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.