On the Moving Coordinate System and Euler-Savary Formula in Affine Cayley-Klein Planes

Abstract

In this present paper, we will take three affine Cayley-Klein planes into consideration: A , P ò ò and  P ò . The plane  P ò is a fixed plane relative to two other moving affine Cayley-Klein (CK)-planes. We will describe one-parameter motions A A / , /  ò ò ò ò P P and /  ò ò P P and discuss the relationship between the motions A A / , /  ò ò ò ò P P and /  ò ò P P by evaluating their derivative formulae, velocity vectors and pole points. Also, we will observe moving coordinate system and after that, we will examine the canonical relative system for one-parameter planar motions in the affine CK-planes by using the notions of moving coordinate system. Moreover, Euler-Savary formula, which gives the relationship between the
curvatures of trajectory curves, will be obtained with the help of canonical relative system for oneparameter motions in affine CK-planes planes by using the method given by H. R. Müller in 1956 [1].

Keywords

• Cayley-Klein planes
• one-parameter planar motion
• moving coordinate system
• kinematics
• Euler-Savary Formula

On the Moving Coordinate System and Euler-Savary Formula in Affine Cayley-Klein Planes

Abstract

Bu çalışmada A , P ò ò ve  P ò üç afin Cayley-Klein düzlemi gözönüne alınmıştır.  P ò düzlemi diğer iki hareketli afin Cayley-Klein (CK)-düzlemine göre sabittir. Çalışmada bir parametreli
A A / , /  ò ò ò ò P P ve/  ò ò P P hareketleri tarif edilecek; türev formülleri, hız vektörleri ve pol noktaları elde edilerek A A / , /  ò ò ò ò P P ve /  ò ò P P hareketleri arasındaki ilişki tartışılacaktır. Ayrıca afin (CK)-düzlemlerinde hareketli koordinat sistemi araştırılarak bu hareketli koordinat sisteminin kavramları ile kavramları bir parametreli hareketler için kanonik izafe sistemi incelenecektir. Bu ifadelere ek olarak, kanonik izafe sistemi yardımıyla afin (CK)-düzlemlerinde bir parametreli hareketler için yörünge eğrilerinin eğrilikleri arasındaki ilişkiyi veren Euler Savary formülü H. R. Müller tarafında 1956 yılında verilen metodla elde edilecektir [1].

Keywords

• Cayley-Klein düzlemleri
• bir parametreli düzlemsel hareket
• hareketli koordinat sistemi
• kinematik
• Euler-Savary formülü

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