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Name of the animation parameter

`ParameterBegin` , `ParameterEnd` , `ParameterName` | Optional | MuPAD^{®} 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 = a`

,
and _{min}`ParameterEnd = a`

in
the definition of the object._{max}

The attribute `ParamterRange = `a_{min}`..`a_{max}``

is
equivalent to setting both `ParameterBegin = a`

and _{min}```
ParameterEnd
= a
```

._{max}

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
= a
```

, _{min}`ParameterEnd = a`

._{max}

The attribute `ParamterRange = `a_{min}`..`a_{max}``

serves
as a short cut for setting both `ParameterBegin = a`

and _{min}```
ParameterEnd
= a
```

._{max}

The values `a`

and _{min}`a`

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 _{max}`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

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

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.

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

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))

We define an animated 2D function plot:

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

We swap the role of the independent variable * x* and
the animation parameter

[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"

plot(f)

delete f:

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

`eigenvals`

.
This procedure uses `option remember`

, because it is called
thrice with the same arguments by the procedures `f`_{1}

, `f`_{2}

, `f`_{3}

that
produce the smallest, the middle, and the largest eigenvalue, respectively,
as functions of the parameters eigenvals := proc(x, a) option remember; local A; begin A:= matrix([[1, a, x ], [a, 2, a*x], [x, a*x, 3 ]]): sort(numeric::eigenvalues(A)): end_proc: 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:

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