We consider the family of lines that are area bisectors of a polygon (possibly with holes) in the plane. We say that two bisectors of a polygon P are combinatorially distinct if they induce di erent partitionings of the vertices of P . We derive an algebraic characterization of area bisectors. We then show that there are simple polygons with n vertices that have (n) combinatorially distinct are...

Introduction. The present investigation is concerned with an axiomatic analysis of the four fundamental theorems of Euclidean geometry which assert that each of the following triplets of lines connected with a triangle is copunctual: the medians, the altitudes, the perpendicular bisectors, and the bisectors of the angles. The general framework for our discussion will be provided by an affine pl...

We characterize the topological configurations of points and lines that may arise when placing n on a circle drawing perpendicular bisectors sides corresponding convex cyclic n-gon. also provide exact asymptotic formulas describing random realizable configuration, obtained either by sampling uniformly at or configuration random.

In this note, we completely describe the shape of the bisector of two given points in a two-dimensional normed vector space. More precisely, we show that, depending on the position of two given points with respect to the shape of the unit circle, the following holds: the bisector of a non-strict pair of points consists of two cones and a curve, which has properties similar to those of bisectors...

We discuss the concept of the bisector of a segment in a Minkowski normed n-space. We prove that all bisectors are topological images of a plane of the embedding Euclidean 3-space iff the shadow boundaries of the unit ball K are topological circles. To a conjectured proving strategy for dimensions n, we introduce the concept of general parameter sphere of the unit ballK, corresponding to a dire...

We present an algorithm for generating Voronoi cells for a set of planar piecewise C1-continuous closed rational curves, which is precise up to machine precision. The algorithm starts with the symbolically generated bisectors for pairs of C1-continuous curve segments (C(t),Ci(r)). The bisectors are represented implicitly in the tr-parameter space. Then, they are properly trimmed after being spl...

By pure analogy with a usual three-dimensional origami, we can fold a regular octahedral material along flat faces in 4-space. The octahedron comprises four congruent tetrahedra. We will show a procedure to fold a tetrahedron along bisectors of the dihedral angles. This procedure demonstrates that each two of four surfaces of the given tetrahedron coincide with each other and that the point of ...