Director circle

{{Short description|Circle formed by all 90° crossings of tangents of an ellipse or hyperbola}}

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 $\sqrt{a^2 + b^2}$ , where $a$ and $b$ 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.{{harvnb|Akopyan|Zaslavsky|2007|pages=12–13}}

The director circle of a hyperbola has radius $\sqrt{a^2 - b^2}$ , 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 $\sqrt{2}$ times the radius of the original circle.

Generalization

More generally, for any collection of points {{mvar|P_i}}, weights {{mvar|w_i}}, and constant {{mvar|C}}, one can define a circle as the locus of points {{mvar|X}} such that

$\sum_i w_i \, d(X,P_i)^2 = C.$

The director circle of an ellipse is a special case of this more general construction with two points {{math|P₁}} and {{math|P₂}} at the foci of the ellipse, weights {{math|1=w₁ = w₂ = 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|P₁}} and {{math|P₂}} is a fixed constant {{mvar|r}}, is another special case, with {{math|1=w₁ = 1}}, {{math|1=w₂ = –r²}}, 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.{{harvnb|Faulkner|1952|page=83}}

Notes

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