let
r=[a(u−sinu),1−cosu,bu]
so
r′ (single deriverative of r) =[a(1−cosu),sinu,0]
r′′=[asinu,cosu,0]
r′′′=acosu,−sinu,0
Now :
Curvature =∣r′∣3∣r′×r′′∣
Now: r′×r′′= i[−bcosu]−j[−absinu]+k[a(cosu−cos2u)−asin2u]
=(−bcosu)i+(absinu)j+(acosu−a)k
∣r′×r′′∣=b2(cos2u+a2sin2u)+a2+a2(cos2u−2cosu)
∣r′∣=a2(1+cos2u)−2acosu+sin2u+b
Curvature =∣r′∣3∣r′×r′′∣=(a2(1+cos2u)−2acosu+sin2u+b)3/2b2(cos2u+a2sin2u)+a2+a2(cos2u−2cosu)
Torsion =∣r′×r′′∣2[r′,r′′,r′′′]
[r′,r′′,r′′′]=[a(1−cosu)(0)]−[sinu(0)]+b(−asin2u−acos2u)=−ab
so Torsion =b2(cos2u+a2sin2u)+a2+a2(cos2u−2cosu)−ab
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