| dbo:abstract |
In geometry, the limiting points of two disjoint circles A and B in the Euclidean plane are points p that may be defined by any of the following equivalent properties: * The pencil of circles defined by A and B contains a degenerate (radius zero) circle centered at p. * Every circle or line that is perpendicular to both A and B passes through p. * An inversion centered at p transforms A and B into concentric circles. The midpoint of the two limiting points is the point where the radical axis of A and B crosses the line through their centers. This intersection point has equal power distance to all the circles in the pencil containing A and B. The limiting points themselves can be found at this distance on either side of the intersection point, on the line through the two circle centers. From this fact it is straightforward to construct the limiting points algebraically or by compass and straightedge.An explicit formula expressing the limiting points as the solution to a quadratic equation in the coordinates of the circle centers and their radii is given by Weisstein. Inverting one of the two limiting points through A or B produces the other limiting point. An inversion centered at one limiting point maps the other limiting point to the common center of the concentric circles. (en) |
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wiki-commons:Special:FilePath/Apollonian_circles.svg?width=300 |
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| dbo:wikiPageWikiLink |
dbr:Power_distance dbr:Quadratic_equation dbr:Perpendicular dbr:Inversive_geometry dbr:Radical_axis dbc:Inversive_geometry dbc:Circles dbr:Euclidean_plane dbr:Concentric dbr:Pencil_of_circles dbr:Compass_and_straightedge dbr:File:Apollonian_circles.svg |
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dbt:Reflist dbt:Elementary-geometry-stub |
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dbc:Inversive_geometry dbc:Circles |
| gold:hypernym |
dbr:P |
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yago:WikicatCircles yago:Abstraction100002137 yago:Attribute100024264 yago:Circle113873502 yago:ConicSection113872975 yago:Ellipse113878306 yago:Figure113862780 yago:PlaneFigure113863186 dbo:Album yago:Shape100027807 |
| rdfs:comment |
In geometry, the limiting points of two disjoint circles A and B in the Euclidean plane are points p that may be defined by any of the following equivalent properties: * The pencil of circles defined by A and B contains a degenerate (radius zero) circle centered at p. * Every circle or line that is perpendicular to both A and B passes through p. * An inversion centered at p transforms A and B into concentric circles. (en) |
| rdfs:label |
Limiting point (geometry) (en) |
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freebase:Limiting point (geometry) wikidata:Limiting point (geometry) https://global.dbpedia.org/id/fR6x yago-res:Limiting point (geometry) |
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wikipedia-en:Limiting_point_(geometry)?oldid=965592462&ns=0 |
| foaf:depiction |
wiki-commons:Special:FilePath/Apollonian_circles.svg |
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wikipedia-en:Limiting_point_(geometry) |
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dbr:Limiting_point |
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dbr:Apollonian_circles dbr:Inversive_geometry dbr:Circle_of_antisimilitude dbr:Limiting_point |
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wikipedia-en:Limiting_point_(geometry) |
