ParameterName, ParameterBegin, ParameterEnd, ParameterRange

Name of the animation parameter

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Value Summary

ParameterBegin, ParameterEnd, ParameterNameOptionalMuPAD® expression
ParameterRange[ParameterBegin .. ParameterEnd]Range of arithmetical expressions


Typically, animations are triggered by passing an equation of the form a = `a_{min}`..`a_{max}` in the definition of an object.

This is equivalent to passing the attributes ParameterName = a, ParameterBegin = amin, and ParameterEnd = amax in the definition of the object.

The attribute ParamterRange = `a_{min}`..`a_{max}` is equivalent to setting both ParameterBegin = amin and ParameterEnd = amax.

Animations are defined object by object, not frame by frame.

In most cases, the user will define animations by passing an equation of the form a = `a_{min}`..`a_{max}` in the definition of an object. Any equation of this form that is not essential for the definition of a static version of the object is interpreted as an animation parameter and an animation range.

Passing such an equation is equivalent to setting the three attributes

ParameterName = a, ParameterBegin = amin, ParameterEnd = amax.

The attribute ParamterRange = `a_{min}`..`a_{max}` serves as a short cut for setting both ParameterBegin = amin and ParameterEnd = amax.

The values amin and amax are the parameter values at the beginning and the end of the real time range in which an object is animated. This time range is set by the attributes TimeBegin and TimeEnd, respectively.

The parameter range `a_{min}`..`a_{max}` is mapped linearly to this time interval.

The name of the animation parameter may be an identifier or an indexed identifier. This parameter is a `global variable' that may be present in other quantities or attributes defining the object.

The definition of an object may involve procedures rather than symbolic expressions. E.g., a 2D function plot may be defined by plot::Function2d(f, x = x_0..x_1), where f is a procedure accepting one numerical argument x from the plot range between x0 und x1.

In an animated version plot::Function2d(f, x = x_0..x_1, a = `a_{min}`..`a_{max}`), the function f will be called with two arguments x and a. Thus, f may be defined as a function accepting two parameters x, a or as a function with one parameter x, using the animation parameter a as a global variable.

Each animated object has its own animation parameter and range `a_{min}`..`a_{max}`. It is not necessary that several animated objects in a scene use the same parameter name. It is not used to synchronize the animations.

The synchronization is determined uniquely by the linear correspondence between the animation range `a_{min}`..`a_{max}` and the real time span of the animation set by the attributes TimeBegin and TimeEnd of the object.


Example 1

The definition of a static 2D function plot involves the specification of one range (for the x variable):

plot(plot::Function2d(sin(x), x = 0 .. 2*PI))

When a “surplus equation” a = `a_{min}`..`a_{max}` is passed, this is interpreted as a call to animate the function. The animation parameter may turn up in the expression defining the function:

plot(plot::Function2d(sin(x + a), x = a .. 2*PI, a = 0..PI))

A static function plot in 3D requires two ranges (for the x and the y variable):

plot(plot::Function3d(sin(x)*sin(y), x = 0 .. PI, y = 0..PI))

Now, a third equation a = `a_{min}`..`a_{max}` triggers an animation:

plot(plot::Function3d(sin(x + a)*sin(y - a), x = 0 .. PI, 
                      y = 0..PI, a = 0..PI))

Example 2

We define an animated 2D function plot:

f := plot::Function2d(x^3 + a*x, x = -1..1, a = 0..2):

We swap the role of the independent variable x and the animation parameter a:

[f::XName, f::ParameterName] := [f::ParameterName, f::XName]:
[f::XRange, f::ParameterRange] := 
                              [f::ParameterRange, f::XRange]:

The function now is drawn as a function of a for various values of the “time” x:


delete f:

Example 3

We demonstrate the use of procedures in the definition of animated functions.

We wish to plot the eigenvalues of a matrix that depends on two parameters x and a. The eigenvalues are computed numerically in the procedure eigenvals. This procedure uses option remember, because it is called thrice with the same arguments by the procedures f1, f2, f3 that produce the smallest, the middle, and the largest eigenvalue, respectively, as functions of the parameters x and a:

eigenvals := 
   proc(x, a) 
     option remember;
     local A;
     A:= matrix([[1,  a,  x ],
                 [a,  2, a*x],
                 [x, a*x, 3 ]]):
f1:= (x, a) -> eigenvals(x, a)[1]:
f2:= (x, a) -> eigenvals(x, a)[2]:
f3:= (x, a) -> eigenvals(x, a)[3]:
plot(plot::Function2d(f1, x = -2..2, a = 0..2,
                      Color = RGB::Red),
     plot::Function2d(f2, x = -2..2, a = 0..2,
                      Color = RGB::Green),
     plot::Function2d(f3, x = -2..2, a = 0..2,
                      Color = RGB::Blue)):

delete eigenvals, f1, f2, f3: