Related papers: The equal tangents problem
The displacement and deviation vectors in spaces (manifolds), the tangent bundle of which is endowed with a transport along paths, are introduced. In case these spaces are equipped with a linear connection, the deviation equations (between…
A Tangle is a smooth simple closed curve formed from arcs (or ``links'') of circles with fixed radius. Most previous study of Tangles has dealt with the case where these arcs are quarter-circles, but Tangles comprised of thirds and sixths…
We develop a systematic method for computing the angle combinations at all vertices in an edge-to-edge tiling of the sphere by pentagons with the same five angles. The method is a useful and necessary step in many tiling problems about…
If A and B are two points in the plane, with B lower and to the right of A, then we may consider the trajectory of an object travelling from A to B under the influence of gravity. The search for the trajectory minimising the time taken by…
Let a polygon be composed of equal rectangles. We find all quadratic irrationals r for which the polygon can be tiled by similar rectangles with given side ratio r.
If a line cuts randomly two sides of a triangle, the length of the segment determined by the points of intersection is also random. The object of this study, applied to a particular case, is to calculate the probability that the length of…
For given finite system of convex polygons in the plane which have no transversal, find such homothety transformations of polygons (having fixed centres inside given polygons) with minimal similarity ratio c>1 that the transformed system…
We investigate the lines tangent to four triangles in R^3. By a construction, there can be as many as 62 tangents. We show that there are at most 162 connected components of tangents, and at most 156 if the triangles are disjoint. In…
Given a regular $n$-gon on the plane, it is evident that from any point on the plane, taken as a center, one can draw $n$ concentric circles such that each circle passes through one of the vertices of the polygon. Naturally, this raises the…
Non-relativistic particles that are effectively confined to two dimensions can in general move on curved surfaces, allowing dynamical phenomena beyond what can be described with scalar potentials or even vector gauge fields. Here we…
We consider the problem of testing, for a given set of planar regions $\cal R$ and an integer $k$, whether there exists a convex shape whose boundary intersects at least $k$ regions of $\cal R$. We provide a polynomial time algorithm for…
Two flat sub-Lorentzian problems on the Martinet distribution are studied. For the first one, the attainable set has a nontrivial intersection with the Martinet plane, but for the second one it does not. Attainable sets, optimal…
We explore convex shapes $S$ in the Euclidean plane which have the following property: there is a circle $C$ such that the angle between the two tangents from any point of $C$ to $S$ is constant equal to $\alpha$. A dynamical formulation…
Every body knows that identical regular triangles or squares can tile the whole plane. Many people know that identical regular hexagons can tile the plane properly as well. In fact, even the bees know and use this fact! Is there any other…
In this work we present some classes of models whose the corresponding two coupled first-order nonlinear equations can be put into a linear form, and consequently be solved completely. In these cases the so-called trial orbit method is…
In this note we investigate the behavior of the polar tangential angle of a general plane curve, and in particular prove its monotonicity for certain curves of monotone curvature. As an application we give (non)existence results for an…
We study the problem of finding a triangulation T of a planar point set S such as to minimize the expected distance between two points x and y chosen uniformly at random from S. By distance we mean the length of the shortest path between x…
John Conway's Circle Theorem is a gem of plane geometry. The six points formed by continuing the sides of a triangle beyond every vertex by the length of its opposite side, are concyclic. The theorem has attracted several proofs. We present…
Congruent polygons are congruent in angles as well as in edge lengths. We concentrate on the angle aspect, and investigate how tilings of the sphere by congruent pentagons can be determined by the angle information only. We also investigate…
The orbital boundary value problem, also known as Lambert Problem, is revisited. Building upon Lancaster and Blanchard approach, new relations are revealed and a new variable representing all problem classes, under L-similarity, is used to…