Kuaterniyon Bakış Açısı ile Doğru ve Hiperdüzlem Boyunca Yansımalar

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

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

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.

Keywords:

Quaternions, Reflections, Eigenvalues, Eigenvectors, Rigid Motions,

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