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Re: Getting started with 3D cardioid

  • To: mathgroup at smc.vnet.net
  • Subject: [mg112035] Re: Getting started with 3D cardioid
  • From: David Thomson <aetherwizard at gmail.com>
  • Date: Thu, 26 Aug 2010 06:47:35 -0400 (EDT)
  • References: <i4r4ab$jt2$1@smc.vnet.net> <i4t54d$fgj$1@smc.vnet.net>

On Aug 23, 1:40 am, Helen Read <h... at together.net> wrote:
> On 8/22/2010 8:14 AM,David Thomsonwrote:
>
> > I'm new to Mathematica, sorry.  I would like to draw a cardioid
> > rotated about its polar axis.  I checked the Mathematica wiki and it
> > did not cover this topic (that I could find).  Can someone help me to
> > understand how to do this in Mathematica?
>
> The thing to do is represent the cardioid parametrically.
>
> For example, if you start with this cardioid:
>
> r[t_] = 1 - Cos[t]
>
> PolarPlot[r[t], {t, 0, 2 \[Pi]}]
>
> Then express the polar curve parametrically, like so.
>
> x[t_] = r[t] Cos[t];
> y[t_] = r[t] Sin[t];
>
> Note that due to symmetry, you need only revolve the top half (t from 0
> to Pi) around the polar axis.
>
> RevolutionPlot3D[{x[t], y[t]}, {t, 0, \[Pi]}, RevolutionAxis -> x]
>
> --
> Helen Read
> University of Vermont

Thank you, Helen.  That worked beautifully.  Not only did it produce
the 3D shape I was looking for, but also with the proper
proportions.

I tried to use a different geometry, but the parametric equation did
not give the results I was expecting.
r[t_] = t^-1 Sin[t + 1]
PolarPlot[r[t], {t, 0, 2 \[Pi]}]

In the above case, I'm looking for something with the appearance of a
horn.

Afterward, I want to learn how to position the cardioid origin on top
of the point of the horn.  This is a project I am actually building
and would like to use Mathematica to model it so I can make
illustrations.

Dave


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