Metrical Coordinates in Non-Euclidean Geometry

Spatial Analysis in curved spaces with Non-Euclidean Geometry

The ultimate goal of spatial information, both as part of technology and as science, is to answer questions and issues related to space, place, and location. Therefore, geometry is widely used for description, storage, and analysis. Undoubtedly, one of the most essential features of spatial information is geometric features, and one of the most obvious types of analysis is the geometric type an...

Euclidean Geometry before non-Euclidean Geometry

In [3], in my argument for the primacy of Euclidean geometry on the basis of rigid motions and the existence of similar but non-congruent triangles, I wrote the following: A: “The mobility of rigid objects is now recognized as one of the things every normal human child learns in infancy, and this learning appears to be part of our biological progaramming.” B. “. . . we are all used to thinking ...

Metrical relations in barycentric coordinates

Let ∆ be the area of the fundamental triangle ABC of barycentric coordinates and let α = cotA, β = cotB, γ = cotC. The vectors vi = [xi, yi, zi] (i = 1, 2) have the scalar product 2∆(αx1x2 + βy1y2 + γz1z2). This fact implies all important formulas about metrical relations of points and lines. The main and probably new results are Theorems 1 and 8.

Non - Euclidean Geometry ,

The uses of homogeneous barycentric coordinates in plane euclidean geometry

The notion of homogeneous barycentric coordinates provides a powerful tool of analysing problems in plane geometry. In this paper, we explain the advantages over the traditional use of trilinear coordinates, and illustrate its powerfulness in leading to discoveries of new and interesting collinearity relations of points associated with a triangle.