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This example shows the graph of the hyperbolic paraboloid z = x 2 y 2 from different viewpoints. (Graphs are grouped together for easy comparison.) Plot3D[x2 y2, {x, 5, 5}, {y, 5, 5}, BoxRatios 1] Plot3D[x2 y2, {x, 5, 5}, {y, 5, 5}, BoxRatios 1, ViewPoint {2, 2, 2}] Plot3D[x2 y2, {x, 5, 5}, {y, 5, 5}, BoxRatios 1, ViewPoint {1.5, 2.6, 1.5}] Plot3D[x2 y2, {x, 5, 5}, {y, 5, 5}, BoxRatios 1, ViewPoint Front] Plot3D[x2 y2, {x, 5, 5}, {y, 5, 5}, BoxRatios 1, ViewPoint Top] Plot3D[x2 y2, {x, 5, 5}, {y, 5, 5}, BoxRatios 1, ViewPoint Right]
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Three-Dimensional Graphics
Once plotted, a three-dimensional object can be rotated three-dimensionally by clicking on the object and dragging the mouse. Dragging with the [CTRL] , [ALT] , or [OPTION] key depressed allows you to zoom in or out, and dragging with the [SHIFT] key depressed allows the graph to be moved horizontally or vertically in your note-
book. Clicking on the object produces a rectangular boundary. Dragging a corner of this rectangle allows you to resize the object; dragging near but inside the rectangle allows you to rotate the object two-dimensionally.
Curves and surfaces defined parametrically can be plotted using ParametricPlot3D.
ParametricPlot3D[{x[t], y[t], z[t]}, {t, tmin, tmax}] plots a space curve in three dimensions for tmin t tmax. ParametricPlot3D[{x[s, t], y[s, t], z[s, t]}, {s, smin, smax}, {t, tmin, tmax}] plots a surface in three dimensions.
EXAMPLE 6
ParametricPlot3D[{Cos[t], Sin[t], t/4}, {t, 0, 4o}]
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EXAMPLE 7
ParametricPlot3D[{Sin[s + t], Cos[s + t], s}, {s, 2, 2}, {t, 2, 2}]
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Three-Dimensional Graphics
Plot3D allows you to plot surfaces expressed by equations in rectangular coordinates. Special surfaces, called surfaces of revolution, can be drawn using the command RevolutionPlot3D. (For additional options that provide more flexibility, please see SurfaceOfRevolution, which is discussed in Section 5.3.)
RevolutionPlot3D[f[x], {x, xmin, xmax}] plots the surface generated by rotating the curve z = f(x) , xmin x xmax, completely around the z-axis. RevolutionPlot3D[f[x], {x, xmin, xmax}, {p, pmin, pmax}] plots the surface generated by rotating the curve z = f(x) , xmin x xmax, around the z-axis for min max where is the angle measured counterclockwise from the positive x-axis. RevolutionPlot3D[{f[t],g[t]}, {t, tmin, tmax}] generates a plot of the surface generated by rotating the curve x = f(t), z = g(t), tmin t tmax, completely around the z-axis. RevolutionPlot3D[{f[t],g[t]}, {t, tmin, tmax},{p, pmin, pmax}] generates a plot of the surface generated by the curve x = f(t), z = g(t), tmin t tmax, around the z-axis for min max where is the angle measured counterclockwise from the positive x-axis.
EXAMPLE 8 Sketch the surface of revolution generated when the curve z =
x , 0 x 4, is rotated about the z-axis. First we draw the two-dimensional generating curve and then the corresponding surface of revolution. (Graphs are placed side by side for easy comparison.) Plot[ x , {x, 0, 4}, AspectRatio 1, AxesLabel {"x", "z"}] RevolutionPlot3D[ x , {x,0,4}, BoxRatios 1, ViewPoint {1, 5, 2}, AxesLabel {"x", "y", "z"}]
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Cylindrical and spherical coordinate systems are useful for solving problems involving cylinders, spheres, and cones. Point P has cylindrical coordinates (r, q, z) where r and q are the polar coordinates of the projection of P in the x y plane. Since the distance from P to the z-axis is r, the surface z = z(r, q ) is a surface of revolution.
RevolutionPlot3D[z[r, p], {r, rmin, rmax}] generates a plot of the surface z = z(r, q ) described in cylindrical coordinates for rmin r rmax. RevolutionPlot3D [z[r, p], {r, rmin, rmax}, {p, pmin, pmax}] generates a plot of the surface z = z(r, q ) for rmin r rmax, min max.
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