A study on densities of sets of points and lines

Abstract

In this paper, firstly, our purpose is to give the relationship among the densities of the sets of collinear points, the relationship among the densities of the sets of noncollinear points, and the relationship among the densities of the sets of the intersecting lines in Euclidean plane, respectively. In addition to that, we define the density formulas for the sets of points and lines under the two‐parameter planar Euclidean motion. By means of these results, we obtain essential properties that explain the connection among the densities of the sets of points and among the densities of the sets of intersecting lines under the two‐parameter planar Euclidean motion.

Ruled Surfaces and Tangent Bundle of Pseudo-Sphere of Natural Lift Curves

“…are the derivatives of () t  and () vt , respectively, see [5]. Here D refers the Levi-Civita connection in 3…”

“…In 3 , a ruled surface is a curved surface represented by moving a straight line along a space curve called the base curve [4]. In literature, ruled surfaces have been widely studied by different authors [5][6][7][8][9][10].…”

A Study on the Harmonic Evolute Surfaces of Quasi Binormal Surfaces

“…The ruled surface generated by the motion of a straight line along a curve is one of the most atractive surfaces to study since the application of surfaces to physics and engineering is countless and these surfaces are the easiest of all surfaces to parametrize. After it was initially discovered by Gaspard Mongea, many researchers have studied ruled surfaces [1][2][3][4]. Besides ruled surfaces, harmonic evolute surfaces of various surfaces have been also studied [5][6][7][8].…”