Frégier Families of Conics Michael Woltermann Washington and Jefferson College Washington, PA 15301...
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Transcript of Frégier Families of Conics Michael Woltermann Washington and Jefferson College Washington, PA 15301...
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.)
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).
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.
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.