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Make Code128 In None Using Barcode encoder for Online Control to generate, create Code128 image in Online applications. GTIN  12 Decoder In VB.NET Using Barcode recognizer for Visual Studio .NET Control to read, scan read, scan image in .NET framework applications. Animation effects can be produced quickly and easily through the use of the Animate command. This command displays several different graphics images rapidly in succession, producing the illusion of movement. The form of the command is Animate[expression, {k, m, n, i}] where expression is any Mathematica command with parameter k which varies from m to n in increments of i (optional; if omitted, i varies continuously from m to n). The following example gives an interesting animated description of the behavior of the odd powers of xn as n gets larger. EXAMPLE 40
Animate[Plot[xk, {x, 1, 1}, PlotRange { 1, 1}, Ticks False], {k, 1, 19, 2}] The speed of the animation and the direction are easily controlled by clicking on the , , and buttons. The animation can be paused, using the button. To allow the user more control over the animation, the Manipulate command can be used. Manipulate works very much the same way as Animate except it allows the user to control the parameter directly with a slider. Manipulate[expression, {k, m, n, i}] EXAMPLE 41
Manipulate[Plot[xk, {x, 1, 1}, PlotRange { 1, 1}, Ticks False] {k, 1, 19, 2}] , Click here for animation controls Click here for an options menu
A convenient way of controlling expressions involving integer parameters is by clicking on radio buttons. This can be accomplished with the option ControlType RadioButton. TwoDimensional Graphics
EXAMPLE 42
Manipulate[Plot[xk, {x, 1, 1}, PlotRange { 1, 1}, Ticks False], {k, 1, 19, 2}, ControlType RadioButton] expression may involve two or more parameters. In this case the form of the command is
Animate[expression,{k1, m1, n1, i1}, {k2, m2, n2, i2},...] Manipulate[expression,{k1, m1, n1, i1}, {k2, m2, n2, i2},...] Each parameter can be controlled independently (speed, direction, pause). EXAMPLE 43
Animate[Plot[a Sin[b x], {x, 0, 2 o}, PlotRange { 10, 10}] , {a, 0, 10}, {b, 0, 10}] 10 5 1 5 10 EXAMPLE 44 This animation shows a circle of varying radius whose center varies from ( 1, 1) to (1, 1). Pause each variable (x, y, r) to see the effect. Animate[Graphics[Circle[{Sin[x], Cos[y]}, r], Axes True, PlotRange {{ 2, 2}, { 2, 2}}], {x, 0, 2 o}, {y, 0, 2 o}, {r, 0, 1}] TwoDimensional Graphics
x y r
Animate and Manipulate are not limited to the presentation of graphics. We will use these commands in other contexts in later chapters. SOLVED PROBLEMS
4.23 Construct an animation of the Spiral of Archimedes, r = q as q varies from 8 to 10 .
SOLUTION
Animate[PolarPlot[ , { , 0, 8o + e} Ticks False, , PlotRange {{ 10 o, 10 o} { 10 o, 10 o}}] {e, 0, 2 o}] , , TwoDimensional Graphics
4.24 Use Manipulate to simulate a point rolling along a sine curve from 0 to 2 .
SOLUTION
First we construct the sine curve. sincurve = Plot[Sin[x], {x, 0, 2 o}, Ticks False] Now we animate the sequence of points as red disks of radii 0.05. Manipulate[Show[sincurve, Graphics[{Red, Disk[{x, Sin[x]}, 0.05]}], PlotRange {{0, 2 o}, { 1, 1}}, AspectRatio Automatic], {x, 0, 2 o}]. Move the slider to control the movement of the disk.
C HA PTE R 5
ThreeDimensional Graphics
5.1 Plotting Functions of Two Variables
A function of two variables may be viewed as a surface in threedimensional space. The simplest command for plotting a surface is Plot3D. Plot3D[f[x, y], {x, xmin, xmax}, {y, ymin, ymax}] plots a threedimensional graph of the function f[x, y] above the rectangle xmin x xmax, ymin y ymax. Plot3D[{f1[x, y], f2[x, y],...}, {x, xmin, xmax}, {y, ymin, ymax}] plots several surfaces on one set of axes. Mathematica s default axis orientation is as shown in the figure to the right. This is somewhat different from what appears in many calculus textbooks. EXAMPLE 1 z
2 1 y 0 1 2 Plot3D[Sin[x y], {x, o, o}, {y, o, o}] x 1.0 0.5 0.0 0.5 1.0 0 2 0 2 2 The option PlotPoints specifies the number of points to be used in each direction to produce the graph. Unlike twodimensional graphics, the default for a threedimensional plot is PlotPoints 15. This often leads to graphs with ragged surfaces. Increasing PlotPoints will alleviate this condition. Plotpoints n specifies that n initial sample points should be used in each direction. Additional points are selected by adaptive algorithms. PlotPoints {nx, ny} specifies that nx and ny initial sample points are to be used along the xaxis and yaxis, respectively.

