The Ballet of Triangle Centers on the Elliptic Billiard
A bevy of new phenomena relating to (i) the shape of 3-periodics and (ii) the kinematics of certain Triangle Centers constrained to the Billiard boundary are explored, specifically the non-monotonic motion some can display with respect to 3- periodics.
INVERSIVE TRIANGLE IN THE ELLIPTIC BILLIARD
Inverting the vertices of elliptic billiard N-periodics with respect to a circle centered on one focus yields a new “focus-inversive” family inscribed in Pascal’s Limaçon. The following are some of…
A Theory for Locus Ellipticity of Poncelet 3-Periodic Centers
A theory which predicts the ellipticity of a triangle center’s locus over a Poncelet 3-periodic family and derives a (long) expression for the loci of the incenter and excenters over a generic Poncelets 3- periodic family, showing they are roots of a quartic.
The talented Mr. Inversive Triangle in the elliptic billiard
Inverting the vertices of elliptic billiard N-periodics with respect to a circle centered on one focus yields a new "focus-inversive" family inscribed in Pascal's Limacon. The following are some of…
New Properties of Triangular Orbits in Elliptic Billiards
- MathematicsAm. Math. Mon.
Some of the proofs omitted from the introduction of new invariants in the one-dimensional family of 3-periodic orbits in the elliptic billiard are presented as well as a few new related facts.
A THEORY FOR LOCUS ELLIPTICITY OF PONCELET TRIANGLE CENTERS
We present a theory which predicts when the locus of a triangle center is an ellipse over a Poncelet family of triangles: this happens if the triangle center is a fixed affine combination of…
Circuminvariants of 3-Periodics in the Elliptic Billiard
A Circumconic passes through a triangle's vertices; an Inconic is tangent to the sides. We introduce the Circumbilliard: the circumellipse of a generic triangle which is an Elliptic Billiard (EB) to…
Poncelet triangles: a theory for locus ellipticity
- MathematicsBeiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry
A theory which predicts if the locus of a triangle center over certain Poncelet triangle families is a conic or not is presented, while predicting its monotonicity with respect to the motion of a vertex of the triangle family.
Invariant Center Power and Elliptic Loci of Poncelet Triangles
- MathematicsJournal of Dynamical and Control Systems
It is shown that for any concentric pair, the power of the center with respect to either circumcircle or Euler’s circle is invariant and if a triangle center of a 3-periodic in a generic nested pair is a fixed linear combination of barycenter and circumcenter, its locus over the family is an ellipse.
Loci and Envelopes of Ellipse-Inscribed Triangles
It is proved that if a triangle center is a fixed linear combination of barycenter and orthocenter, its locus over the family is an ellipse and over the 1d family of linear combinations, loci centers sweep a line.
SHOWING 1-10 OF 57 REFERENCES
Loci of Triangular Orbits in an Elliptic Billiard: Elliptic? Algebraic?
- Mathematics, PhysicsArXiv
A systematic method is presented to prove 29 out of the first 100 Centers listed in Clark Kimberling's Encyclopedia are elliptic and derive conditions under which loci are algebraic.
Elliptic Billiards and Ellipses Associated to the 3-Periodic Orbits
- Mathematics, PhysicsAm. Math. Mon.
The canonical equations of the centers of circumscribed and inscribed circles to the triangles that are the 3-periodic orbits of an elliptic billiard are ellipses and the geometric locus defined by the barycenters of the edges of billiard triangles is an ellipse.
Can the Elliptic Billiard Still Surprise Us?
It is shown that Monge's Orthoptic Circle's close relation to 4-periodic Billiard trajectories is well-known and its geometry provided clues with which to generalize 3- periodic invariants to trajectories of an arbitrary number of edges.
On the incenters of triangular orbits in elliptic billiard
We consider 3-periodic orbits in an elliptic billiard. Numerical experiments conducted by Dan Reznik have shown that the locus of the centers of inscribed circles of the corresponding triangles is an…
Poncelet Porisms and Beyond: Integrable Billiards, Hyperelliptic Jacobians and Pencils of Quadrics
Introduction to Poncelet Porisms.- Billiards - First Examples.- Hyper-Elliptic Curves and Their Jacobians.- Projective geometry.- Poncelet Theorem and Cayley's Condition.- Poncelet-Darboux Curves and…
On the integrability of Birkhoff billiards
- MathematicsPhilosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
In this survey, we provide a concise introduction to convex billiards and describe some recent results, obtained by the authors and collaborators, on the classification of integrable billiards,…
Triangle Centers as Functions
We consider a kind of problem that appears to be new to Euclidean geometry, since it depends on an understanding of a point as a function rather than a position in a two-dimensional plane. Certain…
Dan Reznik’s identities and more
- MathematicsEuropean Journal of Mathematics
Dan Reznik found, by computer experimentation, a number of conserved quantities associated with periodic billiard trajectories in ellipses. We prove some of his observations using a non-standard…
Geometry and billiards
- Mathematics, Physics
Motivation: Mechanics and optics Billiard in the circle and the square Billiard ball map and integral geometry Billiards inside conics and quadrics Existence and non-existence of caustics Periodic…
Direct Least Square Fitting of Ellipses
- Computer ScienceIEEE Trans. Pattern Anal. Mach. Intell.
This work presents a new efficient method for fitting ellipses to scattered data by minimizing the algebraic distance subject to the constraint 4ac-b/sup 2/=1, which incorporates the ellipticity constraint into the normalization factor.