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ParametricPlot3D[{Cos[5t], Sin[3t], Sin[t]}, {t, 0, 2 o}] ParametricPlot3D[{Cos[3t], Sin[3t], Sin[t]}, {t, 0, 2 o}] ParametricPlot3D[{Cos[2t], Sin[5t], Sin[2t]}, {t, 0, 2 o}]
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x = (4 + sin s) cos t 5.8 Sketch the torus defined by y = (4 + sin s)sin t z = cos s
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x[t_] = (4 + Sin[s])Cos[t]; y[t_] = (4 + Sin[s])Sin[t]; z[t_] = Cos[s]; g1 = ParametricPlot3D[{x[t], y[t], z[t]}, {s, 0, 2 o}, {t, 0, 2o}, Mesh False]
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5 1.0 0.5 0.0 0.5 1.0 5
x = (4 + sin 20 t ) cos t 5.9 (Continuation.) Sketch the space curve y = (4 + sin 20 t )sin t z = cos 20 t
SOLUTION
0 t 2
This curve is called a toroidal spiral since it lies on the surface of a torus (let s = 20t).
x[t_] = (4 + Sin[20t])Cos[t]; y[t_] = (4 + Sin[20t])Sin[t]; z[t_] = Cos[20t]; g2 = ParametricPlot3D[{x[t], y[t], z[t]}, {t, 0, 2o}]
5 1.0 0.5 0.0 0.5 1.0 5
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5.10 (Continuation.) Sketch the torus and the toroidal spiral on the same set of axes.
SOLUTION
Show[g1, g2]
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5.11 Sketch the graph of a ribbon one unit wide having the shape of a sine curve from 0 to 4 .
SOLUTION
x = t We can represent this surface parametrically: y = s z = sin t
0 s 1, 0 t 4p.
ParametricPlot3D[{t, s, Sin[t]}, {s, 0, 1}, {t, 0, 4 o}, Axes False]
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5.12 Draw the ice cream cone formed by the cone z = 3 x 2 + y 2 and the upper half of the sphere x 2 + y 2 + (z 9)2 = 9. Use cylindrical coordinates.
SOLUTION
In cylindrical coordinates the cone has the equation z = 3 r and the hemisphere has the equation z = 9 + 9 r 2 . cone = RevolutionPlot3D[3 r, {r, 0, 3}, BoxRatios 1]; hemisphere = RevolutionPlot3D[ 9+ 9 r 2 ], {r, 0, 3}]; Show[cone, hemisphere, PlotRange All, BoxRatios {1,1,2}]
2 2 0 2 0 2
5.13 Sketch the graph of the following surface given in spherical coordinates: = 1 + sin 4 sin , 0 2 , 0
SOLUTION
SphericalPlot3D[1 + Sin[4 p]Sin[e], {e, 0, o}, {p, 0, 2o}]
1 0 1
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5.2 Other Graphics Commands
A level curve of a function of two variables, f ( x, y), is a two-dimensional graph of the equation f ( x, y) = k for some fixed value of k. A contour plot is a collection of level curves drawn on the same set of axes. The Mathematica command ContourPlot draws contour plots of functions of two variables. The contours join points on the surface having the same height. The default is to have contours corresponding to a sequence of equally spaced values of the function.
ContourPlot[f[x, y], {x, xmin, xmax}, {y, ymin, ymax}] draws a contour plot of f(x, y) in a rectangle determined by xmin, xmax, ymin, and ymax.
Contour plots produced by Mathematica are drawn shaded, in such a way that regions with higher values of f(x, y) are drawn lighter. As with all Mathematica graphics commands, options allow you to control the appearance of the graph.
Contours n allows you to determine the number of contours to be drawn. The default is ten equally spaced curves. Contours {k1, k2,...} draws contours corresponding to function values k1, k2, . . . ContourShading False turns off shading. This option is particularly useful if your monitor or printer does not handle grayscales well. ContourLines False eliminates the lines that separate the shaded contours. PlotPoints n controls how many points will be used in each direction in an adaptive algorithm to plot each curve. The default is 15. (The default for two-dimensional graphics is 25.)
A complete list of options and their default values can be obtained using the command Options[ContourPlot].
EXAMPLE 11 Obtain contour plots of the paraboloid z = x2 + y2. Note that the level curves are all circles x2 + y2 = k. (Plots are placed side by side for easy comparison.)
ContourPlot[x2 + y2, {x, 10, 10}, {y, 10, 10}] ContourPlot[x2 + y2, {x, 10, 10}, {y, 10, 10}, ContourLines False] ContourPlot[x2 + y2, {x, 10, 10}, {y, 10, 10}, ContourShading False]
10 10
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A density plot shows the values of a function at a regular array of points. Lighter regions have higher values.
DensityPlot[f[x, y], {x, xmin, xmax}, {y, ymin, ymax}] draws a density plot of f(x, y) in a rectangle determined by xmin, xmax, ymin, and ymax.
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