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The discrete analog of ContourPlot3D is ListContourPlot3D.
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ListContourPlot3D[array] draws a contour plot of the values in array, a three-dimensional array of numbers representing the values of a function. Contours n is an option that draws contours at n equally spaced levels. The default is Contours 3. Contours {k1, k2,. . .} draws contours corresponding to function values k1, k2, . . . DataRange {{xmin, xmax}, {ymin, ymax}, {zmin, zmax}} labels the x, y, and z axes from xmin to xmax, ymin to ymax, and zmin to zmax, respectively. The default is DataRange Automatic, which assigns values starting with 1.
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This example generates a discrete set of values of the function f(x, y, z) = x2 + y2 + z2 and draws two contour plots of f(x, y, z) = k for k = .5 and k = 1.5. The surfaces generated are spheres, but the larger sphere is drawn in a box that is too small to contain it completely. The result is that the inner sphere is partially visible in the picture.
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list = Table[x2 + y2 + z2, {x, 1, 1, .25}, {y, 1, 1, .25}, {z, 1, 1, .25}]; ListContourPlot3D[list, DataRange {{ 1, 1}, { 1, 1}, { 1, 1}}, Contours {.5, 1.5}]
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5.14 Obtain a contour plot of f(x, y) = sin x + sin y on the square 4 x, y 4 and compare it to the three-dimensional graph of the function.
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SOLUTION (Graphs are placed side by side for easy comparison.)
Plot3D[Sin[x] + Sin[y], {x, 4 o, 4 o}, {y, 4 o, 4 o}] ContourPlot[Sin[x] + Sin[y], {x, 4 o, 4 o}, {y, 4 o, 4 o}]
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2 1 0 1 2 10 0 10 10
10 5 0 5 10
5.15 Compare a contour plot and a density plot for the function f(x, y) = sin xy over the rectangle x, y .
SOLUTION (Graphs are placed side by side for easy comparison.)
ContourPlot[Sin[x y], {x, o, o}, {y, o, o}] DensityPlot[Sin[x y], {x, o, o}, {y, o, o}]
3 2 1 0 1 2 3 3 2 1 0 1 2 3
3 2 1 0 1 2 3 3 2 1 0 1 2 3
5.16 Obtain a contour plot and a density plot of the discrete function Quotient[x, y] as x and y range from 1 to 10.
SOLUTION (Graphs are placed side by side for easy comparison.)
list = Table[Quotient[x, y], {x, 1, 10}, {y, 1, 10}]; ListContourPlot[list] ListDensityPlot[list]
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5.17 Let f(x, y, z) = 5x2 + 2y2 + z2. Draw the level surfaces f(x, y, z) = k for k = 1, 4, 9, 16, and 25. Sketch the surfaces only for y 0 so that all the surfaces will be visible.
SOLUTION
ContourPlot3D[5 x2 + 2 y2 + z2, {x, 5, 5}, {y, 0, 5}, {z, 5, 5}, Contours {1, 4, 9, 16, 25}]
4 2 0 5
5 5 0 5
5.18 Generate a 5 5 array of random integers between 1 and 10 and construct a three-dimensional list plot of these values.
SOLUTION
list = Table[Random[Integer, {0, 10}], {x, 1, 5}, {y, 1, 5}]; ListPlot3D[list]
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10 5 5 4 0 1 2 3 4 5 1 2 3
5.19 Draw hyperbolic cylinders x2 y2 = k, k = 0, 2, and 5, by computing f(x, y, z) = x2 y2 at integer values between 5 and 5 for each variable and using ListContourPlot3D.
SOLUTION
We use integer values of x, y, and z to construct our list. list = Table[x2 y2, {z, 5, 5}, {y, 5, 5}, {x, 5, 5}]; ListContourPlot3D[list, Contours {0, 2, 5}, DataRange {{ 5, 5}, { 5, 5}, { 5, 5}}]
5 5
5 5 0 5
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