Reflections with Respect to Line and Hyperplane from Quaternionic Point of View

Yıl 2019, Cilt: 9 Sayı: 3, 1612 - 1621, 01.09.2019

Melek Erdoğdu

https://doi.org/10.21597/jist.461015

Öz

In this study, the reflections in and are investigated by unit quaternions. Firstly, a linear transformation is defined to describe reflections in with respect to the plane passing through the origin and orthogonal to the quaternion. Then some examples are given to discuss obtained results. Similarly, two linear transformations are stated which correspond to the reflection in with respect to the hyperplane passing through the origin and a reflection with respect to the line in the direction of the quaternion. Finally, the matrix representaions of these reflections are found and the eigenvalues, eigenvectors of them are given to analyse the geometric meaning in terms of the components of the quaternion for each case.

Anahtar Kelimeler

Quaternions, Reflections, Eigenvalues, Eigenvectors, Rigid Motions

Kuaterniyon Bakış Açısı ile Doğru ve Hiperdüzlem Boyunca Yansımalar

Öz

Bu çalışmada,  ve  uzayında
yansımalar birim kuaterniyonlar ile incelenmiştir. İlk olarak,  uzayında orjinden geçen ve kuaterniyona dik
doğrultudaki doğru boyunca yansımayı belirten bir lineer dönüşüm
tanımlanmıştır. Ardından, ortaya çıkan sonuçlar örneklendirilmiştir. Benzer
şekilde, uzayında orjinden geçen
hiperdüzlem ve kuaterniyon doğrultusundaki doğru boyunca yansımalara karşılık
gelen dönüşümler tanıtılmıştır. Son olarak bu yansıma dönüşümlerinin matris
temsilleri elde edilmiş ve her durum için bu özdeğer ve özvekörlerin
hesaplanması ile geometrik yorumlar kuaterniyon katsayıları ile analiz
edilmiştir. 

Anahtar Kelimeler

Kuaterniyonlar, Yansımalar, Özdeğerler, Özvektörler, Katı hareketler

Kaynakça

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