plot::Function3d – 3D function graphs

plot::Function3d creates the 3D graph of a function in 2 variables.

Calls:

plot::Function3d(f, Options)

plot::Function3d(f, x = $`x_{min}`$ .. $`x_{max}`$ , y = $`y_{min}`$ .. $`y_{max}`$ , <a = amin .. amax>, Options)

Parameters:

f:	the function: an arithmetical expression or a piecewise object in the independent variables , and the animation parameter a. Alternatively, a MuPAD procedure that accepts 2 input parameter , or 3 input parameters , , and returns a numerical value when the input parameters are numerical. f is equivalent to the attribute Function.
x:	the first independent variable: an identifier or an indexed identifier. x is equivalent to the attribute XName.
$`x_{min}`$ .. $`x_{max}`$ :	the plot range in direction: $`x_{min}`$ , $`x_{max}`$ must be numerical real values or expressions of the animation parameter . If not specified, the default range x = -5 .. 5 is used. $`x_{min}`$ .. $`x_{max}`$ is equivalent to the attributes XRange, XMin, XMax.
y:	the second independent variable: an identifier or an indexed identifier. y is equivalent to the attribute YName.
$`y_{min}`$ .. $`y_{max}`$ :	the plot range in direction: $`y_{min}`$ , $`y_{max}`$ must be numerical real values or expressions of the animation parameter . If not specified, the default range y = -5 .. 5 is used. $`y_{min}`$ .. $`y_{max}`$ is equivalent to the attributes YRange, YMin, YMax.

Details:

The expression f(x, y) is evaluated at finitely many points , in the plot range. There may be singularities. Although a heuristics is used to find a reasonable range when singularities are present, it is highly recommended to specify a range via ViewingBoxZRange = $`z_{min}`..`z_{max}`$ with suitable numerical real values $`z_{min}`$ , $`z_{max}`$ . Cf. example 2.
Animations are triggered by specifying a range $a = `a_{min}`..`a_{max}`$ for a parameter a that is different from the indedependent variables x, y. Thus, in animations, the -range $x = `x_{min}`..`x_{max}`$ , the -range $y = `y_{min}`..`y_{max}`$ as well as the animation range $a = `a_{min}`..`a_{max}`$ must be specified. Cf. example 3.
The function f is evaluated on a regular equidistant mesh of sample points determined by the attributes XMesh and YMesh (or the shorthand-notation for both, Mesh). 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 XMesh and YMesh or set AdaptiveMesh = n with some (small) positive integer n. This may result in up to times as many triangles as used with AdaptiveMesh = 0, potentially more when f has non-isolated singularities. Cf. example 4.
The “coordinate lines” (“parameter lines”) are curves on the function graph.

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

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

By default, the parameter lines are visible. They may be “switched off” by specifying XLinesVisible = FALSE and YLinesVisible = FALSE, respectively.
The coordinate lines controlled by XLinesVisible = TRUE/FALSE and YLinesVisible = TRUE/FALSE indicate the equidistant regular mesh set via the Mesh attributes. If the mesh is refined by the Submesh attributes or by the adaptive mechanism controlled by AdaptiveMesh = n, no additional parameter lines are drawn.

As far as the numerical approximation of the function graph is concerned, the settings

,

and

, Submesh = [0, 0]

are equivalent. However, in the first setting, nx parameter lines are visible in the direction, while in the latter setting parameter lines are visible. Cf. example 5.

Example 1

The following call returns an object representing the graph of the function over the region , :

g := plot::Function3d(sin(x^2 + y^2), x = -2..2, y = -2..2)

Call plot to plot the graph:

plot(g)

MuPAD graphics

Functions can also be specified by piecewise objects or procedures:

f := piecewise([x < y, 0], [x >= y, (x - y)^2]):
plot(plot::Function3d(f, x = -2 .. 4, y = -1 .. 3))

MuPAD graphics

f := proc(x, y)
begin
if x + y^2 + 2*y < 0 then
0
else
x + y^2 + 2*y
end_if:
end_proc:
plot(plot::Function3d(f, x = -3 .. 2, y = -2 .. 2))

MuPAD graphics

delete g, f

Example 2

We plot a function with singularities:

f := plot::Function3d(x/y + y/x, x = -1 .. 1, y = - 1 .. 1):
plot(f)

MuPAD graphics

We specify an explicit viewing range for the direction:

plot(f, ViewingBoxZRange = -20 .. 20)

MuPAD graphics

delete f

Example 3

We generate an animation of a parametrized function:

plot(plot::Function3d(sin((x - a)^2 + y^2),
x = -2 .. 2, y = -2 .. 2, a = 0 .. 5))

MuPAD graphics

Example 4

The standard mesh for the numerical evaluation of a function graph does not suffice to generate a satisfying graphics in the following case:

plot(plot::Function3d(besselJ(0, sqrt(x^2 + y^2)),
x = -20 .. 20, y = -20 .. 20))

MuPAD graphics

We increase the number of mesh points. Here, we use XSubmesh and YSubmesh to place 2 additional points in each direction between each pair of neighboring points of the default mesh. This increases the runtime by a factor of :

plot(plot::Function3d(besselJ(0, sqrt(x^2 + y^2)),
x = -20 .. 20, y = -20 .. 20,
Submesh = [2, 2]))

MuPAD graphics

Alternatively, we enable adaptive sampling by setting the value of AdaptiveMesh to some positive value:

plot(plot::Function3d(besselJ(0, sqrt(x^2 + y^2)),
x = -20 .. 20, y = -20 .. 20,
AdaptiveMesh = 2))

MuPAD graphics

Example 5

By default, the parameter lines of a function graph are “switched on”:

plot(plot::Function3d(x^2 + y^2, x = 0 .. 1, y = 0 .. 1))

MuPAD graphics

The parameter lines are “switched off” by setting XLinesVisible, YLinesVisible:

plot(plot::Function3d(x^2 + y^2, x = 0 .. 1, y = 0 .. 1,
XLinesVisible = FALSE,
YLinesVisible = FALSE))

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The number of parameter lines are determined by the Mesh attributes:

plot(plot::Function3d(x^2 + y^2, x = 0 .. 1, y = 0 .. 1,
Mesh = [5, 12]))

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When the mesh is refined via the Submesh attributes, the numerical approximation of the surface becomes smoother. However, the number of parameter lines is not increased:

plot(plot::Function3d(x^2 + y^2, x = 0 .. 1, y = 0 .. 1,
Mesh = [5, 12],
XSubmesh = 1, YSubmesh = 2))

MuPAD graphics