Parametric Surface Plots

## Description

We can use parametric_plot3d to graph a parametric surface, where each of the coordinates $x$, $y$, $z$ is determined by a function of one or two variables (the parameters, typically $u$ and $v$). As an example, we plot the three-dimensional torus,

(1)
\begin{align} \mathbf x(u, v) = ( \cos(u)(2 + \cos v), \sin(u)(2 + \cos v), \sin v), \end{align}

where $0\leq u \leq 2 \pi$ and $0\leq v \leq 2 \pi$.

## Sage Cell

#### Code

u, v = var('u, v')
f_x(u, v) = cos(u)*(2 + cos(v))
f_y(u, v) = sin(u)*(2 + cos(v))
f_z(u, v) = sin(v)
parametric_plot3d([f_x, f_y, f_z], (u, 0, 2*pi), (v, 0, 2*pi), aspect_ratio=1)


## Tags

Primary Tags: Plotting: Three-dimensional plots

Secondary Tags: Three-dimensional plots: Parametric plots of surfaces

None