let
r = [ a ( u − s i n u ) , 1 − c o s u , b u ] r=[a(u-sinu),1-cosu,bu] r = [ a ( u − s in u ) , 1 − cos u , b u ]
so
r ′ r^{'} r ′ = [ a ( 1 − c o s u ) , s i n u , 0 ] =[a(1-cosu),sinu,0] = [ a ( 1 − cos u ) , s in u , 0 ]
r ′ ′ = [ a s i n u , c o s u , 0 ] r^{''}=[asinu,cosu,0] r ′′ = [ a s in u , cos u , 0 ]
r ′ ′ ′ = a c o s u , − s i n u , 0 r^{'''}=acosu,-sinu,0 r ′′′ = a cos u , − s in u , 0
Now :
Curvature = ∣ r ′ × r ′ ′ ∣ ∣ r ′ ∣ 3 =\dfrac{|r^{'} \times r^{''}|}{|r^{'}|^3} = ∣ r ′ ∣ 3 ∣ r ′ × r ′′ ∣
Now: r ′ × r ′ ′ = r^{'} \times r^{''}= r ′ × r ′′ = i [ − b c o s u ] − j [ − a b s i n u ] + k [ a ( c o s u − c o s 2 u ) − a s i n 2 u ] i[-bcosu]-j[-absinu]+k[a(cosu-cos^2u)-asin^2u] i [ − b cos u ] − j [ − ab s in u ] + k [ a ( cos u − co s 2 u ) − a s i n 2 u ]
= ( − b c o s u ) i + ( a b s i n u ) j + ( a c o s u − a ) k =(-bcosu)i +(absinu)j+(acosu-a)k = ( − b cos u ) i + ( ab s in u ) j + ( a cos u − a ) k
∣ r ′ × r ′ ′ ∣ = b 2 ( c o s 2 u + a 2 s i n 2 u ) + a 2 + a 2 ( c o s 2 u − 2 c o s u ) |r^{'} \times r^{''}|= \sqrt{b^2(cos^2u+a^2sin^2u)+a^2+a^2(cos^2u-2cosu)} ∣ r ′ × r ′′ ∣ = b 2 ( co s 2 u + a 2 s i n 2 u ) + a 2 + a 2 ( co s 2 u − 2 cos u )
∣ r ′ ∣ = a 2 ( 1 + c o s 2 u ) − 2 a c o s u + s i n 2 u + b |r^{'}|= \sqrt{a^2(1+cos^2u)-2acosu+sin^2u+b} ∣ r ′ ∣ = a 2 ( 1 + co s 2 u ) − 2 a cos u + s i n 2 u + b
Curvature = ∣ r ′ × r ′ ′ ∣ ∣ r ′ ∣ 3 = b 2 ( c o s 2 u + a 2 s i n 2 u ) + a 2 + a 2 ( c o s 2 u − 2 c o s u ) ( a 2 ( 1 + c o s 2 u ) − 2 a c o s u + s i n 2 u + b ) 3 / 2 = \dfrac{|r^{'} \times r^{''}|}{|r^{'}|^3} = \dfrac{\sqrt{b^2(cos^2u+a^2sin^2u)+a^2+a^2(cos^2u-2cosu)}}{(a^2(1+cos^2u)-2acosu+sin^2u+b)^{3/2}} = ∣ r ′ ∣ 3 ∣ r ′ × r ′′ ∣ = ( a 2 ( 1 + co s 2 u ) − 2 a cos u + s i n 2 u + b ) 3/2 b 2 ( co s 2 u + a 2 s i n 2 u ) + a 2 + a 2 ( co s 2 u − 2 cos u )
Torsion = [ r ′ , r ′ ′ , r ′ ′ ′ ] ∣ r ′ × r ′ ′ ∣ 2 = \dfrac{[r^{'},r^{''},r^{'''}]}{|r^{'} \times r^{''}|^2} = ∣ r ′ × r ′′ ∣ 2 [ r ′ , r ′′ , r ′′′ ]
[ r ′ , r ′ ′ , r ′ ′ ′ ] = [ a ( 1 − c o s u ) ( 0 ) ] − [ s i n u ( 0 ) ] + b ( − a s i n 2 u − a c o s 2 u ) = − a b [r^{'},r^{''},r^{'''}]=[a(1-cosu)(0)] -[sinu(0)] +b(-asin^2u-acos^2u)= -ab [ r ′ , r ′′ , r ′′′ ] = [ a ( 1 − cos u ) ( 0 )] − [ s in u ( 0 )] + b ( − a s i n 2 u − a co s 2 u ) = − ab
so Torsion = − a b b 2 ( c o s 2 u + a 2 s i n 2 u ) + a 2 + a 2 ( c o s 2 u − 2 c o s u ) = \dfrac{-ab}{b^2(cos^2u+a^2sin^2u)+a^2+a^2(cos^2u-2cosu)} = b 2 ( co s 2 u + a 2 s i n 2 u ) + a 2 + a 2 ( co s 2 u − 2 cos u ) − ab
#340153 on Dec 2023
Comments