[pstricks] Volumes/Solids of Revolution Examples?

Tue Nov 22 07:46:22 CET 2016

Dougherty, Michael schrieb:

> the illustration and then some code leaking into the final document.
> Probably a dumb question, but any idea what I'm missing?  Here are the
> errors.
>
>
>
> ! Package xkeyval Error: `PointName' undefined in families
> `,pstricks,pst-plot,
> pst-node,pst-3d,pstricks-add,pst-

\usepackage{pst-eucl}

Is needed

Herbert

> See the xkeyval package documentation for explanation.
> Type  H <return>  for immediate help.
>  ...
>
> l.34 ...PointSymbol=none,unit=1cm,arrowscale=1.55}
>
> ?
>
> ! Package xkeyval Error: `PointSymbol' undefined in families
> `,pstricks,pst-plo
> t,pst-node,pst-3d,pstricks-add,pst-func'.
>
> See the xkeyval package documentation for explanation.
> .
> .
> .
>
> ________________________________________
> From: PSTricks [pstricks-bounces at tug.org] on behalf of Hubert Lam
> [hubert at lamfamily.info]
> Sent: Monday, November 21, 2016 5:32 PM
> To: Graphics with PSTricks
> Subject: Re: [pstricks] Volumes/Solids of Revolution Examples?
>
> Hi Michael/all,
>
> Here is the code for the cone that I showed in a previous post. It does
> use \psVolume but with an opacity setting :)
>
>
> \newcommand{\pstEllipse}[5][]{%
> \psset{#1}
> \parametricplot{#4}{#5}{#2\space t cos mul #3\space t sin mul}}
>
> \begin{center}
> \psset{PointName=none,PointSymbol=none,unit=1cm,arrowscale=1.55}
>         \begin{pspicture}(-2,-3)(5,4)
>         \pstGeonode(0,0){O}(4,0){A}(-2,0){B}(0,2.4){T}(0,-2.4){B}
>         \psaxes[arrows=->,labels=none,ticks=none](0,0)(-2.5,-3)(4.75,3.5)
>         \psVolume[fillcolor=blue!30,fillstyle=solid,opacity=0.4,linecolor=gray,linestyle=solid](0.6,1){4}{-0.6
> x mul 2.4 add}
>         \psVolume[fillcolor=green!30,fillstyle=solid,opacity=0.4,linecolor=gray,linestyle=solid](2.4,2.8){4}{-0.6
> x mul 2.4 add}
>         \uput{\pslabelsep}[135](T){$(0,r)$}
>         \uput{\pslabelsep}[90](A){$(h,0)$}
>         \pstLineAB[linewidth=1.15pt]{A}{T}
>         \uput{4.9\psxunit}[0](0,0){$x$}
>         \uput{3.75\psyunit}[90](0,0){$y$}
>         \rput(-1,0){\pstEllipse[arrows=->,arrowscale=1]{0.2}{0.5}{165}{-165}}
>         \pcline[arrows=|<*->|*,offset=-0.5em](B)(A|B)
>             \ncput*[fillcolor=magenta!10]{$h$}
>             \pcline[linestyle=dashed,linecolor=gray](A|B)(A)
>         \pcline[arrows=|<*->|*,offset=2](O)(T)
>             \ncput*[fillcolor=magenta!10]{$r$}
>         \pstEllipse[linecolor=gray,linestyle=dashed,arrowscale=1.55]{0.4}{2.4}{0}{360}
>         \pstLineAB[linecolor=gray]{B}{A}
>         \end{pspicture}
>         \end{center}
>
>
> -----Original Message-----
> From: PSTricks [mailto:pstricks-bounces at tug.org] On Behalf Of Dougherty,
> Michael
> Sent: Sunday, November 20, 2016 10:48 AM
> To: Graphics with PSTricks <pstricks at tug.org>
> Subject: Re: [pstricks] Volumes/Solids of Revolution Examples?
>
> Thank you all for the responses.  Those are all useful.
>
> To answer Hubert's question:  While I wouldn't mind having "smooth"
> solids, with gradient effects and all of that, I do like all of the
> suggestions I've seen so far.  I've been so far keeping my graphics
> simple, and trying to do everything on PSTricks instead of importing
> pictures generated elsewhere.  I should also mention that I've been
> assuming my book would be in black and white (and gray).  I want it to be
