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Visualizing 3d Surfaces

The graph of a function z=f(x,y) of two independent variables x and y is a surface in the 3d space. Sometimes it is difficult or impossible to express z explicitly in terms of x and y. In such cases a surface is given in the implicit form F(x,y,z)=0. Examples for an explicitly given functions are for implicitly given surfaces or .

Maple provides the plot3d and implicitplot3d commands to display 3d wiremesh plots of surfaces.

```  > with(plots):
> plot3d(x^2/4+y^2/9,x=-4..4,y=-4..4);
> implicitplot3d(x^2/4+y^2/9-z^2/(1/4)=-1,x=-8..8,y=-8..8,z=-2..2,
orientation=[45,80],axes=boxed,style=patchcontour);
```

The secret of getting nice, informative plots is proper scaling. Often you will have to experiment with various x,y and z domains. To get a better, more informative view, you can rotate the plots by holding down the left mouse button and dragging the box. To redisplay the plot press the middle mouse button.

It is not necessary to enter the ``orientation", ``axes", ``style" and ``color" options from the command line. They can be selected from the menus of the plot window. Experiment with various options.

Below you will find the commands to display the quadratic surfaces from p.645 of the textbook. Note that there are various ways to define and plot the surfaces. You will use similar commands in your exercises. To make sure that you know what functions you plot, write down with pencil on paper the formulas for the functions in the usual non-Maple form

```  > implicitplot3d(x^2/4+y^2/9-z^2=0,x=-3..3,y=-3..3,z=-1..1,
axes=boxed,style=patchcontour);

> implicitplot3d(x^2/4+y^2/16-z^2/16-z^2=1,x=-5..5,y=-5..5,z=-1..1,
orientation=[45,80],axes=boxed,style=patchcontour);

> implicitplot3d(x^2/16+y^2/9+z^2/4=1,x=-4..4,y=-4..4,z=-4..4,
axes=boxed,style=patch);

> implicitplot3d(x^2+y^2+z^2=1,x=-1..1,y=-1..1,z=-1..1,axes=boxed,
style=patchcontour);

> Sphere:=(x,y,z)->x^2+y^2+z^2=1;

> implicitplot3d(Sphere(x,y,z),x=-1..1,y=-1..1,z=-1..1,
style=patchcontour);