       Re: Archimedes' Spiral

• To: mathgroup at smc.vnet.net
• Subject: [mg85636] Re: Archimedes' Spiral
• From: "David Park" <djmpark at comcast.net>
• Date: Sat, 16 Feb 2008 03:32:05 -0500 (EST)
• References: <fp3uad\$9gf\$1@smc.vnet.net>

```Lea,

We can obtain the parametrization for Archimedes spiral from  Alfred Gray's
excellent book 'Modern Differential Geometry of Curves and Surfaces with
Mathematica: Second Edition'.

archimedesspiral[n_, a_][t_] := a t^(1/n) {Cos[t], Sin[t]}

With[{n = 1, a = 1},
ParametricPlot[archimedesspiral[n, a][t], {t, 0, 6 \[Pi]},
Frame -> True,
Axes -> False,
PlotLabel -> "Archimedes Spiral",
Epilog -> {Text[HoldForm["a" == a], Scaled[{.80, .95}], {-1, 0}],
Text[HoldForm["n" == n], Scaled[{.80, .9}], {-1, 0}]},
BaseStyle -> {FontSize -> 12}]
]

For those who have the Presentations package we can also draw the curve as a
complex expression in the complex plane and dispense with Epilog.

Needs["Presentations`Master`"]

With[{n = 1, a = 1},
Draw2D[
{ComplexCurve[a t^(1/n) \[ExponentialE]^(\[ImaginaryI] t), {t, 0, 6
\[Pi]}],
Text[HoldForm["a" == a], Scaled[{.80, .95}], {-1, 0}],
Text[HoldForm["n" == n], Scaled[{.80, .9}], {-1, 0}]},
Frame -> True,
PlotLabel -> "Archimedes Spiral",
BaseStyle -> {FontSize -> 12}]
]

--
David Park
djmpark at comcast.net
http://home.comcast.net/~djmpark/

"Lea Rebanks" <lrebanks at netvigator.com> wrote in message
news:fp3uad\$9gf\$1 at smc.vnet.net...
> Hi All,
>
> I am trying to plot the  Archimedes' Spiral.
>
> I copied this code from a web site. But it didn't work. Any ideas.
>
> ParaPlot[ArchimedeanSpiral[t],
>     {t, 0, 10*2*Pi}, PlotDot ->
>       False, AspectRatio -> Automatic,
>     PlotLabel ->
>       "Archimedes' spiral, r == theta"\
>    , Ticks -> {Range[0, 60, 20],
>         Range[0, 60, 20]},
>     Background -> GrayLevel];
> Do[ParaPlot[Evaluate[
>       ArchimedeanSpiral[i][t]],
>     {t, 0.0001, 5*2*Pi},
>     PlotDot -> False, PlotPoints ->
>       30, AspectRatio -> Automatic,
>     PlotRange -> {{-1, 1}, {-1, 1}}*
>         (5*2*Pi)^i*1.1, PlotLabel ->
>       StringForm["r == theta^``",
>         PaddedForm[N[i], {4, 2}]],
>     Ticks -> {{N[Floor[(4*2*Pi)^i]]},
>         {N[Floor[(4*2*Pi)^i]]}},
>     Background -> GrayLevel],
>   {i, 0, 2, 2/20}]
>
>
>
>
> Many thanks for your help & attention.
> Best Regards - Lea Rebanks...
>
>

```

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