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[2]*Sqrt[1 + z^2])/Sqrt[5 - 3*Cos[2*t]])}, {r -> (4*Sqrt[2]*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[[2]] {(4*Sqrt[2]*Sqrt[1 + z^2]*Cos[t])/Sqrt[5 - 3*Cos[2*t]], (4*Sqrt[2]*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 djmp at earthlink.net http://home.earthlink.net/~djmp/ > -----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 > > Please please someone help me!!! > > > Andy Sokol > Florida Institute of Technology > asokol at fit.edu >