plot::Cylindrical – surfaces in 3D parameterized in cylindrical coordinates

plot::Cylindrical creates parameterized surfaces in 3D, with parametrization in cylindrical coordinates.

Call:

plot::Cylindrical([r, , z], u = $`u_{min}`$ .. $`u_{max}`$ , v = $`v_{min}`$ .. $`v_{max}`$ , <a = amin .. amax>, Options)

Parameters:

r, , z:	the coordinate functions: arithmetical expressions or piecewise objects depending on the surface parameters , and the animation parameter . Alternatively, MuPAD procedures that accept 2 input parameters , or 3 input parameters , , and return a real numerical value when the input parameters are numerical. r, , z are equivalent to the attributes XFunction, YFunction, ZFunction.
u:	the first surface parameter: an identifier or an indexed identifier. u is equivalent to the attributes UName, UMin, UMax.
$`u_{min}`$ .. $`u_{max}`$ :	the plot range for the parameter : $`u_{min}`$ , $`u_{max}`$ must be numerical real values or expressions of the animation parameter . $`u_{min}`$ .. $`u_{max}`$ is equivalent to the attributes URange, UMin, UMax.
v:	the second surface parameter: an identifier or an indexed identifier. v is equivalent to the attribute VName.
$`v_{min}`$ .. $`v_{max}`$ :	the plot range for the parameter : $`v_{min}`$ , $`v_{max}`$ must be numerical real values or expressions of the animation parameter . $`v_{min}`$ .. $`v_{max}`$ is equivalent to the attributes VRange, VMin, VMax.

Details:

plot::Cylindrical creates surfaces parametrized in cylindrical coordinates.
The surface given by a mapping (“parametrization”) is the set of all image points

$ImageSet(matrix([[r(u, v)], [Symbol::phi(u, v)], [z(u, v)]]), u in [`u_{min}`, `u_{max}`], v in [`v_{min}`, `v_{max}`])$

in cylindrical coordinates, which translate to the usual “Cartesian” coordinates as

is referred to as “radius,” as “polar angle,” and as the “height” of a point.
The functions , , are evaluated on a regular equidistant mesh of sample points in the - plane. This mesh is determined by the attributes UMesh, VMesh. By default, the attribute AdaptiveMesh = 0 is set, i.e., no adaptive refinement of the equidistant mesh is used.

If the standard mesh does not suffice to produce a sufficiently detailed plot, one may either increase the value of UMesh, VMesh or USubmesh, VSubmesh,or set AdaptiveMesh = n with some (small) positive integer n. If necessary, up to additional points are placed in each direction of the - plane between adjacent points of the initial equidistant mesh. Cf. example 2.
“Coordinate lines” (“parameter lines”) are curves on the surface.

The phrase “ULines” refers to the curves with the parameter running from $`u_{min}`$ to $`u_{max}`$ , while is some fixed value from the interval $[`v_{min}`, `v_{max}`]$ .

The phrase “VLines” refers to the curves with the parameter running from $`v_{min}`$ to $`v_{max}`$ , while is some fixed value from the interval $[`u_{min}`, `u_{max}`]$ .

By default, the parameter curves are visible. They may be switched off by specifying ULinesVisible = FALSE and VLinesVisible = FALSE, respectively.
The coordinate lines controlled by ULinesVisible = TRUE/FALSE and VLinesVisible = TRUE/FALSE indicate the equidistant mesh in the - plane set via the UMesh, VMesh attributes. If the mesh is refined by the USubmesh, VSubmesh attributes, or by the adaptive mechanism controlled by AdaptiveMesh = n, no additional parameter lines are drawn.

As far as the numerical approximation of the surface is concerned, the settings

, , ,

and

, ,

USubmesh = 0, VSubmesh = 0

are equivalent. However, in the first setting, nu parameter lines are visible in the direction, while in the latter setting parameter lines are visible. Cf. example 2.
Use Filled = FALSE to obtain a wireframe representation of the surface.
If the expressions/functions and/or contain singularities, it is recommended (but not strictly necessary) to use the attribute ViewingBox to set a suitable viewing box. No such precautions are necessary for , although singularities in this function may result in poorly rendered surfaces – in many cases setting the attributes Mesh and/or AdaptiveMesh to higher values will help. Cf. example 3.

Example 1

Using a constant radius for plot::Cylindrical, with the other two functions straight from the surface parameters, results in a right cylinder. This explains the name “cylindrical coordinates”:

plot(plot::Cylindrical([1, phi, z], phi = 0..2*PI, z = -1..1))

MuPAD graphics

Other straightforward examples include cones and paraboloids of revolution:

plot(plot::Cylindrical([r, phi, 2*r], r = 0..1, phi = 0..2*PI))

MuPAD graphics

plot(plot::Cylindrical([r, phi, r^2], r = 0..1, phi = 0..2*PI))

MuPAD graphics

Example 2

Cylindrical surfaces are drawn from evaluations on an equidistant mesh of points. In some cases, the default mesh density is insufficient or otherwise inappropriate:

plot(plot::Cylindrical([cos(phi^2), phi, z],
phi=-2.8..2.8, z=0..1/2))

MuPAD graphics

One possible change to this plot command is to explicitly set the mesh with the attribute Mesh. Note that this setting influences the density of parameter lines:

plot(plot::Cylindrical([cos(phi^2), phi, z],
phi=-2.8..2.8, z=0..1/2,
Mesh = [100, 5]))

MuPAD graphics

To increase the mesh density without introducing additional parameter lines, you can use submesh settings:

plot(plot::Cylindrical([cos(phi^2), phi, z],
phi=-2.8..2.8, z=0..1/2,
VMesh = 5, USubmesh = 3))

MuPAD graphics

Finally, we can also ask plot::Cylindrical to refine the mesh only in areas of higher curvature. In the following example, we allow for additional points between each two neighboring points of the initial mesh:

plot(plot::Cylindrical([cos(phi^2), phi, z],
phi=-2.8..2.8, z=0..1/2,
VMesh = 5, AdaptiveMesh = 3))

MuPAD graphics

Example 3

If the radius- or the -function/expression contains singularities, plot::Cylindrical employs heuristic clipping to select a range to display:

plot(plot::Cylindrical([1/sqrt((phi - PI)^2 + z^2), phi, z],
phi = 0..2*PI, z = -1..1))

MuPAD graphics

While these heuristics work well in many cases, there are also examples where they do not select a useful box:

plot(plot::Cylindrical([1/((phi - PI)^2 + z^2), phi, z],
phi = 0.. 2*PI, z = -1..1))

MuPAD graphics

In these cases, the user should set the range to display explicitly:

plot(plot::Cylindrical([1/((phi - PI)^2+z^2), phi, z],
phi = 0..2*PI, z = -1..1),
ViewingBox = [-2..0.3, -1.5..1.5, -1..1])

MuPAD graphics

Example 4

Since the transformation from cylindrical to orthogonal coordinates is reversible (up to reducing the angle to the range , it is possible to plot any surface with plot::Cylindrical (although this is probably more a curiosity than really useful):

trans := linalg::ogCoordTab[Cylindrical, InverseTransformation]:
cyl := trans(x, y, sin(x^2+y^2))

plot(plot::Cylindrical(cyl, x = -2..2, y = -2..2))

MuPAD graphics

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