Frégier Families of Conics

1. Frégier Families of Conics Michael Woltermann Washington and Jefferson College Washington, PA 15301 JMM Meeting San Diego, CA, Jan., 2013

2. 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’.

3. Frégier’sTheorem • Modern proofs involve things like • Involutivehomographies • 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.

4. A lemma An equation for a pair of perpendicular lines through the origin is . Let the lines be and where □

5. 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 y-axis, an equation of c is (I): • An equation for perpendiculars PQ and PR is (II): . • (I)-(II) is:

6. Johnston’s Proof • Or • 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). □

7. 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.)

8. For example

9. 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

10. 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? ().

11. 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).

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

13. 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’’.

15. 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.

16. FrégierFamilies of Conics • If c is • , then F(c) is • then

17. 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.