# Making sketch and explaning the surface curves

Rotating a circle of the radius $r$ in the $x,z$ plane with center point $(R, 0)$ around the $z$ axis, exist torus, which is defined by

$\begin{pmatrix} (R+r\sin(u))\cos(v) \\ (R+r\sin(u))\sin(v) \\ r\cos(u) \end{pmatrix}$ $u\in \left[0,2\pi\right]$, $v\in \left[0,2\pi\right]$

$r<R$

Explain and sketch the surface curves:

$u=u_{0}\in \left[0,2\pi\right]$, $v\in \left[0,2\pi\right]$

$u\in \left[0,2\pi\right]$,$v=v_{0}\in \left[0,2\pi\right]$

Can someone give me hint how to solve this?

The strategy is now to assume $v = v_{0}$ is constant in the torus parametrization, to put $u = t$, and to write $$\begin{pmatrix} (R + r\sin t)\cos v_{0} \\ (R + r\sin t)\sin v_{0} \\ r \cos t \end{pmatrix} \tag{2}$$ in the form (1), then similarly if $u = u_{0}$ is constant and $v = t$.
In each case, break (2) into "some vector times $\cos t$ plus some (orthogonal) vector times $\sin t$ plus a constant (point)".