Why does Maple plot a different result for a 3D parametric circle for parameter t from -100 to 100 vs 0 to 2$\pi$? I have the following parametric equation in three dimensions: $$\vec{r}(t) = (\cos t) \vec{i} + (\sin t) \vec{j}$$
I believe this is the equation for a circle on the xy-plane, centered at 0, with radius 1.
In Maple, if I plot this equation for values of t from 0 to $2\pi$ it looks fine.

If I plot for t from -100 to 100, it does not look fine:

Also note $0 to 8\pi$

What is happening here?
Here are the commands in Maple:
with(VectorCalculus)
ihat := <1, 0, 0>
jhat := <0, 1, 0>
r := cos(t)*ihat + sin(t)*jhat
plot3d(r, t = -100 .. 100)


This is a sampling artifact.
When you say $t = -100$ to $100$, you are specifying a range of $200$ radians for Maple to plot.  This corresponds roughly to about $32$ full rotations around the circle ($\frac{200}{2\pi} \approx 31.8$).  So you are overlapping the same path many times.
Since Maple is using an internal algorithm to decide how many points along this path to sample for plotting the curve, it sees this large range and chooses some points that it thinks are representative of the behavior of the curve.  It can't choose every point along the path because there are infinitely many.  So it picks and then connects them with lines because it assumes that the behavior of the curve is roughly linear between the points it samples.  The result is this artifact.
There is usually a plotting option to increase the sampling resolution, basically telling Maple to sample more points so that the plot doesn't look so strange.  As long as the curve being plotted is "smooth" (i.e., continuous and differentiable), this usually produces good results even when the plotting range is very large.  As I am not an expert in Maple, preferring to use Mathematica instead, all I can tell you is that in Mathematica, this is specified using the option PlotPoints and MaxRecursion.  The first one increases the number of points to sample, and the second tells the program how many times to recursively bisect between the sampled points if it detects that the plot is not smooth.  Increasing both of these will generally produce better results.
