       RE: Graphing Hyperboloids

• To: mathgroup at smc.vnet.net
• Subject: [mg25590] RE: [mg25582] Graphing Hyperboloids
• From: "David Park" <djmp at earthlink.net>
• Date: Mon, 9 Oct 2000 21:43:30 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

```Andy,

Here is your equation for a hyperboloid of one sheet.

eqn = (x^2/16) + (y^2/4) - z^2 == 1;

To make a nice plot, it is much easier to work with cylindrical coordinates.
This converts the equation to cylindrical coordinates and solves for r as a
function of z and t (theta).

eqn /. {x -> r*Cos[t], y -> r*Sin[t]}
rsols = Simplify[Solve[%, r]]
-z^2 + 1/16*r^2*Cos[t]^2 + 1/4*r^2*Sin[t]^2 == 1
{{r -> -((4*Sqrt*Sqrt[1 + z^2])/Sqrt[5 - 3*Cos[2*t]])},
{r -> (4*Sqrt*Sqrt[1 + z^2])/Sqrt[5 - 3*Cos[2*t]]}}

We want the radius to be positive, so we use the second solution. This
parametrizes the sheet.

sheet1[z_, t_] = {r Cos[t], r Sin[t], z} /. rsols[]
{(4*Sqrt*Sqrt[1 + z^2]*Cos[t])/Sqrt[5 - 3*Cos[2*t]],
(4*Sqrt*Sqrt[1 + z^2]*Sin[t])/Sqrt[5 - 3*Cos[2*t]], z}

This plots it.

ParametricPlot3D[Evaluate[sheet1[z, t]], {z, -3, 3}, {t, 0, 2*Pi},
PlotPoints -> {21, 41}, ImageSize -> 450,
ViewPoint -> {1.598, -2.555, 1.54}];

You could also use ImplicitPlot3D, but the above is much faster and nicer.

David Park

> -----Original Message-----
> From: Andy Sokol [mailto:asokol at fit.edu]
To: mathgroup at smc.vnet.net
>
> Hello at Math Group!
>
> Unfortunately, I am not very skilled with Mathematica, and have been
> assigned a few problems for Calculus 3.  I'm been working on these for
> hours and I'm just absolutely stumped on the last two.  None of my
> classmates have been able to solve it either, so I was searching for
> something to help me and I found you!  This assignment is due tomorrow,
> and so I guess I'm kind of just keeping my fingers crossed that you guys
> may be reading this at 12:30 am.
>
> I really hope this is like a super-easy problem for you...
>
> The problems are:
>
> Graph the hyperboloid of one-sheet:  (x^2 / 16) + (y^2 / 4) - z^2 = 1
>
> Graph the hyperboloid of "one-sheet" (it's written as one sheet on the
> page, but based on the equation I believe that's just a typo and it's of
> two-sheets):  (x^2 / 16) - (y^2 / 4) - z^2 = 1
>
>
>
> Andy Sokol
> Florida Institute of Technology
> asokol at fit.edu
>

```

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