Frégier Families of Conics
Michael Woltermann
Washington and Jefferson College
Washington, 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 Theorem

Modern 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
𝑥2 + 𝜆𝑥𝑦 − 𝑦2 = 0.
−
1
𝑚
Let the lines be 𝑦 = 𝑚𝑥 and 𝑦 =
𝑥.
1
0 = 𝑚𝑥 − 𝑦
𝑥+𝑦
𝑚
1
2
𝑦 where 𝜆 = 𝑚 − . □
𝑚
= 𝑥 2 + 𝜆𝑥𝑦 −
Johnston’s Proof
Let 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 yaxis, an equation of c is (I): 𝑎𝑥 2 + 2ℎ𝑥𝑦 +
𝑏𝑦 2 = 2𝑦.
 An equation for perpendiculars PQ and
PR is (II): 𝑥2 + 𝜆𝑥𝑦 − 𝑦2 = 0.
 (I)- 𝑎(II) is:
2ℎ − 𝑎𝜆 𝑥𝑦 + 𝑏 + 𝑎 𝑦 2 − 2𝑦 = 0.

Johnston’s Proof
Or 𝑦 2ℎ − 𝑎𝜆 𝑥 + 𝑏 + 𝑎 𝑦 − 2 = 0.
 𝑦 = 0 is the tangent line at P.
 2ℎ − 𝑎𝜆 𝑥 + 𝑏 + 𝑎 𝑦 − 2 = 0 is the
equation of QR.
 Its y-intercept (on the normal line at P) is
found by setting 𝑥 = 0, giving a y2
intercept of
independent of 𝜆 (and 𝑚,
𝑏+𝑎
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

•
𝑥2
𝑎2
•
𝑥2
𝑎2
•𝑦
+
𝑦2
𝑏2
= 1, then F(c) is
𝑦2
− 2 = 1,
𝑏
1 2
= 𝑥 ,
4𝑝
then F(c) is
𝑥2
𝑎2
+
𝑥2
𝑎2
then F(c) is 𝑦
𝑦2
𝑏2
=
𝑎2 −𝑏 2
𝑎2 +𝑏 2
𝑦2
𝑎2 +𝑏 2
− 2= 2 2
𝑏
𝑎 −𝑏
1 2
= 𝑥 + 4𝑝.
4𝑝
2
,
2
,
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? (𝑥 2 − 𝑦 2 = ±𝑎2 ).
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 F
Fn(c)= F(Fn-1(c)) for n≥1.
 If c is

•
𝑥2
𝑎2
•
𝑥2
𝑎2
•𝑦

𝑦2
+ 2
𝑏
= 1, then F(c) is
𝑦2
− 2 = 1,
𝑏
1 2
= 𝑥 ,
4𝑝
then F(c) is
𝑥2
𝑎2
𝑥2
𝑎2
then F(c) is 𝑦
What is F-1(c)?
+
𝑦2
𝑏2
=
𝑎2 −𝑏 2
𝑎2 +𝑏 2
𝑦2
𝑎2 +𝑏 2
− 2= 2 2
𝑏
𝑎 −𝑏
1 2
= 𝑥 + 4𝑛𝑝.
4𝑝
2𝑛
,
2𝑛
,
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 Conics

If c is
•
𝑥2
𝑎2
•
𝑥2
𝑎2
•𝑦
•
𝑦2
+ 2
𝑏
= 1, then F(c) is
𝑦2
− 2 = 1,
𝑏
1 2
= 𝑥 ,
4𝑝
then F(c) is
𝑥2
𝑎2
𝑥2
𝑎2
then F(c) is 𝑦
𝑛 = 0, ±1, ±2, ±3, …
+
𝑦2
𝑏2
=
𝑎2 −𝑏 2
𝑎2 +𝑏 2
𝑦2
𝑎2 +𝑏2
− 2= 2 2
𝑏
𝑎 −𝑏
1 2
= 𝑥 + 4𝑛𝑝.
4𝑝
2𝑛
,
2𝑛
,
References
Akopyan, 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.
