**1. Plotting equations that involve ***z ***just to the first power**. If the variable *z* occurs in an equation *F(x, y, z) = *0 just to the first power, then it is easy to solve for *z* as an explicit function of the variables *x* and *y*. You can then plot the graph by using *Mathematica*'s built-in Plot3D command.

**Example 1**. **Paraboloids**. The following routine plots either type of paraboloid: elliptical or hyperbolic. To use it, solve the given equation for *z* and enter the resulting expression in place of the formula in *x* and *y* within the Plot3D command. Altering the plotting window may produce a better image for a particular equation. To do that, change the preliminary values the routine assigns to *a*, *b*, and *c*. As usual, execute the routine by hittting the Enter key after you place the cursor at end of the last line of the blue code.

**a = 2;**

b = 4;

c = 1;

surf = Graphics3D[

Plot3D[c^2(x^2/a^2 - y^2/b^2), {x, -a, a},

{y, -b, b}, AxesLabel -> {x, y, z}]

];

coords = Graphics3D[{

{RGBColor[1, 0, 0],

Line[{{-a, 0, 0}, {a+1, 0, 0}}],

Text["x", {a + 1.25, 0, 0}]},

{RGBColor[1, 0, 0],

Line[{{0, -b, 0},{0, b+2, 0}}],

Text["y", {0, b+2.5, 0}]},

{RGBColor[1, 0, 0],

Line[{{0, 0, -c},{0, 0, c+2}}],

Text["z", {0, 0, c+1}]} }

];

Show[surf, coords, AxesLabel -> {x, y, z}]

**2. Plotting equations that involve **.** **Since the standard graphics package ContourPlot3D by default plots a single level surface *f(x, y, z) = *0, it is general enough to provide plots of any quadric surface. (However, for an equation in which the variable *z* occurs just to the first power, it is quicker and easier to use the method of Section 1 above of solving for *z*.) For equations that contain , the following routine gives a reasonable plot, although the execution is noticeably slower than for routines that use the Plot3D command.

**Example 2**. The following routine plots cones, cylinders, ellipsoids, or hyperboloids of one or two sheets. You will want to alter the plotting window and the coordinate-axis plot to produce a good image for a given equation. That led to the choices for the illustrated hyperboloid of two sheets: -- * *+ */*4 -- */*9 = 1.

Needs["Graphics`ContourPlot3D`"]

a = 1;

b = 2;

c = 3;

surfbr = ContourPlot3D[

-x^2/a^2 + y^2/b^2 - z^2/c^2 - 1,

{x, -3 a, 3 a}, {y, -3 b, 3 b}, {z, -3 c, 3 c},

Axes -> True, AxesLabel -> {x, y, z}];

coords = Graphics3D[{

{RGBColor[1, 0, 0],

Line[{{-2 a, 0, 0}, {5 a + 1, 0, 0}}],

Text["x", {5 a + 1.5, 0, 0}]},

{RGBColor[1, 0, 0],

Line[{{0, -2 b, 0}, {0, 6 b, 0}}],

Text["y", {0, 6 b + 1, 0}]},

{RGBColor[1, 0, 0],

Line[{{0, 0, -2 c}, {0, 0, 2 c + 1}}],

Text["z", {0, 0, 2 c + 2}]}

}

];

Show[surfbr, coords]

**3. Printing a graphical image from **** Mathematica**. To print an image like the above plot of a hyperboloid of two sheets,

• click on the first vertical bar to the right of the plot (to select the image)

• pull down the

Converted by