> inexpensive and easily LaTeX-able.
>
> I'm happy to see ANY examples, and appreciate them all.
>
> Herbert, yours has appeal, and I appreciate you pointing out \psVolume,
> which I now see in your book.  For limited applications (rotations around
> the x-axis) it looks really useful and I probably will use it in an
> example or two.
>
> Hubert, I like yours very much too.  If you want to share some code I'd
> appreciate it.  Trying to get those discs into good perspective has been a
> challenge.  I thought I'd try ellipses but those aren't obviously
> ellipses.
>
> Randy, I might bug you for some of your code.  I especially like the big
> curly braces, and should investigate those!
>
> It's funny that the volume of a cone came up a couple times already.
> Without making it a "solid of revolution" problem I already did just that
> for my book project, and recently (like two days ago):
>
> \documentclass{article}
> \usepackage{pst-func,multicol,fouriernc,amsmath}
>
> \begin{document}
> \begin{multicols}{2}
>
> \underline{Solution}:
> Our illustration is on the right.
> We note that at each height $y\in[0,h]$, the cross sectional area is
> $\pi(\text{rad})^2$, where the radius ``rad'' is a function of $y$.  If we
> give such a circle a ``thickness,''
> or ``height'' $dy$, then it forms an infinitesimally thin cylinder of
> volume  $dV=\pi(\text{rad})^2\,dy$.
>
> What is left to do is to find the radius, labeled ``rad,''
> as a function of $y$, and
> then set up and compute the resulting integral.  From the diagram we can
> see that by a similar triangles argument, $(h-y)/h=(\text{rad})/r$---or
> equivalently $\text{rad}/(h-y)=r/h$.
>>From this we get
> $\text{rad}=(h-y)\frac{r}h$, and so
>
> \begin{center}
> \begin{pspicture}(-2,-.4)(2,5.5)
> \psset{xunit=1.2cm}
> \parametricplot[linestyle=dashed]{0}{180}{t cos 2 mul t sin .5 mul}
> \parametricplot{180}{360}{t cos 2 mul t sin .5 mul}
> \psline(-2,0)(0,5)(2,0)
> \psline{*-}(0,0)(2,0)
> \rput(1,0.2){$r$}
> \psline(0,0)(0,5)
> %\parametricplot{0}{180}%
> %   {t cos 2 mul 3 div 5 2 3 div mul t sin 3 div sub}
> %\parametricplot{180}{360}{t cos 2 mul t sin .5 mul}
> \parametricplot[linestyle=dashed]{0}{180}%
>    {t cos 3 mul 5 div 2 mul t sin .5 mul 3 mul 5 div 2 add}
> \parametricplot{180}{360}%
>    {t cos 3 mul 5 div 2 mul t sin .5 mul 3 mul 5 div 2 add}
> \parametricplot[linestyle=dashed]{0}{180}%
>    {t cos 3 mul 5 div 2 mul t sin .5 mul 3 mul 5 div 2.1 add}
> \parametricplot{180}{360}%
>    {t cos 3 mul 5 div 2 mul t sin .5 mul 3 mul 5 div 2.1 add}
> \psline[linewidth=1.5pt]{*-}(0,2)(0,5)
> \rput[l]{270}(.2,3.5){$h-y$}
> \psline[linewidth=1.5pt]{*-}(0,2)(1.2,2)
> \rput(.625,2.2){rad}
> \psline{<->}(2.2,0)(2.2,5)
> \rput[l](2.4,2.5){$h$}
> \rput[l](.2,1){$y$}
> \psline(-2.3,2)(-1.7,2)
> \psline(-2.3,2.1)(-1.7,2.1)
> \psline{->}(-2,2.4)(-2,2.1)
> \psline{<-}(-2,2.0)(-2,1.7)
> \rput(-2,2.6){$dy$}
> \psline(0,0)(.2,0)(.2,.2)(0,.2)
> \end{pspicture}
> \end{center}
> \end{multicols}
> \begin{align*}V&=\int_0^h
> \pi\left[(h-y)\frac{r}h\right]^2\,dy
> =\pi\cdot\frac{r^2}{h^2}\int_0^h
> (h^2-2hy+y^2)\,dy
> =\pi\cdot\frac{r^2}{h^2}\left.\left[h^2y-hy^2+\frac13y^3\right]\right|_0^h\\
> &=\pi\cdot\frac{r^2}{h^2}\left[h^3-h^3+\frac13h^3\right]-0
> =\pi\cdot\frac{r^2}{h^2}\cdot\frac13h^3=\frac13\pi r^2h.
> \end{align*}
> \end{document}
>
>