Month: September 2016

3D Printing and Maxima

In response to several nearly simultaneous queries, I’ve been working this week on generating a 3D  plot  and creating a mesh file from within Maxima  that is suitable for exporting to a 3D printer.

As a guide, I used this 2012 article from the Mathematical Intelligencer by Henry Segerman.  The idea is to draw a surface using traces made from parametric tubes, then export that to a .PLY format file.

Here’s my prototype for the process of generating the surface, reading in the resulting gnuplot data file, identifying faces appropriately, and writing into a .PLY file.

That isn’t yet ready for the 3D printer, but I could load it into the free program MeshLab, and from there convert to .STL

Here’s my figure from Maxima:

pringle

And the resulting .PLY file loaded into MeshLab

 

 

saddle_meshdraw

Here’s a few seconds of video from the printing process

And 2.5 hours later, the low resolution first-try 3D printed object:

3dprintverision01

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Visualizing Complex Functions by their images over concentric circles

A few weeks ago I posted my Maxima code for visualizing complex function by their images over a grid of line segments in the complex plane.  Here are a few images from an experiment of plotting the image of complex function over concentric circles, centered at the origin, traversed in the counterclockwise direction.  The first image below shows the identity map.  The colors are in ROYGBIV rainbow order, as noted in the legend, which gives the radius of the pre-image circles.  The second image is for the function f(z)=(z-1/2)(z-1).  Notice for the smallest radius pre-image, the image under f is largely unchanged.  As radius grows the image forms a loop then doubles on itself.  Here’s a fun insight:  that looping behavior occurs at radius .75, a circle containing a point z at which f'(z)=0.  More on this later, but its exciting to find a geometric significance of a zero of the derivative, considering how much importance such points have in real calculus.  Finally the third figure shows really interesting behavior for the function f(z)=\frac{1}{(z-1/2)(z-1)}.  Notice again the looping brought about by points at which the derivative is zero.  Notice also the change in direction of the image each time we cross over a pole—that is a root of the denominator.  The maxima code is given below.

/* wxdrawcircleC
 plot the traces of complex valued function _f
 along concentric circles |z|=r for 0<r <=1
 blue - images of _f along horizontal lines
 red - images of _f along vertical lines */
wxdrawcircleC(_f):=block(
[fx,fy,fxt,fyt,pl,j,ncirc,r,theta_shift,lm,l,dx,dy,dxy,v,nt],
ratprint:false,
fx:realpart(rectform(subst(z=x+%i*y,_f))),
fy:imagpart(rectform(subst(z=x+%i*y,_f))),
pl:[],
nt:200,
r:2,
ncirc:7,
theta_shift:%pi/20,
/* get the max modulus for computing length of arrows */
lm:0,
for n:1 thru ncirc do (
l:cons(lm,makelist(cabs(subst(z=r*n/7*exp(%i*t),_f)),t,0,2*%pi,.5)),
 lm:lmax(l)
 ),
 for j:1 thru ncirc do(
 /*first the images of the circles */
 fxt:subst([x=r*cos(t)*j/ncirc,y=r*sin(t)*j/ncirc],fx),
 fyt:subst([x=r*cos(t)*j/ncirc,y=r*sin(t)*j/ncirc],fy),
 pl:cons([parametric(fxt,fyt,t,0,2*%pi)] , pl),
 pl:cons([color=colorlist(j)],pl),
 pl:cons([key=string(r*j/ncirc)],pl),
 /* now the arrows */
 dx:subst(t=3/nt,fxt)-subst(t=0,fxt),
 dy:subst(t=3/nt,fyt)-subst(t=0,fyt),
 dxy:sqrt(dx^2+dy^2),
 v:vector([subst(t=0,fxt),subst(t=0,fyt)],[lm*dx/dxy/20,lm*dy/dxy/20]),
 pl:cons([v],pl),
 pl:cons([head_length=lm,head_angle=1],pl),
 pl:cons([color=colorlist(j)],pl),
 pl:cons([key=""],pl)
 ),
pl:cons([xlabel="",ylabel=""],pl),
pl:cons([nticks=nt],pl),
pl:cons([grid=on],pl),
apply(wxdraw2d,pl)
);