Functions 

Several of the surfaces in our catalog are functions of two variables. We start with elliptical paraboloids. 

>	f := x^2+y^2; plot3d(f, x = -3 .. 3, y = -3 .. 3, axes = normal,  style=patchnogrid, scaling = constrained);

 

Note that our surface looks a little different from the graph usually shown in calculus texts, due to the way we have restricted the x and y values.  If we choose, we can match the z-range to the x and y values to make the graph appear with a circular top. This is done by using the 'view' option as shown below.  

Note that below I've also removed the 'style' option. This changes the way the surface is drawn. You can also do this by clicking on the graph and looking at the new options that appear in the tool bar. The buttons which appear allow you to change the appearance of the surface, change the axes, change the scaling, and also change the orientation and magnification level of the graph. You should experiment with these buttons. 

>	plot3d(f, x = -3 .. 3, y = -3 .. 3, view = 0 .. 9, axes = normal,  scaling = constrained);

 

Next we graph a hyperbolic paraboloid. 

>	g := -x^2+y^2/4; plot3d(g, x = -3 .. 3, y = -3 .. 3, axes = normal);

 

Notice the cross-sections are parabolas opening up when x is fixed and opening down when y is fixed. Click on the contour view to see why this surface is a *hyperbolic* paraboloid (rotate to look down on the graph). 

If one of the variables is missing from an equation, we get a cylinder.  Here is an example of a parabolic cylinder. 

>	h := y^2; plot3d(h, x = -3 .. 3, y = -3 .. 3, style = patchnogrid, axes = normal);

 

Graph a linear function. Play with different values of A, B and F. 

>	A := 1; B := 1; F := 0; k := A*x+B*y+F; plot3d(k, x = -3 .. 3, y = -3 .. 3, axes = normal);

 

Implicit plots 

Other surfaces cannot be expressed as the graph of a (single) function of two variables.  For these we can use the implicit plot command. 

We start with a sphere. Notice that the radius is set inside the 'implicitplot3d' command. This is just by choice. 'm' could just as well have been defined to be the equation itself, i.e. 'm:=x^2+y^2+z^2=1'. 

>	m := x^2+y^2+z^2; implicitplot3d(m = 1, x = -1.5 .. 1.5, y = -1.5 .. 1.5, z = -1.5 .. 1.5, axes = normal, scaling = constrained, grid = [20, 20, 20]);

 

The next surface we sketch is a hyperboloid of one sheet. Change the values of a, b and  c to see different versions. 

>	a := 1; b := 1; c := 1; n := x^2/a^2+y^2/b^2-z^2/c^2; implicitplot3d(n = 1, x = -1.5 .. 1.5, y = -1.5 .. 1.5, z = -1.5 .. 1.5, axes = normal, scaling = constrained, style = patchnogrid);

 

We move on to hyperboloids of two sheets. 

>	a := 1; b := 1; c := 1; n := x^2/a^2+y^2/b^2-z^2/c^2; implicitplot3d(n = -1, x = -3 .. 3, y = -3 .. 3, z = -3 .. 3, axes = normal, scaling = constrained, style = patchnogrid);

 

In the degenerate case (right side of the equation = 0), we obtain at a pair of cones, the geometric foundation for the conic sections.  

>	a := 1; b := 1; c := 1; n := x^2/a^2+y^2/b^2-z^2/c^2; implicitplot3d(n = 0, x = -3 .. 3, y = -3 .. 3, z = -3 .. 3, axes = normal, scaling = constrained, style = patchnogrid, grid = [30, 30, 30]);

 

Planes can also be graphed using an implicit plot. 

>	a := 1; b := 1; c := 1; p := a*x+b*y+c*z; implicitplot3d(p = 0, x = -3 .. 3, y = -3 .. 3, z = -3 .. 3, axes = normal, scaling = constrained);

 

Finally, we graph some cylinders implicitly. 

>	q := x^2+y^2; implicitplot3d(q = 4, x = -3 .. 3, y = -3 .. 3, z = -3 .. 3, axes = normal, scaling = constrained, style = patchnogrid, grid = [20, 20, 20]);

 

>	r := y-x^2; implicitplot3d(r = 0, x = -3 .. 3, y = -3 .. 3, z = -3 .. 3, axes = normal, scaling = constrained, style = patchnogrid);

 

Exercises 

Do the following problems from your textbook. 

11.6 #'s 31-38 

In the instructions, the author suggests you solve for z. Don't do that. Instead, use the implicitplot command if necessary.