Solution:

The equation of lofted surface is,

$\begin{aligned} P(u, v) &=(1-v) P(u, 0)+v P(u, 1) \\ P_{x}(u, v) &=(1-v) r \cos 2 \pi u+v 2 r \cos 2 \pi u \\ &=(1+v) r \cos 2 \pi u \end{aligned}$

$\begin{aligned} P_{y}(u, v) &=(1-v) r \sin 2 \pi u+v 2 r \sin 2 \pi u \\ &=(1+v) r \sin 2 \pi u \\ P_{z}(u, v) &=(1-v)(0)+v(H) \\ &=v \mathrm{H} \end{aligned}$

$\therefore P(u, v)=[(1+v) \operatorname{rcos} 2 \pi u \quad(1+v) r \sin 2 \pi u \quad v H]$