曲線C C C ( x ( θ ) , y ( θ ) ) (x(\theta),y(\theta)) ( x ( θ ) , y ( θ )) x ′ ( θ ) = a ( 1 − cos θ ) , y ′ ( θ ) = a sin θ . x'(\theta)=a(1-\cos\theta),\qquad y'(\theta)=a\sin\theta. x ′ ( θ ) = a ( 1 − cos θ ) , y ′ ( θ ) = a sin θ . 0 ≦ θ ≦ 2 π 0\leq\theta\leq2\pi 0 ≦ θ ≦ 2 π sin ( θ / 2 ) ≧ 0 \sin(\theta/2)\geq0 sin ( θ /2 ) ≧ 0 x ′ ( θ ) 2 + y ′ ( θ ) 2 = a ( 1 − cos θ ) 2 + sin 2 θ = 2 a sin θ 2 . \displaystyle \sqrt{x'(\theta)^2+y'(\theta)^2}=a\sqrt{(1-\cos\theta)^2+\sin^2\theta}=2a\sin\frac{\theta}{2}. x ′ ( θ ) 2 + y ′ ( θ ) 2 = a ( 1 − cos θ ) 2 + sin 2 θ = 2 a sin 2 θ . (1) ∫ 0 2 π 2 a sin θ 2 d θ = 8 a . \displaystyle \text{(1)}\quad \int_0^{2\pi}2a\sin\frac{\theta}{2}\,d\theta=8a. (1) ∫ 0 2 π 2 a sin 2 θ d θ = 8 a .
次に0 < θ < 2 π 0<\theta<2\pi 0 < θ < 2 π sin ( θ / 2 ) > 0 \sin(\theta/2)>0 sin ( θ /2 ) > 0 ( x ′ ( θ ) , y ′ ( θ ) ) = 2 a sin θ 2 ( sin θ 2 , cos θ 2 ) . \displaystyle (x'(\theta),y'(\theta))=2a\sin\frac{\theta}{2}\left(\sin\frac{\theta}{2},\cos\frac{\theta}{2}\right). ( x ′ ( θ ) , y ′ ( θ )) = 2 a sin 2 θ ( sin 2 θ , cos 2 θ ) . ( sin θ 2 , cos θ 2 ) \displaystyle \left(\sin\frac{\theta}{2},\cos\frac{\theta}{2}\right) ( sin 2 θ , cos 2 θ ) y y y ( − cos θ 2 , sin θ 2 ) \displaystyle \left(-\cos\frac{\theta}{2},\sin\frac{\theta}{2}\right) ( − cos 2 θ , sin 2 θ ) A B = 1 AB=1 A B = 1 B = A + ( − cos θ 2 , sin θ 2 ) , \displaystyle B=A+\left(-\cos\frac{\theta}{2},\sin\frac{\theta}{2}\right), B = A + ( − cos 2 θ , sin 2 θ ) , (2) B = ( a ( θ − sin θ ) − cos θ 2 , a ( 1 − cos θ ) + sin θ 2 ) . \displaystyle \text{(2)}\quad B=\left(a(\theta-\sin\theta)-\cos\frac{\theta}{2},\ a(1-\cos\theta)+\sin\frac{\theta}{2}\right). (2) B = ( a ( θ − sin θ ) − cos 2 θ , a ( 1 − cos θ ) + sin 2 θ ) . θ = 0 , 2 π \theta=0,2\pi θ = 0 , 2 π ( − 1 , 0 ) (-1,0) ( − 1 , 0 ) ( 2 π a + 1 , 0 ) (2\pi a+1,0) ( 2 π a + 1 , 0 )
最後に、B B B ( X ( θ ) , Y ( θ ) ) (X(\theta),Y(\theta)) ( X ( θ ) , Y ( θ )) X ′ ( θ ) = 2 a sin 2 θ 2 + 1 2 sin θ 2 = ( 2 a sin θ 2 + 1 2 ) sin θ 2 , \displaystyle X'(\theta)=2a\sin^2\frac{\theta}{2}+\frac12\sin\frac{\theta}{2}=\left(2a\sin\frac{\theta}{2}+\frac12\right)\sin\frac{\theta}{2}, X ′ ( θ ) = 2 a sin 2 2 θ + 2 1 sin 2 θ = ( 2 a sin 2 θ + 2 1 ) sin 2 θ , Y ′ ( θ ) = 2 a sin θ 2 cos θ 2 + 1 2 cos θ 2 = ( 2 a sin θ 2 + 1 2 ) cos θ 2 . \displaystyle Y'(\theta)=2a\sin\frac{\theta}{2}\cos\frac{\theta}{2}+\frac12\cos\frac{\theta}{2}=\left(2a\sin\frac{\theta}{2}+\frac12\right)\cos\frac{\theta}{2}. Y ′ ( θ ) = 2 a sin 2 θ cos 2 θ + 2 1 cos 2 θ = ( 2 a sin 2 θ + 2 1 ) cos 2 θ . a > 0 a>0 a > 0 sin ( θ / 2 ) ≧ 0 \sin(\theta/2)\geq0 sin ( θ /2 ) ≧ 0 X ′ ( θ ) 2 + Y ′ ( θ ) 2 = 2 a sin θ 2 + 1 2 . \displaystyle \sqrt{X'(\theta)^2+Y'(\theta)^2}=2a\sin\frac{\theta}{2}+\frac12. X ′ ( θ ) 2 + Y ′ ( θ ) 2 = 2 a sin 2 θ + 2 1 . (3) ∫ 0 2 π ( 2 a sin θ 2 + 1 2 ) d θ = 8 a + π . \displaystyle \text{(3)}\quad \int_0^{2\pi}\left(2a\sin\frac{\theta}{2}+\frac12\right)d\theta=8a+\pi. (3) ∫ 0 2 π ( 2 a sin 2 θ + 2 1 ) d θ = 8 a + π .
