arXiv subjects
Publications and source records attributed to Ronaldo A. Garcia.
We tour several Euclidean properties of Poncelet triangles inscribed in an ellipse and circumscribing the incircle, including loci of triangle centers and envelopes of key objects. We also show that a number of degenerate behaviors are triggered by the presence of an equilateral triangle in the family.
We describe four special families of ellipse-inscribed Poncelet triangles about the incircle which maintain certain triangle centers stationary and which also display interesting conservations.
We show that the focus of the Kiepert in-parabola remains stationary over a family of circle-inscribed Poncelet triangles which contain an equilateral triangle.
We prove that over a Poncelet triangle family interscribed between two nested ellipses $\mathcal{E},\mathcal{E}_c$, (i) the locus of the orthocenter is not only a conic, but it is axis-aligned and homothetic to a $90^o$-rotated copy of $\mathcal{E}$, and (ii) the locus of the isogonal conjugate of a fixed point $P$ is also a conic (the expected degree was four); a parabola (resp. line) if $P$ is on the (degree-four) envelope of the circumcircle (resp. on $\mathcal{E}$). We also show that the envelope of both the circumcircle and radical axis of incircle and circumcircle contain a conic component if and only if $\mathcal{E}_c$ is a circle. The former case is the union of two circles!
In this paper are studied the simplest qualitative properties of asymptotic lines of a plane field in Euclidean space. These lines are the integral curves of the null directions of the normal curvature of the plane field, on the closure of the hyperbolic region, where the Gaussian curvature is negative. When the plane field is completely integrable, these curves coincides with the classical asymptotic lines on surfaces.
In this paper we discuss the behavior of the curvature lines of a transversal eqüiaffine vector field along a surface in $3$-space at isolated umbilical points.
In this work it will be analyzed $η$-principal cycles (compact leaves) of one dimensional singular foliations associated to a plane field $Δ_η$ defined by a unit and normal vector field $η$ in $ \mathbb E^3$. The leaves are orthogonal to the orbits of $η$ and are the integral curves corresponding to directions of extreme normal curvature of the plane field $Δ_η$. % It is shown that, generically, given a $η$-principal cycle it can be make hyperbolic (the derivative of the first return of the Poincaré map has all eigenvalues disjoint from the unit circle) by a small deformation of the vector field $η$. Also is shown that for a dense set of unit vector fields, with the weak $C^r$-topology of Whitney, the $η$-principal cycles are hyperbolic.
We prove that a finite type curve is an $ξ$-asymptotic line (without parabolic points) of a suitable plane field. It is also given an explicit example of a hyperbolic closed finite type $ξ$-asymptotic line. These results obtained here are generalizations, for plane fields, of the results of V. Arnold [4].
The locus of centers of inscribed circles in triangles, the 3-periodic orbits of an elliptic billiard, is also an ellipse. In this work we obtain the canonical equation of this ellipse, complementing the previous results obtained by O. Romaskevich.