호의 길이와 곡률

속도 벡터는

$$\mathbf{v}(\theta)=x'(\theta)\,\hat{\boldsymbol{\imath}}+y'(\theta)\,\hat{\boldsymbol{\jmath}} = \cos\big(\tfrac{1}{2}\pi\theta^2\big)\,\hat{\boldsymbol{\imath}} + \sin\big(\tfrac{1}{2}\pi\theta^2\big)\,\hat{\boldsymbol{\jmath}}$$

이다. 따라서 속력은

$$\frac{\mathrm{d}s}{\mathrm{d}\theta}(\theta)= |\mathbf{v}(\theta)|=1 \implies s(\theta)=\theta+s(0)$$

이다. $\theta=0$ 일 때 $s(\theta)$ 가 0 이므로 $s(\theta)=\theta$ 이고, 따라서

$$\hat{\mathbf{T}}(s)=\mathbf{v}(s) = \cos\big(\tfrac{1}{2}\pi s^2\big)\,\hat{\boldsymbol{\imath}} + \sin\big(\tfrac{1}{2}\pi s^2\big)\,\hat{\boldsymbol{\jmath}}$$

이므로

$$\kappa (s)=\left|\frac{\mathrm{d}\hat{\mathbf{T}}}{\mathrm{d}s}(s)\right| =\big|-\pi s\sin\big(\tfrac{1}{2}\pi s^2\big)\,\hat{\boldsymbol{\imath}} +\pi s \cos\big(\tfrac{1}{2}\pi s^2\big)\,\hat{\boldsymbol{\jmath}}\big| =\pi s$$

이다.