# Director circle

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{{Short description|Circle formed by all 90° crossings of tangents of an ellipse or hyperbola}}
thumb|upright=1.35|An ellipse, its minimum bounding box, and its director circle.
In [geometry](/source/geometry), the '''director circle''' of an [ellipse](/source/ellipse) or [hyperbola](/source/hyperbola) (also called the '''[orthoptic](/source/Isoptic) circle''' or '''Fermat–Apollonius circle''') is a [circle](/source/circle) consisting of all points where two [perpendicular](/source/perpendicular) [tangent line](/source/tangent_line)s to the ellipse or hyperbola cross each other.

==Properties==
The director circle of an ellipse [circumscribes](/source/Circumscribed_circle) the [minimum bounding box](/source/minimum_bounding_box) of the ellipse.  It has the same center as the ellipse, with radius <math display="inline">\sqrt{a^2 + b^2}</math>, where <math>a</math> and <math>b</math> are the [semi-major axis](/source/semi-major_axis) and [semi-minor axis](/source/semi-minor_axis) of the ellipse. Additionally, it has the property that, when viewed from any point on the circle, the ellipse spans a [right angle](/source/right_angle).<ref>{{harvnb|Akopyan|Zaslavsky|2007|pages=12–13}}</ref>

The director circle of a hyperbola has radius <math display="inline">\sqrt{a^2 - b^2}</math>, and so, may not exist in the [Euclidean plane](/source/Euclidean_plane), but could be a circle with imaginary radius in the [complex plane](/source/complex_plane).

The director circle of a circle is a concentric circle having radius <math display="inline">\sqrt{2}</math> times the radius of the original circle.

==Generalization==
More generally, for any collection of points {{mvar|P<sub>i</sub>}}, weights {{mvar|w<sub>i</sub>}}, and constant {{mvar|C}}, one can define a circle as the locus of points {{mvar|X}} such that
<math display="block">\sum_i w_i \, d(X,P_i)^2 = C.</math>

The director circle of an ellipse is a special case of this more general construction with two points {{math|''P''<sub>1</sub>}} and {{math|''P''<sub>2</sub>}} at the foci of the ellipse, weights {{math|1=''w''<sub>1</sub> = ''w''<sub>2</sub> = 1}}, and {{mvar|C}} equal to the square of the major axis of the ellipse. The [Apollonius circle](/source/Circle), the locus of points {{mvar|X}} such that the ratio of distances of {{mvar|X}} to two foci {{math|''P''<sub>1</sub>}} and {{math|''P''<sub>2</sub>}} is a fixed constant {{mvar|r}}, is another special case, with {{math|1=''w''<sub>1</sub> = 1}}, {{math|1=''w''<sub>2</sub> = –''r''<sup> 2</sup>}}, and {{math|1=''C'' = 0}}.

==Related constructions==
In the case of a [parabola](/source/parabola) the director circle degenerates to a straight line, the [directrix](/source/Directrix_(conic_section)) of the parabola.<ref>{{harvnb|Faulkner|1952|page=83}}</ref>

==Notes==
{{reflist}}

==References==
*{{citation
 | last1 = Akopyan | first1 = A. V.
 | last2 = Zaslavsky | first2 = A. A.
 | title = Geometry of Conics
 | publisher = [American Mathematical Society](/source/American_Mathematical_Society)
 | series = Mathematical World | volume = 26
 | year = 2007 | isbn = 978-0-8218-4323-9 }}
*{{citation | first=Luigi | last=Cremona | authorlink=Luigi Cremona | title=Elements of Projective Geometry | location=Oxford | publisher=Clarendon Press | year=1885 | page=369}}
*{{citation|first=T. Ewan|last=Faulkner | year=1952 | title=Projective Geometry | publisher=Oliver and Boyd | place=Edinburgh and London}}
*{{citation
 | last = Hawkesworth | first = Alan S.
 | doi = 10.2307/2968867
 | issue = 1
 | journal = The American Mathematical Monthly
 | mr = 1516260
 | pages = 1–8
 | title = Some new ratios of conic curves
 | volume = 12
 | year = 1905 | jstor = 2968867 }}
*{{citation | first=Sidney Luxton | last=Loney | authorlink=S. L. Loney | title=The Elements of Coordinate Geometry | publisher=Macmillan and Company, Limited | location=London | year=1897 | page=365}}
*{{citation|first=George Albert |last=Wentworth |title=Elements of Analytic Geometry|publisher=Ginn & Company |year=1886|page=150}}

Category:Conic sections
Category:Circles

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