plot::Spherical – surfaces in 3D parameterized in spherical coordinates

plot::Spherical creates parametrized surfaces in 3D, with parametrization in spherical coordinates.

Call:

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

Parameters:

r, , :	the coordinate functions: arithmetical expressions or piecewise objects depending on the surface parameters , and the animation parameter a. 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, , are equivalent to the attributes XFunction, YFunction, ZFunction.
u:	the first surface parameter: an identifier or an indexed identifier. u is equivalent to the attribute UName.
$`u_{min}`$ .. $`u_{max}`$ :	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::Spherical creates surfaces parametrized in spherical coordinates.
The surface given by a mapping (“parametrization”) is the set of all image points

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

is referred to as “radius”, as “azimuthal angle”, and is known as “polar angle.”
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 3.
“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 (nu - 1)*(mu + 1) + 1 parameter lines are visible. Cf. example 3.
Use Filled = FALSE to obtain a wireframe representation of the surface.
If the expression/function contains 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 and , although singularities in these functions may result in poorly rendered surfaces – in many cases setting the attributes Mesh and/or AdaptiveMesh to higher values will help. Cf. example 6.

Example 1

Spherical coordinates get their name from the fact that, with a constant radius, the parameterize a sphere:

plot(plot::Spherical([1, u, v], u = 0..2*PI, v = 0..PI))

MuPAD graphics

plot(plot::Spherical([1, u, v], u = 0..PI, v = 0..2*PI))

MuPAD graphics

Example 2

The following plot demonstrates that spherical plots can exhibit singular surface features even with differentiable parameterizations; in this case, the rim in the middle is actually a border of both the left- and the right-hand part:

plot(plot::Spherical(
       [(phi^2*thet), phi, thet^2],
       phi = -PI..PI, thet=0..0.25*PI,
       Mesh = [40,40], Submesh=[3,0],
       Color = [0.9$3], FillColorType=Flat, LineColor=[0.8$3]),
     Axes = None, CameraDirection = [1, 0, 0])

MuPAD graphics

Example 3

For oscillating parameterizations or other surfaces with fine details, the default mesh may be too coarse. As stated above, the three attributes Mesh, Submesh, and AdaptiveMesh can be used for improving plots of these objects.

First, note that the following plot is not rendered with a sufficient resolution:

surf := plot::Spherical([4+sin(5*(u+v)), u, v], u = 0..PI, v = 0..2*PI):
plot(surf, Axes = None)

MuPAD graphics

Setting Mesh to twice its default, we get a smoother surface with additional parameter lines:

surf::Mesh := [50, 50]:
plot(surf, Axes = None)

MuPAD graphics

Almost the same effect, but without the additional parameter lines, can be achieved by setting Submesh = [1, 1]:

delete surf::Mesh:
surf::Submesh := [1, 1]:
plot(surf, Axes = None)

MuPAD graphics

It is also possible to use adaptive mesh refinement in areas where neighboring patches have an angle of more than 10 degrees. While this option is mostly useful for surfaces which require refinement only in some parts, it is certainly feasible with a plot like this, too (but increasing Submesh is faster):

delete surf::Submesh:
surf::AdaptiveMesh := 2:
plot(surf, Axes = None)

MuPAD graphics

Example 4

The radius function may also take on negative values. With radius functions of changing sign, spherical surfaces often do self-intersect:

plot(plot::Spherical(
       [sin(phi^2*thet), phi, thet],
         phi = -PI..PI, thet = 0..0.5*PI,
       Mesh = [40, 20], Submesh=[0, 3]))

MuPAD graphics

Example 5

The angular functions ( and ) are not limited in value:

plot(plot::Spherical([r, r, thet], r = 0..9, thet = -PI..PI,
                     Mesh = [60, 60], Filled = FALSE),
     Axes = None,
     plot::Camera([100, 100, 50], [0,0,0], 0.1))

MuPAD graphics

Note that we used an explicit plot::Camera object here because the automatic camera is always placed such that all of an object is visible, even when using CameraDirection. To get a “closer” look, use the interactive manipulation possibilities or an explicit camera.

Example 6

Singularities in the radius function are heuristically handled:

plot(plot::Spherical([1/(u + v), u, v], u = 0..PI, v = 0..PI))

MuPAD graphics

However, the heuristics fails for some examples:

plot(plot::Spherical([1/(u + v)^2, u, v], u = 0..PI, v = 0..PI))

MuPAD graphics

In cases like this, we recommend setting a viewing box explicitly with the attribute ViewingBox:

plot(plot::Spherical([1/(u + v)^2, u, v], u = 0..PI, v = 0..PI),
ViewingBox = [-1/10..0.7, 0..1/4, -0.2..0.3])

MuPAD graphics

Example 7

By setting one of the parameter ranges to a degenerate interval, it is possible to draw curves on a spherical surface:

f := (u, v) -> [1 + u/10, u, v]:
surface := plot::Spherical(f(u,v), u = 0..2, v = 0..2,
                           FillColor = RGB::Grey, FillColorType = Flat):
curve := plot::Spherical(f((1 + sin(u)), (1 + sin(2*u))),
                         u = 0..2*PI, v = 0..0, Mesh = [200, 1],
                         LineColor = RGB::Red, LineWidth = 1):
plot(surface, curve)

MuPAD graphics

Example 8

While the transformation from spherical to Cartesian coordinates is not invertible, there are at least two ways of expressing each Cartesian point in spherical coordinates and any surface parameterizable in Cartesian coordinates can also be plotted using plot::Spherical (although this is probably more a curiosity than really useful):

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

[sqrt(x^2 + y^2 + sin(x^2 + y^2)^2), arccos(x/sqrt(x^2 + y^2)) + sign(y)*(sign(y) - 1)*(PI - arccos(x/sqrt(x^2 + y^2))), arccos(sin(x^2 + y^2)/sqrt(x^2 + y^2 + sin(x^2 + y^2)^2))]

plot(plot::Spherical(spher, x = -2..2, y = -2..2))

MuPAD graphics

Example 9

Last but not least we can also produce animations with the help of plot::Spherical. The following shows a deformation from a general spherical object to a sphere. We have used the animation parameter a inside of the argument for the sine function to obtain a slight rotation during the deformation process:

plot(
plot::Spherical(
[1+a*sin(3*Phi+a)*sin(2*Theta),Phi,Theta],
Theta=0..PI, Phi=0..2*PI, a=5..0
)
)

MuPAD graphics

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