If we ask how to tell if two triangles are different or equal, Elementary school students will use the superposition of the figures to insist that they are the same if they match, Middle school students will use conditions for congruent triangles learned in class. But since this is not a proof, we were necessary to device more rigorous way, accordingly we introduced the isometry. So our study has a purpose to classify these isometries and starting with the familiar Euclidean Geometry, we will confirm the structure of isometries of unfamiliar Euclidean Geometry. Through this study, students acquire mathematical rigor and by learning the structure of the new Geometry, they will erase vagueness about Geometry. Finally, our purpose of this study was to further enhance our understanding of mathematics and Geometry.
Since Euclidean Geometry was covered a lot in college calculus and geometry classes after going through elementary, middle and high school, students are familiar to Euclidean Geometry. So in the case of Euclidean Geometry, our purpose is arranging the important parts strictly and exploring structure of isometry of Euclidean Geometry. As a result, we will show that in Euclidean Geometry, isometries have translation, reflection, rotation, glide reflection and every isometry of Euclidean Geometry is expressed by their composition.
We chose Spherical Geometry for first step of Non-Euclidean Geometry. Since we already know that sphere is a subset of Euclidean 3-dimensional space, students can approach it more familiarly than other non-Euclidean Geometry and it is natural in the connection with the previous Euclidean Geometry. By observing structure of Spherical Geometry in detail, we make sure students are aware. And based on this, we understand structure of isometries of Spherical Geometry. As a result, we will show that in Spherical Geometry, isometries have reflection, rotation, glide reflection and isometry of Spherical Geometry consist of orthogonal transformations.
When many books and papers explain Hyperbolic Geometry, they use Klein model. However our paper will use Poincaré model that is relatively recent model. Since Poincaré model’s structure is expressed by Euclidean Geometry, students can observe more easily. For this reason, we prepare for Poincaré model by introducing linear fractional transformation first. In this process, we will show that Hyperbolic Geometry is different to Euclidean Geometry. Especially, distance will have a very fresh visual effect to student. Based on the above, we will show that in Hyperbolic Geometry, isometries have translation, reflection, rotation, parallel displacement. In this process, they are familiar and very different from isometries of Euclidean Geometry. Finally, any isometry of Hyperbolic Geometry is expressed by composition of linear fractional transformation and reflection.
This study helps students develop mathematical rigor and at this same time by understanding new geometry easily more than existing book or papers, student will be able to erase vagueness about geometry. Finally, we hope student to increase their interest in mathematics and geometry.