Here on Wolfram Alpha parametric graph plotter, http://www.wolframalpha.com/widgets/view.jsp?id=ddaa2332531af389fba463a032fcec9d, I plotted this parametric function:
$x(t)=Re(t^{i\pi})$, $y(t)=Im(t^{i\pi})$, $z(t)=0$ only from $t=0$ to $1$. So, unless $t^{i\pi}=-1$ is satisfied by some $t$ in $[0,1]$, the point $(Re(e^{i\pi}),Im(e^{i\pi}),0$), i.e. $(-1,0,0)$ shouldn't be on the curve, right?
But the plot is this:
This seems to contain the point. This plot is wrong, isn't it?
On
To expand on Lord Shark's comment, you'll have $t^{i\pi} = -1$ whenever $t = e^{k}$ for odd $k$. So on the interval $[0,1]$, you'll have the values $e^{-1}, e^{-3}, e^{-5},\ldots$ (numerically $0.37, 0.050, 0.0067,\ldots$).
To see this, recall that $(-1)^k = -1$ if $k$ is odd. So $$ t^{\pi i} = (e^k)^{\pi i} = e^{\pi i k} = (e^{\pi i})^k = (-1)^k = -1. $$
Just transform:
$$t^{i\pi}=e^{i\pi \ln t}=-1$$ This is true, if the angle in the exponent is $\pi + 2k\pi$ for integer $k$. So,
$$\ln t = \ldots-1,1,3,5,\ldots$$ or $$t=\ldots e^{-1},e,e^3,e^5\ldots$$
Explanation for irregular plot:
Plotting programs just put into the formula different values of parameters and connect them with lines. The jagged line shows that the plotting program is making steps that are too big, so it connects points that are very far apart, and miss the curve that happens in the middle.
Why in this case? Plotting $t^{i\pi}=e^{i\pi \ln t}$ is just plotting exponentials of imaginary numbers, which we know (also from the Euler's formula), that you just get the unit circle. Just change the variable:
$$e^{i\pi \ln t } = e^{i\phi};\quad \phi=\pi \ln t$$
Plotting programs (unless they have some smart logic to detect if it needs more points), will use equally spaced values for the parameter. If you take for example, $t=0.01,0.02,0.03,\ldots,0.99,1.00,1.01\ldots$, you get $$\phi=-14.47,-12.29,-11.02,\ldots,-0.0316,0,0.0312,\ldots$$ Notice that for small $t$, the steps are huge (difference between first two steps is more than two (radians), which is more than $90^\circ$, so you are connecting points across more than quarter of a circle). For values of $t$ around $1$, it is making much smaller steps and gets a smoother curve.
This shows this parameter is not really got for plotting: no matter how closely you space the values of $t$, the pole of the $\ln t$ function at $t=0$ makes sure that the smallest values will be very far apart, and you are wasting a lot of time calculating very close points when $t$ is large.
If you simply change the variable and plot with equal steps in $\phi$, you get a perfectly smooth circle even with only 30 points around the circle.