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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?

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$\newcommand{\Reals}{\mathbf{R}}\newcommand{\Vec}[1]{\mathbf{#1}}\newcommand{\Basis}{\Vec{e}}\newcommand{\Ctr}{\Vec{p}}$Hint: If $\Basis_{1}$ and $\Basis_{2}$ are orthogonal unit vectors in $\Reals^{n}$, if $\Ctr_{0}$ is a point of $\Reals^{n}$, and if $r_{0} > 0$ is real, then the set of points of the form $$ \Vec{x}(t) = \Ctr_{0} + (r_{0}\cos t) \Basis_{1} + (r_{0}\sin t) \Basis_{2},\quad 0 \leq t \leq 2\pi, \tag{1} $$ is the circle of center $\Ctr_{0}$ and radius $r_{0}$ lying in the plane parallel to $\Basis_{1}$ and $\Basis_{2}$.

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)".

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