plot::XRotate – surfaces of revolution around x-axis

plot::ZRotate – surfaces of revolution around z-axis

$plot::XRotate(f, x = `x_{min}`..xmax)$ creates a surface of revolution by rotating the function graph with around the -axis.

$plot::ZRotate(f, x = `x_{min}`..xmax)$ creates a surface of revolution by rotating the function graph with around the -axis.

→ Examples

Calls:

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

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

Parameters:

f:	the function: an arithmetical expression or a piecewise object in the independent variable and the animation parameter . Alternatively, a MuPAD procedure that accepts 1 input parameter or 2 input parameters , and returns a real numerical value when the input parameters are numerical. f is equivalent to the attribute Function.
x:	the independent variable: an identifier or an indexed identifier. x is equivalent to the attribute XName.
$`x_{min}`$ .. $`x_{max}`$ :	the plot range: $`x_{min}`$ , $`x_{max}`$ must be numerical real values or expressions of the animation parameter . $`x_{min}`$ .. $`x_{max}`$ is equivalent to the attributes XRange, XMin, XMax.

Details:

plot::XRotate rotates the graph of the function around the -axis, creating a surface of revolution. The slice of the surface parallel to the - plane at a point is a circle of radius .
plot::ZRotate rotates the graph of the given function around the -axis, creating another surface of revolution. The slice of the surface parallel to the - plane at a point consists of circles with radii given by the solutions of .
The range of the rotation can be restricted with the attributes AngleBegin, AngleEnd, AngleRange. The surface of revolution will only span over the given range of the rotation angle.
Surfaces of revolution are parametrized surfaces. The first surface parameter is , the second is the rotation angle. Surfaces of revolution react to most of the graphical attributes that surfaces of type plot::Surface react to. For example, use Mesh, Submesh to control the numerical mesh or use ULinesVisible, VLinesVisible to switch the parameter lines on/off.

Example 1

By default, plot::XRotate displays a complete revolution, just as if an object was created on a lathe:

plot(plot::XRotate(2 - sin(x), x = 0..2*PI))

MuPAD graphics

This rotation is insensitive to negative values: The surfaces of revolution of and are identical:

plot(plot::Scene3d(plot::XRotate(sin(x), x = 0..2*PI)),
plot::Scene3d(plot::XRotate(abs(sin(x)), x = 0..2*PI)),
Layout = Horizontal)

MuPAD graphics

This symmetry is broken when not performing a whole revolution:

plot(plot::Scene3d(plot::XRotate(sin(x), x = 0..2*PI,
                                 AngleRange = -PI/2..PI/2)),
     plot::Scene3d(plot::XRotate(abs(sin(x)), x = 0..2*PI,
                                 AngleRange = -PI/2..PI/2)),
     Layout = Horizontal)

MuPAD graphics

plot::XRotate can be animated, like almost every plot object:

plot(plot::XRotate(sin(x + a), x = 0 .. 2*a + PI,
AngleRange = 0 .. PI + a/2,
a = 0..2*PI))

MuPAD graphics

Example 2

As for plot::ZRotate, most of the points from above hold here, too. Obviously, the symmetry for a whole revolution is now with respect to the values, not the function values:

plot(plot::Scene3d(plot::ZRotate(exp(x), x = -1..1)),
plot::Scene3d(plot::ZRotate(exp(-x), x = -1..1)),
Layout = Horizontal)

MuPAD graphics

plot(plot::Scene3d(plot::ZRotate(exp(x), x = -1..1,
                                 AngleRange = -a..a,
                                 a = 0..PI)),
     plot::Scene3d(plot::ZRotate(exp(-x), x = -1..1,
                                 AngleRange = -a..a,
                                 a = 0..PI)),
     Layout = Horizontal)

MuPAD graphics

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