Ikrar Pramudya, Bagus Aqil Saputra, Ira Kurniawati

ANALYSIS OF THE COMPOSITION OF TWO PLANE ISOMETRIES: TRANSLATION, REFLECTION, AND ROTATION USING A LINEAR ALGEBRA APPROACH

Introduction

Analysis of the composition of two plane isometries: translation, reflection, and rotation using a linear algebra approach. Explore the composition of two plane isometries (translation, reflection, rotation) using a linear algebra approach. Discover theorems describing their properties, offering an alternative to axiomatic geometry.

Abstract

Geometric transformation is a branch of mathematics that focuses on the transformation (bijective function) of geometric objects in a plane or transformation in . One of the topics discussed in geometric transformation is the properties of the composition of two isometries. Generally, topics in geometric transformation are studied using an axiomatic geometry approach that requires strong geometric visualization skills.  This research aims to study the composition of two isometries through an alternative approach, specifically linear algebra.  The study focuses on deriving the properties of the composition of two isometries, namely translations, reflections, and rotations, while considering their algebraic forms.  The research methodology employed is literature study and deductive reasoning in accordance with mathematical syllogism. The results of this research are theorems that state the properties of the composition of two isometries.  Based on the findings of this study, it can be concluded that the properties of the composition of two isometries can be derived using a linear algebra approach.

Review

The paper titled "ANALYSIS OF THE COMPOSITION OF TWO PLANE ISOMETRIES: TRANSLATION, REFLECTION, AND ROTATION USING A LINEER ALGEBRA APPROACH" presents a compelling objective: to re-examine the properties of composing fundamental plane isometries through an alternative methodological framework. While the study of geometric transformations, particularly the composition of translations, reflections, and rotations, is a cornerstone of mathematics, it is traditionally approached via axiomatic geometry, which often necessitates strong geometric visualization. This research proposes to derive these properties using linear algebra, aiming to offer an analytical and potentially more accessible pathway to understanding these foundational concepts. The core strength of this research lies in its departure from conventional approaches by utilizing linear algebra. The authors intend to investigate the algebraic forms of these isometries and their compositions, employing a methodology rooted in literature study and deductive reasoning, which aligns well with the rigorous demands of mathematical research. This algebraic approach could serve to demystify complex geometric interactions by translating them into a system of equations and matrix operations. Such a focus promises to not only provide an alternative means of proof but also potentially enhance pedagogical methods for teaching geometric transformations. The stated outcome of the research is a set of theorems describing the properties of composed isometries, all derived within the linear algebra framework. The conclusion that these properties are indeed derivable through this alternative approach confirms the viability and potential advantages of the chosen methodology. This work makes a valuable contribution by demonstrating the power of linear algebra to formalize and prove geometric principles, potentially offering a new lens for researchers and students alike. It would be particularly impactful if the paper articulates any novel insights or simplifications that emerge specifically from this algebraic perspective, rather than merely reproducing known results through a different method.

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