Copyright © 1995, 1997, 2003 by James F. Hurley, Department of Mathematics, University of Connecticut, Unit 3009, Storrs, CT 06269-3009. All rights reserved

**1. Basics**. The standard package Graphics`ParametricPlot3D`contains commands for 3-dimensional plotting of regions with cylindrical-coordinate descriptions. This notebook discusses cylindrical-coordinate plotting.

The cylindrical coordinate system is just the hybrid that results from crossing polar coordinates in the *xy*-plane with the ordinary vertical Cartesian coordinate *z*. To distinguish them readily from Cartesian coordinates, the cylindrical coordinates of points in this notebook are in *square* brackets. An expression of the form *P*(*x, y, z*) = *P*[*r,θ, z*] means that a point *P*(*x, y, z*) has cylindrical coordinates *r, θ,* and *z*.

It is easy to generate plots of the basic cylindrical surfaces *z = c, r = c, *and *θ *= *k, *where *c *and *k *are constants*. *Note that the plot of a horizontal plane *z = *3 looks circular, not like the parallelogram one sees in a Cartesian plot. That is so because the intersection of a *cylinder* *r* = *c* with the horizontal plane *z = k *is a circle of radius *c.* As usual, to produce such a plot, position your cursor at the end of the last line of code below, and hit the Enter key (or press Shift-Return).* *

**Needs["Graphics`ParametricPlot3D`"]**

CylindricalPlot3D[3, {r, 0, 4}, {theta, 0, 2 Pi},

AxesLabel -> {x, y, z}]

*Mathematica*'s CylindricalPlot3D command can only plot surfaces that are the graphs of equations *z = f(r, θ)*, so to plot the cylindrical-coordinate equation *r = c* it is best to *parametrize* the surface. A natural way to do so is to let *x = c *cos *θ*, *y = c *sin *θ*, and *z = v, *where *θ* varies over [0, 2π] and *v* ranges over any convenient interval [*a, b*]. The following plot uses that approach. To use it, note that you do not have to call the graphics *package* ParametricPlot3D.

**ParametricPlot3D[{4 Cos[u], 4 Sin[u], v}, **

{u, 0, 2 Pi}, {v, 0, 5},

AxesLabel -> {x, y, z} ]

A convenient way to plot *θ = k* is also via a parametrization of the surface, which in this case is a vertical plane through the line* θ = k* in the *xy-*plane. In this case, making a parametrization is somewhat more challenging. One way is to let *x = u *cos* k, y = u *sin *k, *and *z = v,* where again *u *and* v* range over some convenient intervals.

**ParametricPlot3D[{u Cos[1], u Sin[1], v}, {u, 0, 4}, **

{v, 0, 4}, AxesLabel -> {x, y, z}]

**Needs["Graphics`ParametricPlot3D`"]**

CylindricalPlot3D[3, {r, 2, 3}, {theta, Pi/6, Pi/3}];

CylindricalPlot3D[4, {r, 2, 3}, {theta, Pi/6, Pi/3}];

ParametricPlot3D[{u Cos[Pi/6], u Sin[Pi/6], v},

{u, 2, 3}, {v, 3, 4}];

ParametricPlot3D[{u Cos[Pi/3], u Sin[Pi/3], v},

{u, 2, 3}, {v, 3, 4}];

ParametricPlot3D[{2 Cos[u], 2 Sin[u], v},

{u, Pi/6, Pi/3}, {v, 3, 4}]

ParametricPlot3D[{3 Cos[u], 3 Sin[u], v},

{u, Pi/6, Pi/3}, {v, 3, 4},

AxesLabel -> {x, y, z}]

Show[%, %%, %%%, %%%%, %%%%%, %%%%%%,

AxesLabel -> {x, y, z}]

**2. Examples**. To illustrate the process of using *Mathematica* to plot regions with cylindrical-coordinates descriptions, we examine two such regions.

**Example 1**. The region *D* in the *xz*-plane between the *x*-axis and the graphs of *z = *
${x}^{2}$
and *x = *2 is revolved about the *z*-axis to form a solid of revolution popular in second-semester calculus volume problems*.*

**Plot[x^2, {x, 0, 2}, **

PlotStyle -> Text[FontForm["D",{"Times-Italic",

12}],{1.5, 1} ] ];

ListPlot[{{2, 0}, {2, 4}}, PlotJoined -> True,

PlotStyle -> RGBColor[1, 0, 0] ];

Show[%, %%, AxesLabel -> {x, z} ]

**Solution**. The above code produces a figure that shows the region *D. Fact: *Revolving the graph of *x = *
${z}^{1/2}$
about the *z-*axis generates a paraboloid of revolution with Cartesian equation
${x}^{2}+\phantom{\rule{0.3em}{0.3ex}}\uf39f{y}^{2}$
*= z,* In cylindrical coordinates that becomes *z = *
${r}^{2}$
*. *Similarly, revolution of the line *x = *2 about the *z*-axis generates the right circular cylinder whose equation is
${x}^{2}+\phantom{\rule{0.3em}{0.3ex}}\uf39f{y}^{2}$
* = *4*, *that is,
${r}^{2}$
* = *4*.* In the presence of the restriction *r >= *0, that simplifies to *r = *2*.* A parametric representation of this surface results from letting

*x* = 2 cos *u*, *y* = 2 sin *u*, and *z = v*.

The idea is that the circular portion sweeps out as *u* varies over [0, 2π], and the vertical part as *v* varies. The following routine plots the paraboloid and the cylinder. Watching the pictures generate helps you see how the region *S* lies between the two surfaces whose plots follow: inside the cylinder *r = *2 and outside the paraboloid *z = *
${r}^{2}$
*. (*The final plot shows only half the region, so that hidden-line removal does not erase the inner surface.) Execute the routine to watch the final plot take shape.

**Needs["Graphics`ParametricPlot3D`"]**

CylindricalPlot3D[r^2, {r, 0, 2}, {theta, 0, Pi}];

ParametricPlot3D[{2 Cos[u], 2 Sin[u], v},

{u, 0, Pi},{v, 0, 4}]

Show[%, %%, AxesLabel -> {x, y, z}]

**Example 2.** Plot the region *U *bounded by the graphs of *z = *0, *z = y*, and
${x}^{2}+\phantom{\rule{0.3em}{0.3ex}}\uf39f{y}^{2}$
* = *1. **Solution. ** The following code produces a *Mathematica* plot of half the region. As before, it is instructive to watch the final plot evolve. Execute the code to do so:

**Needs["Graphics`ParametricPlot3D`"]**

CylindricalPlot3D[r Sin[theta], {r, 0, 2}, {theta, 0, Pi}] ;

ParametricPlot3D[{Cos[u], Sin[u], v}, {u, 0, Pi},

{v, 0, 2}];

Show[%, %%, AxesLabel -> {x, y, z}]

The sequence of figures shows clearly that the region *U *is symmetric with respect to the *xy*-plane: the portion of *U * above that plane is congruent to the portion below it.

Converted by