On The Investigating Cycle Properties In The Galilean Plane G^2

Abstract

The introduction of the Galilean plane within the affine plane parallels the familiar concepts of the Euclidean plane, extending the realm of geometric exploration. The fundamental concepts of lines, triangles, squares, and circles are important in both planes, allowing for a smooth transition between these mathematical environments. The noteworthy aspect is the discovery that cycles in the Galilean plane have properties similar to circles in the Euclidean plane. This paper contributes to the mathematical literature by carefully deriving and establishing features of cycles in the Galilean plane, exhibiting their startling resemblance to Euclidean circles. The use of the inscribed angle as an alternative definition of the circle is particularly insightful, providing a faster and more intuitive explanation of some findings than the usual definition. Such comparative assessments not only broaden our understanding of various geometries but also give us chances to streamline the learning process. The paper argues for the inclusion of Galilean geometry in the high school curriculum by highlighting these parallels. It implies that exposing students to various geometrical systems not only broadens their mathematical perspectives but also fosters a larger and more inclusive vision of the subject, potentially inspiring increased interest and acknowledgment of Galilean geometry among students.

Keywords

• cycle
• e²
• g²
• non-euclidean geometry
• special line