directions of curvature are rotated from the $x$-$y$ axes, which we choose as the principal axes of inertia (Fig. \[fig:coordinate\]). $\theta/a$ and $\phi/a$ are the principal curvatures at the bottom, namely at $(0,0,a)^{t}$. Now, we can formulate the contact point condition (1); the components of the contact point vector $\bm{r}$ should satisfy Eq. (\[eq:def-z\]), and the normal vector of the surface at C should be parallel to the vertical vector $\bm{u}$. Thus we have $$\bm{u} \parallel \nabla f,$$ which gives the relation $$\frac{\bm{r}_{\perp}}{a} = \frac{1}{u_{z}} \hat{R}(\xi)\hat{\Theta}^{-1}\hat{R}^{-1}(\xi)\bm{u}_{\perp} \label{eq:def-xy},$$ where $\bm{a}_{\perp}$ represents the $x$ and $y$ components of a vector $\bm{a}$ in the body-fixed co-ordinate.

Before we proceed, we introduce a dotted derivative of a vector $\bm{a}$ defined as the time derivative of the vector components in the body-fixed co-ordinate. This is related to the time derivative by $$\frac{d\bm{a}}{dt} = \dot{\bm{a}} + \bm{\omega} \times \bm{a}.$$ Note that the vertical vector $\bm{u}$ does not depend on time, thus we have $$\frac{d\bm{u}}{dt} = \dot{\bm{u}} + \bm{\omega} \times \bm{u} = \bm{0}. \label{eq:diff-u}$$ These conditions, i.e., the no-slip condition (\[eq:no-slip\]), the conditions of the contact point (\[eq:def-z\]) and (\[eq:def-xy\]), and the vertical vector condition (\[eq:diff-u\]), close the equations of motion (\[eq:em-1\]) and (\[eq:em-2\]).

Following Garcia and 