Singular Frégier Conicsin Non-Euclidean Geometry

Hans-Peter Schröcker

Unit Geometry and CADUniversity of Innsbruck

17th International Conference on Geometry and GraphicsBeijing, August 4–8, 2016

Geometrie

Overview

Frégier Conics in Euclidean GeometryFrégier Points and ConicsBasic PropertiesSingular Frégier Loci

Frégier Conics in Hyperbolic GeometryConstruction, Basic PropertiesSingular Frégier Loci for:

– General Conics– Parabolas– Circles

Frégier Conics in Elliptic Geometry

Frégier’s Theorem

TheoremAll hypotenuses of right triangles with fixed right angle vertex p andinscribed into a conic C are incident with a point f .

Frégier Locus

PropositionThe locus of Frégier points f for varying point p ∈ C is a conicsection F .

I If C is an ellipse or a hyperbola, it is homothetic to F .I If C is a parabola, it is a translate of F .

Singular Frégier Loci

p

f = F

C

f ppf

C

PropositionThe locus of Frégier points f for varying points p ∈ C is singular ifeither

I C is a circle (scale factor zero, circle center) orI C is an equilateral hyperbola (scale factor ∞, infinite projective

line segment).

The Frégier Point in Hyperbolic Geometry

CN

pp

f

Construction/Computation of Frégier Points

p

I

I

ii

ii

ff

C

FC

N

pf F

i

ii

I

I

PropositionThe Frégier point f is the pole of the line F that connects theintersection points of C with the isotropic tangents through p.

Frégier Conics in Hyperbolic Geometry

TheoremIn general, the Frégier locus in elliptic/hyperbolic geometryis a conic section.

Question: For which conics is the Frégier locus singular?

N

C

general

N

C

parabola

N

C

circle

N C

osculatingparabola

NC

horocycle

Frégier Conics in Hyperbolic Geometry

TheoremIn general, the Frégier locus in elliptic/hyperbolic geometryis a conic section.

Question: For which conics is the Frégier locus singular?

N

C

general

N

C

parabola

N

C

circle

N C

osculatingparabola

NC

horocycle

General Conics

p1p1

p2p2

p3p3

f1f1

f2f2

f3f3

f3f3

NN

C1C1

C2C2

C3C3

F1F1

F2F2

Threeone-parametricfamilies ofincongruentgeneral conicswith singularFrégier locus.

Parabolas

One one-parametric familyof incongruent parabolaswith singular Frégier locus.

Circles

Thales’ Theorem inhyperbolic geometry holdstrue for

I infinite line segmentsand

I circles (equidistantcurves) of radius12 ln(3 + 21/2).

Frégier Conics in Elliptic Geometry

I Frégier’s Theorem holds true, Frégier locus is a conic(in general).

I Two real families of conics with singular Frégier locus.I No real circles with singular Frégier locus.

Summary and Conclusions

Elliptic and Hyperbolic GeometryI In general, the Frégier locus is a conic section.I Singular Frégier loci are always line segments.

Hyperbolic GeometryI Conics with singular Frégier locus:

I three families of incongruent general conicsI one family of incongruent parabolasI two circles

Elliptic GeometryI Two real families of incongruent conics with singular

Frégier locus.I No singular Frégier locus for circles.