Frégier Families of Conics

Michael WoltermannWashington and Jefferson CollegeWashington, PA 15301

JMM Meeting San Diego, CA, Jan., 2013

Frégier’s Theorem

If from a point P on a conic any two perpendicular lines are drawn cutting the conic in points Q and R, then line QR meets the normal at P at a fixed point P’.

Frégier’s TheoremModern proofs involve things like

◦Involutive homographies◦Good paramatrizations◦Polar correspondence

An analytic proof (for an ellipse) by John Casey (1893) finds equation of lines in terms of eccentric angles.

An analytic proof for any conic section by W.J.Johnston (1893) is fairly straightforward.

A lemma

An equation for a pair of perpendicular lines through the origin is .

Let the lines be and where □

Johnston’s ProofLet P be a point on conic c.With P as origin and the tangent

line as the x-axis and the normal line as the y-axis, an equation of c is (I):

An equation for perpendiculars PQ and PR is (II): .

(I)- (II) is:

Johnston’s ProofOr is the tangent line at P. is the equation of QR.Its y-intercept (on the normal line

at P) is found by setting giving a y-intercept of independent of (and , the slope of PQ). □

How to Find P’Let P0 be the point of intersection

(other than P) of the conic c with the line through P parallel to the directrix.

P’ is the intersection of the normal line at P with the line through P0 and the center of c. (The center of a parabola is the ideal point on its axis.)

For example

What is the locus of P’?As P moves on a conic c, P’

moves along a conic F(c).If c is • , then F(c) is

• then

The locus of P’

In other words, F(c) is dilated (or translated) image of c.

But not pointwise.

Some Exceptions

What happens if c is a circle?What happens if c is a rectangular

(equilateral) hyperbola? ().

Some Properties of F(c)c and F(c) have the same

eccentricity.c and F(c) have the same center.If c is a parabola, the lengths of

the latus rectum of both c and F(c) are the same.

If c is a hyperbola c and F(c) have the same asymptotes.

If c and d are conjugate hyperbolas, so are F(c) and F(d).

Iterating FFn(c)= F(Fn-1(c)) for n≥1.If c is • , then F(c) is

• then What is F-1(c)?

Finding P from P’Let c’ be a conic, P’ be on c’, O

the center of c’.Reflect P’ about the major axis of

c’ to point P’’.Construct normal to c’ at P’Reflect the normal about the line

through P’ parallel to the directrix to line m.

P is the intersection of m and line OP’’.

Why?An analytic proof is easy.Show that if P’ is the Frégier point

of P relative to a conic c, then the construction above takes P’ back to P.

Consider central conics and parabolas separately.

Frégier Families of ConicsIf c is • , then F(c) is

• then

ReferencesAkopyan, A.V. and Zaslavsky, A.A.; Geometry

of Conics; AMS, 2007.Casey, John; A Treatise on the analytical

geometry of the point, line, circle, and conic sections; Dublin U. Press, 1893.

Frégier involution by orthogonals from a conic-point; http://www.math.uoc.gr/

Johnston, W.J.; An Elementary Treatise on Analytical Geometry; Clarendon Press, 1893

Wells, D.; The Penguin Dictionary of Curious and Interesting Geometry; Penguin, 1991.