Parallel bundles in planar map geometries

TL;DR

This paper investigates the behavior of parallel bundles in planar map geometries, a type of Smarandache geometry, revealing their unique characteristics influenced by elliptic and hyperbolic points.

Contribution

It introduces the concept of parallel bundles in planar map geometries and characterizes their properties, extending classical parallel line concepts to complex geometries.

Findings

Parallel bundles exhibit distinct behaviors in elliptic and hyperbolic regions.

Characteristics of parallel bundles are derived for Smarandache geometries.

The study broadens understanding of parallel concepts beyond Euclidean geometry.

Abstract

Parallel lines are very important objects in Euclid plane geometry and its behaviors can be gotten by one's intuition. But in a planar map geometry, a kind of the Smarandache geometries, the sutation is complex since it may contains elliptic or hyperbolic points. This paper concentrates on the behavior of parallel bundles, a generazation of parallel lines in plane geometry and obtains characteristics for for parallel bundles.