International Scholarly Research Notices

Research Article

Rigid Body Trajectories in Different 6D Spaces

Table 2

Approximate displacement functions for small displacements.
ℝ 3 ⋉ 𝑆 𝑂 ( 3 ) 𝑆 𝑂 ( 4 ) 𝑆 𝐸 ( 3 ) 𝑆 𝑂 ( 1 , 3 )
𝑥 ( 𝑡 ) 𝑣 𝑥 𝑡 s i n ( 𝑣 𝑥 𝑡 ) 𝑣 𝑥 𝑡 s i n h ( 𝑣 𝑥 𝑡 )
𝑦 ( 𝑡 ) 𝑣 𝑦 ( s i n ( 𝑤 𝑡 + 𝑓 ) − s i n ( 𝑓 ) ) / 𝑤 𝑣 𝑦 ( s i n ( 𝑤 4 𝑡 + 𝑓 ) − s i n ( 𝑓 ) ) / 𝑤 4 𝑣 𝑦 ( s i n ( 𝑤 5 𝑡 + 𝑓 ) − s i n ( 𝑓 ) ) / 𝑤 5 𝑣 𝑦 ( s i n ( 𝑤 6 𝑡 + 𝑓 ) − s i n ( 𝑓 ) ) / 𝑤 6
𝑧 ( 𝑡 ) 𝑣 𝑦 ( c o s ( 𝑓 ) − c o s ( 𝑤 𝑡 + 𝑓 ) ) / 𝑤 𝑣 𝑦 ( c o s ( 𝑓 ) − c o s ( 𝑤 4 𝑡 + 𝑓 ) ) / 𝑤 4 𝑣 𝑦 ( c o s ( 𝑓 ) − c o s ( 𝑤 5 𝑡 + 𝑓 ) ) / 𝑤 5 𝑣 𝑦 ( c o s ( 𝑓 ) − c o s ( 𝑤 6 𝑡 + 𝑓 ) ) / 𝑤 6
𝜃 1 ( 𝑡 ) 𝑤 t 𝑤 𝑡 𝑤 𝑡 𝑤 𝑡
𝜃 2 ( 𝑡 ) 0 0 0 0
𝜃 3 ( 𝑡 ) 0 0 0 0
𝑤 4 = ( 2 − 𝑜 1 / 𝑚 ) 𝑤 𝑤 5 = 2 𝑤 𝑤 6 = ( 2 + 𝑜 1 / 𝑚 ) 𝑤