This demo includes two graphical display windows.

The window "Curve: X(t)" displays the curve X(t) and the point X(t0).

The window "Perturbed Unit Point to Curve" shows the unit vector curve (X(t)-P)/||X(t)-P||, but perturbed slightly so the viewer can see its path on the unit circle. The point P is a hotspot point so can be dragged around.
We can read the winding number of the curve around the point P by the degree of the unit circle mapping (X(t)-P)/||X(t)-P||, i.e., intuitively how many times the curve goes around the unit circle in a positive direction.