In the three-dimensional orthogonal coordinate system $x y z$, the unit vectors along the $x , y$, and $z$ directions are $\mathbf { i } , \mathbf { j }$, and $\mathbf { k }$, respectively. Using the parameter $\theta ( 0 \leq \theta \leq \pi )$, we define two curves by their vector functions $\mathbf { P } ( \theta )$ and $\mathbf { Q } ( \theta )$ :
$$\begin{aligned} & \mathbf { P } ( \theta ) = x ( \theta ) \mathbf { i } + y ( \theta ) \mathbf { j } \\ & \mathbf { Q } ( \theta ) = \mathbf { P } ( \theta ) + z ( \theta ) \mathbf { k } \end{aligned}$$
where
$$\begin{aligned} & x ( \theta ) = \frac { 3 } { 2 } \cos ( \theta ) - \frac { 1 } { 2 } \cos ( 3 \theta ) \\ & y ( \theta ) = \frac { 3 } { 2 } \sin ( \theta ) - \frac { 1 } { 2 } \sin ( 3 \theta ) \end{aligned}$$
Here, $z ( \theta )$ is a continuous function satisfying $z ( 0 ) > 0$ and $z ( \pi ) < 0$, and the curve parametrized by $\mathbf { Q } ( \theta )$ lies on the sphere of radius 2, centered at the origin $( 0,0,0 )$ of the coordinate system. The positive direction of a curve corresponds to increasing values of the parameter $\theta$. Note that the curvature is the reciprocal of the radius of curvature. Answer the following questions.
I. As $\theta$ is varied from 0 to $\pi$, calculate the arc length of the curve parametrized by $\mathbf { P } ( \theta )$.
II. Obtain $z ( \theta )$.
III. Let $\alpha$ be the angle between the tangent of the curve parametrized by $\mathbf { Q } ( \theta )$ and the unit vector $\mathbf { k }$. Calculate $\cos ( \alpha )$.
IV. Find the curvature $\kappa _ { P } ( \theta )$ of the curve parametrized by $\mathbf { P } ( \theta )$. Here, $\theta = 0$ and $\theta = \pi$ are excluded.
V. Let $\kappa _ { Q } ( \theta )$ be the curvature of the curve parametrized by $\mathbf { Q } ( \theta )$. Express $\kappa _ { Q } ( \theta )$ in terms of $\kappa _ { P } ( \theta )$ and $\alpha$. Here, $\theta = 0$ and $\theta = \pi$ are excluded.