Related papers: Area of triangles associated with a curve II
We consider quadrangles of perimeter $2$ in the plane with marked directed edge. To such quadrangle $Q$ a two-dimensional plane $\Pi\in\mathbb{R}^4$ with orthonormal base is corresponded. Orthogonal plane $\Pi^\bot$ defines a plane…
Tightness of a triangulated manifold is a topological condition, roughly meaning that any simplexwise linear embedding of the triangulation into euclidean space is "as convex as possible". It can thus be understood as a generalization of…
We consider a system of three surfaces, graphs over a bounded domain in ${\mathbb R}^2$, intersecting along a time-dependent curve and moving by mean curvature while preserving the pairwise angles at the curve of intersection (equal to…
The space of shapes of a polyhedron with given total angles less than 2\pi at each of its n vertices has a Kaehler metric, locally isometric to complex hyperbolic space CH^{n-3}. The metric is not complete: collisions between vertices take…
Given a point set, mostly a grid in our case, we seek upper and lower bounds on the number of curves that are needed to cover the point set. We say a curve covers a point if the curve passes through the point. We consider such coverings by…
The author has been interested in regions surrounded by real algebraic curves of degree $1$ or $2$ in the plane. The author is mainly interested in their shapes and combinatorics. This is a fundamental and natural problem in mathematics…
We consider the motion by mean curvature of an $n$-dimensional graph over a time-dependent domain in $\mathbb{R}^n$, intersecting $\mathbb{R}^n$ at a constant angle. In the general case, we prove local existence for the corresponding…
We say that a simple, closed curve $\gamma$ in the plane has bounded convex curvature if for every point $x$ on $\gamma$, there is an open unit disk $U_x$ and $\varepsilon_x>0$ such that $x\in\partial U_x$ and $B_{\varepsilon_x}(x)\cap…
We will use toric degenerations of the projective plane ${{\mathbb{P}}^ 2}$ to give a new proof of the triple points interpolation problems in the projective plane. We also give a complete list of toric surfaces that are useful as…
The primary objects of study in the ``knot theory of complex plane curves'' are C-links: links (or knots) cut out of a 3-sphere in the complex plane by complex plane transverse and totally tangential. Transverse C-links are naturally…
We show that every smooth closed curve C immersed in Euclidean 3-space satisfies the sharp inequality 2(P+I)+V >5 which relates the numbers P of pairs of parallel tangent lines, I of inflections (or points of vanishing curvature), and V of…
We study the relationship between the areas of the consecutive quadrilaterals cut from a convex quadrilateral in the plane by means of a finite or infinite number of straight lines intersecting two of its opposite sides. Moreover, we obtain…
For a semisimple Lie group $G$ with parabolic subgroups $Q\subset P\subset G$, we associate to a parabolic geometry of type $(G,P)$ on a smooth manifold $N$ the correspondence space $\Cal CN$, which is the total space of a fiber bundle over…
An approach to defining quadratic implicit curves is to prescribe two tangent lines and a secant line going through the points of tangency. This paper will show that this method can be generalized to a higher number of tangents, resulting…
Every three-connected planar graph with n vertices has a drawing on an O(n^2) x O(n^2) grid in which all faces are strictly convex polygons. These drawings are obtained by perturbing (not strictly) convex drawings on O(n) x O(n) grids. More…
A topologically minimal surface may be isotoped into a normal form with respect to a fixed triangulation. If the intersection with each tetrahedron is simply connected, then the pieces of this normal form are triangles, quadrilaterals, and…
We present an algorithm for the following problem. Given a triangulated 3-manifold M and a (possibly non-simple) closed curve on the boundary of M, decide whether this curve is contractible in M. Our algorithm runs in space polynomial in…
We consider arrangements of axis-aligned rectangles in the plane. A geometric arrangement specifies the coordinates of all rectangles, while a combinatorial arrangement specifies only the respective intersection type in which each pair of…
A lattice equable quadrilateral is a quadrilateral in the plane whose vertices lie on the integer lattice and which is equable in the sense that its area equals its perimeter. This paper treats the tangential and extangential cases. We show…
We present a criterion when six points chosen on the sides of a triangle belong to the same conic. Using this tool we show how the two geometrical gems - celebrated Poncelet's theorem of projective geometry and incredible Morley's theorem…