How can I plot my curve, so it can be seen that it is a spiral?

Question

I want to plot the parametric curve $ [\exp(-1/(1-t) )\cos(1/(1-t)) , \, \exp(-1/(1-t)) \sin(1/(1-t)) ]$ for $ t \in [0,1] $ in Maple or Matlab. It should look like a spiral which winds infinitley often around $(0,0)$, but when I plot it, all I can get is an arc which goes to $ (0,0) $ and does not look like a spiral. Is there a way to plot this curve, to see the behaviour of a spiral?

Answer 1

This works fine for me in MATLAB R2016b, as far as I know it doesn't use anything version-specific. I don't know Maple well enough to give a Maple version.

t = linspace(0,1,1000);
x = exp(-1./(1-t)).*cos(1./(1-t));
y = exp(-1./(1-t)).*sin(1./(1-t));
plot(x,y)

This gave me the following graph, which shows more of the spiral the more I zoom in. The spiral shrinks so fast that you can't see more than about this much of it any any given zoom level. For completeness sake, here is the link to the Desmos demo showing the same.

Answer 2

In Maple,

ee := [exp(-1/(1-t))*cos(1/(1-t)),
       exp(-1/(1-t))*sin(1/(1-t))]:

plot([ee[], t=0..1]);

You could also make an interactive app, to illustrate zooming in on that. For example,

ee := [exp(-1/(1-t))*cos(1/(1-t)),exp(-1/(1-t))*sin(1/(1-t))]:
ff := eval(ee,t=a):
Explore(plots:-display(
    plots:-textplot([ff[], sprintf("t = %.3g",a)],
                    align={piecewise(sin(1/(1-a))<0,'below','above')}),
    plots:-textplot([0, 0, sprintf("t = 1.00",a)],
                    align={piecewise(sin(1/(1-a))<0,'below','above'),
                           piecewise(sin(1/(1-a))<0,'left','right')}),
    plots:-pointplot([ff,[0,0]],color=blue,symbol=solidcircle,symbolsize=6), 
    plot([ee[], t=0 .. 99*a/100]),
    plot([ee[], t=a..1.0]),
    plot([ee[], t=99*a/100..a])
    ,scaling=unconstrained, axes=box
    , size=[600,500], gridlines
    , title=typeset([x,y]=ee,"\n\n t = ",0.0..1.00,"\n")
    ,labels=["",cat("",seq(" ",i=5-log[10](1/4*exp(1/(1-a)))..0,-1))]
    ,view=map(u->map(`*`,u,(4*exp(-1/(1-a)))),[-0.3..0.3,-0.3..0.3])
         ),
     parameters=[[a=0.0 .. 0.95, width=400, label=""]]):

If I'm lucky then that could be viewed (with a movable slider) as spiral zoom on the Maple Cloud.