{{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, the '''director circle''' of an ellipse or hyperbola (also called the '''orthoptic circle''' or '''Fermat–Apollonius circle''') is a circle consisting of all points where two perpendicular tangent lines to the ellipse or hyperbola cross each other.
==Properties== The director circle of an ellipse circumscribes the 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 and 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.<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, but could be a circle with imaginary radius in the 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, 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 the director circle degenerates to a straight line, the directrix 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 | 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