of some classical triangle inequalities |publisher=Forum Geometricorum |volume=12 |year=2012 |pages=197–209 |url=http://forumgeom.fau.edu/FG2012volume12/FG201217index.html |website=forumgeom.fau.edu}}</ref>{{rp|p. 198}}<br />
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The circumcircle and the incircle of a [[regular polygon|regular ''n''-gon]], and the regular ''n''-gon itself, are concentric. For the circumradius-to-inradius ratio for various ''n'', see [[Bicentric polygon#Regular polygons]]. The same can be said of a [[regular polyhedron]]'s [[insphere]], [[midsphere]] and [[circumsphere]].<br />
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The region of the plane between two concentric circles is an [[Annulus (mathematics)|annulus]], and analogously the region of space between two concentric spheres is a [[spherical shell]].<ref name="apostol">{{citation|title=New Horizons in Geometry|volume=47|series=Dolciani Mathematical Expositions|first=Tom|last=Apostol|author-link=Tom Apostol|publisher=Mathematical Association of America|year=2013|isbn=9780883853542|page=140|url=https://books.google.com/books?id=PUVvwfjhjvMC&pg=PA140}}.</ref><br />
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For a given point ''c'' in the plane, the set of all circles having ''c'' as their center forms a [[Apollonian circles|pencil of circles]]. Each two circles in the pencil are concentric, and have different radii. Every point in the plane, except for the shared center, belongs to exactly one of the circles in the pencil. Every two disjoint circles, and every hyperbolic pencil of circles, may be transformed into a set of concentric circles by a [[Möbius transformation]].<ref>{{citation|title=Complex Numbers and Geometry|series=MAA Spectrum|first=Liang-shin|last=Hahn|publisher=Cambridge University Press|year=1994|isbn=9780883855102|page=142|url=https://books.google.com/books?id=s3nMMkPEvqoC&pg=PA142}}.</ref><ref>{{citation|title=Geometry|first1=David A.|last1=Brannan|first2=Matthew F.|last2=Esplen|first3=Jeremy J.|last3=Gray|publisher=Cambridge University Press|year=2011|isbn=9781139503709|pages=320–321|url=https://books.google.com/books?id=UlrmKjIjrzQC&pg=PA320}}.</ref><br />
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== Applications and examples ==<br />
The [[Capillary wave|ripple]]s formed by dropping a small object into still water naturally form an expanding system of concentric circles.<ref>{{citation|title=Waves and Ripples in Water, Air, and Æther: Being a Course of Christmas Lectures Delivered at the Royal Institution of Great Britain|first=Sir John Ambrose|last=Fleming|author-link=John Ambrose Fleming|publisher=Society for Promoting Christian Knowledge|year=1902|page=20|url=https://books.google.com/books?id=BiRJAAAAIAAJ&pg=PA20}}.</ref> Evenly spaced circles on the targets used in [[target archery]]<ref>{{citation|title=Archery: Steps to Success|first1=Kathleen|last1=Haywood|first2=Catherine|last2=Lewis|publisher=Human Kinetics|year=2006|isbn=9780736055420|page=xxiii|url=https://books.google.com/books?id=-EFySHOgGmIC&pg=PR23}}.</ref> or similar sports provide another familiar example of concentric circles.<br />
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[[Coaxial cable]] is a type of electrical cable in which the combined neutral and earth core completely surrounds the live core(s) in system of concentric cylindrical shells.<ref>{{citation|title=Fiber Optics Standard Dictionary|first=Martin|last=Weik|publisher=Springer|year=1997|isbn=9780412122415|page=124|url=https://books.google.com/books?id=s56JS2WkXE4C&pg=PA124}}.</ref><br />
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[[Johannes Kepler]]'s ''[[Mysterium Cosmographicum]]'' envisioned a cosmological system formed by concentric regular polyhedra and spheres.<ref>{{citation|title=Geometry and Its Applications|first=Walter A.|last=Meyer|edition=2nd|publisher=Academic Press|year=2006|isbn=9780080478036|page=436|url=https://books.google.com/books?id=ez6H5Ho6E3cC&pg=PA436}}.</ref><br />
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Concentric circles have been used on firearms surfaces as means of holding lubrication or reducing friction on components, similar to [[Engine turning|jewelling]].<ref>{{citation|url=https://americanhandgunner.com/handguns/behind-enemy-lines-sterling-haydens-registered-magnum/|title=Behind Enemy Lines: Sterling Hayden's Registered Magnum|magazine=American Handgunner|first=Dave|last=Elliot|year=2018}}</ref><br />
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Concentric circles are also found in [[Diopter sight|diopter sights]], a type of mechanic sights commonly found on target rifles. They usually feature a large disk with a small-diameter hole near the shooter's eye, and a front globe sight (a circle contained inside another circle, called ''tunnel''). When these sights are correctly aligned, the point of impact will be in the middle of the front sight circle.<br />
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<gallery mode=packed><br />
File:2006-01-14 Surface waves.jpg|[[Capillary wave|Ripples]] in water<br />
File:Histology of a Pacinian corpuscle.jpg|[[Histology]] of a [[Pacinian corpuscle]], in a typical expanding circular pattern.<br />
File:Wooden Piling - dendrochronolgy.jpg|Tree rings, as can be used for [[Dendrochronology|tree-ring dating]]<br />
</gallery><br />
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==See also==<br />
{{div col}}<br />
*[[Centered cube number]]<br />
*[[Homoeoid]]<br />
*[[Focaloid]]<br />
*[[Circular symmetry]]<br />
*[[Magic circle (mathematics)]]<br />
*[[Osculating circle]]<br />
*[[Spiral]]<br />
{{div col end}}<br />
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==References==<br />
{{reflist}}<br />
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==External links==<br />
*Geometry: [http://www.mathopenref.com/concentric.html Concentric circles demonstration] With interactive animation<br />
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[[Category:Corrosion prevention]]<br />
[[Category:Geometric centers]]<br />
[[Category:Visual motifs]]</div>imported>David Eppstein