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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).
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.
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.