# Parametric equations for a hyperbolic paraboloid

1. Sep 15, 2005

### Townsend

I need to find a set of parametric equations for a hyperbolic paraboloid. The hint is that I should review some trigonometric identities that involve differences of squares that equal 1.

The equation is:
$$\frac{y^2}{2}- \frac{x^2}{4} - \frac{z^2}{9} = 1$$

And what I have is

$$y= \sqrt{2}*sec(t)*sec(s)$$
$$x=2*tan(t)*sec(s)$$
$$z=3*tan(s)$$

I am then suppose to write the maple code and send it to my instructor. The problem is that when I do the plot3d with those equations I get a strange looking thing that looks nothing like what a hyperbolic paraboloid should look like. I did the implicitplot3d for the equation to see what it should look like so I know I am way off.

Can anyone offer me any hints?

Thanks

2. Sep 15, 2005

### Townsend

My equations might be right but I have the wrong domain for s and t. This is the code
Code (Text):

with(plots);

plot3d([2*tan(t)*sec(s), sqrt(2)*sec(t)*sec(s), 3*tan(s)], t=-4*Pi..4*Pi, s=-4*Pi..4*Pi);

And compare that with
Code (Text):

restart; with(plots);

implicitplot3d(y^2/2-x^2/4-z^2/9=1, x=-10..10, y=-10..10, z=-10..10, grid=[20,20,20]);

3. Sep 16, 2005

### Townsend

bump...

Can anyone offer me any ideas?

4. Sep 16, 2005

### Tom Mattson

Staff Emeritus
Your equations are correct. And you're right about having a problem with your domains. The sec and tan functions both have multiple vertical asymptotes in the interval $[-4\pi,4\pi]$, so you'll have to do something about that.

5. Sep 16, 2005

### Townsend

Thanks Mr. Mattson

I finally got the graphs to look reasonable. You're right about the asymptotes being the problem of course. If I go from -Pi/4 to Pi/4 for both s and t, things look good. To get both sheets I used the following code

Code (Text):

with(plots);

surface1:=plot3d([2*tan(t)*sec(s), sqrt(2)*sec(t)*sec(s), 3*tan(s)], t=-P/4i..Pi/4, s=-Pi/4..Pi/4):
surface1:=plot3d([2*tan(t)*sec(s), -sqrt(2)*sec(t)*sec(s), 3*tan(s)], t=-P/4i..Pi/4, s=-Pi/4..Pi/4):
display(surface1, surface2);

But thanks again Tom...