Related papers: The equal tangents problem
We study plane curves over finite fields whose tangent lines at smooth $\mathbb{F}_q$-points together cover all the points of $\mathbb{P}^2(\mathbb{F}_q)$.
A fundamental problem in computational geometry is to compute an obstacle-avoiding Euclidean shortest path between two points in the plane. The case of this problem on polygonal obstacles is well studied. In this paper, we consider the…
Interpreting tangency as a limit of two transverse intersections, we obtain a concrete formula to enumerate smooth degree $d$ plane curves tangent to a given line at multiple points with arbitrary order of tangency. Extending that idea, we…
A notion of random walks for circle packings is introduced. The geometry behind this notion is discussed, together with some applications. In particular, we obtain a short proof of a result regarding the type problem for circle packings,…
The traditional study of plane and space algebraic curves by looking at their tangent vectors, curvatures and torsions provides geometric, but unfortunately not sufficient information about individual curves in order to be able to…
We investigate the 2-center problem for arbitrary strictly convex, centrally symmetric curves instead of usual circles. In other words, we extend the 2-center problem (from the Euclidean plane) to strictly convex normed planes, since any…
The point of this short note concerns with two facts on the scheme of secant loci. The first one is an attempt to describe the tangent cone of these schemes globally and the second one is a comparision on the dimension of the tangent spaces…
Let $ l =[l_0,l_1]$ be the directed line segment from $l_0\in {\mathbb R}^n$ to $l_1\in{\mathbb R}^n.$ Suppose $\bar l=[\bar l_0,\bar l_1]$ is a second segment of equal length such that $l, \bar l$ satisfy the "two sticks condition": $\|…
We study the variety of common tangents for up to four quadric surfaces in projective three-space, with particular regard to configurations of four quadrics admitting a continuum of common tangents. We formulate geometrical conditions in…
We study the sets of planes in an even dimensional real vector space $V$ which are simultaneously stabilised by a pair of complex structures on $V$. We completely describe these sets of planes for pairs of orthogonal complex structures.…
It is well known that the area $U$ of the triangle formed by three tangents to a parabola $X$ is half of the area $T$ of the triangle formed by joining their points of contact. In this article, we consider whether this property and similar…
We establish sufficient conditions for existence of curves minimizing length as measured with respect to a degenerate metric on the plane while enclosing a specified amount of Euclidean area. Non-existence of minimizers can occur and…
We compute explicitly the limits of tangents of a quasi-ordinary singularity in terms of its special monomials. We show that the set of limits of tangents of Y is essentially a topological invariant of Y .
We study closed smooth convex plane curves $\Gamma$ enjoying the following property: a pair of points $x,y$ can traverse $\Gamma$ so that the distances between $x$ and $y$ along the curve and in the ambient plane do not change; such curves…
The Gaussian Moat Problem asks whether it is possible to walk from the origin to infinity in the complex plane using only Gaussian primes as stepstones and steps of bounded length. We prove that this is not possible.
We construct and classify, in the case of two complex dimensions, the possible tangent cones at points of limit spaces of non-collapsed sequences of K\"ahler-Einstein metrics with cone singularities.
We compute the limit of tangents of an arbitrary surface. We obtain as a byproduct an embedded version of Jung's desingularization theorem for surface singularities with finite limits of tangents.
In this paper, we consider the rectilinear one-center problem on uncertain points in the plane. In this problem, we are given a set $P$ of $n$ (weighted) uncertain points in the plane and each uncertain point has $m$ possible locations each…
A differentiable curve y = y(x) is determined by its tangent lines and is said to be the envelope of its tangent lines. The coefficients of the curve's tangent lines form a curve in another space, called the dual space. There is a…
The time evolution of plane waves in the presence of a 1-dimensional square quantum barrier is considered. Comparison is made between the cases of an infinite and a cut-off (shutter) initial plane wave. The difference is relevant when the…