A Catalog of Functions and Surfaces Worksheet by Russell Blyth - blythrd@slu.edu QyZJKHJlc3RhcnRHJSpwcm90ZWN0ZWRHIiIiLUkld2l0aEc2IjYjSSZwbG90c0c2JEYkSShfc3lzbGliR0YoISIi

Several of the surfaces in our catalog are functions of two variables. We start with elliptical paraboloids. a := 1; b := 1; f := x^2/a^2+y^2/b^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. plot3d(f, x = -3 .. 3, y = -3 .. 3, view = 0 .. 9, axes = normal, style = patchnogrid, scaling = constrained);

1. Set a = 1 and b = 2 and then set the x-range, the y-range and view so that the top edge of the graphed part of the surface is a familiar smooth curve. Click on the contour view to see why this surface is an *elliptical* paraboloid. a := 1; b := 2; f := x^2/a^2+y^2/b^2; plot3d(f, x = -3 .. 3, y = -3 .. 3, axes = normal, style = patchnogrid, scaling = constrained);Qyg+SSJhRzYiIiIiRiY+SSJiR0YlIiIjRiYtSSdwbG90M2RHRiU2KEkiZkdGJS9JInhHRiU7ISIkIiIkL0kieUdGJUYwL0klYXhlc0dGJUknbm9ybWFsRyUqcHJvdGVjdGVkRy9JJnN0eWxlR0YlSSxwYXRjaG5vZ3JpZEdGJS9JKHNjYWxpbmdHRiVJLGNvbnN0cmFpbmVkR0YlRiY=

Next we graph a hyperbolic paraboloid. a := 1; b := 1; g := -x^2/a^2+y^2/b^2; plot3d(g, x = -3 .. 3, y = -3 .. 3, style = patchnogrid, 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 we 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);

2. Graph the linear function which has a slope of -2 in the x-direction (for fixed values of y) and a slope of 0.75 in the y-direction (for fixed values of x) and which passes through the point (1, -3, 2). You may find the "scaling=constrained" option useful.

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 the ellipsoid, of which a sphere is a special case. a := 1; b := 1; c := 1; m := x^2/a^2+y^2/b^2+z^2/c^2; implicitplot3d(m = 1, x = -1.5 .. 1.5, y = -1.5 .. 1.5, z = -1.5 .. 1.5, axes = normal, scaling = constrained, style = patchnogrid, grid = [20, 20, 20]);

3. Change the values of a, b and c used in the ellipsoid equation until you produce a football-shaped surface. You may want to adjust the x,y and z ranges as well. a := 1; b := 1; c := 1; m := x^2/a^2+y^2/b^2+z^2/c^2; implicitplot3d(m = 1, x = -1.5 .. 1.5, y = -1.5 .. 1.5, z = -1.5 .. 1.5, axes = normal, scaling = constrained, style = patchnogrid, color = brown, 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);

4. Graph a hyperboloid of one sheet which extends along the x-axis instead of along the z-axis. Once graphed, click on the contour button to see where the name "hyperboloid" comes from.

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]);

5. Even with the grid option set to a higher than default value, the point at (0,0,0) is still difficult for Maple to plot. Plot the cones lying on their sides to see if that helps.

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);

6. Graph the plane with x-intercept at -2, y intercept at 3 and z-intercept at -4.

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);