Torus in differential geometry.

Question

I want to write separately parametrizations (surface patches) $\sigma$ for torus when (1) x-axis rotation in the first part of the picture and (2) y-axis rotation in the second part of the picture.

But I cannot. Please show me how to write?

Answer 1

First, we parameterize the circle: $$\begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} b + a \cos u \\ 0 \\ a \sin u \end{pmatrix}$$ Now we need to rotate this around the $z$ axis. Given an arbitrary point $p$, we can rotate it around the $z$-axis by this parameterization: $$\begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} \cos v & -\sin v & 0 \\ \sin v & \cos v & 0 \\ 0 & 0 & 1\end{pmatrix} \begin{pmatrix} p_x \\ p_y \\ p_z \end{pmatrix} = \begin{pmatrix} p_x \cos v - p_y \sin v \\ p_x \sin v + p_y \cos v \\ p_z \end{pmatrix} $$ If we apply this to our circle, we get: $$\begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} (b + a \cos u) \cos v - 0 \sin v \\ (b + a \cos u) \sin v + 0 \cos v \\ a \sin u \end{pmatrix} = \begin{pmatrix} (b + a \cos u) \cos v \\ (b + a \cos u) \sin v \\ a \sin u \end{pmatrix}$$

Can you extend this to the second example as well?