And it gets worse. If we try and move a point to edit our curve, the nearby segments deform in reflex action, creating obnoxious kinks. To get past these problems, we need a higher grade of curve. These provide a lot more control over local curvature. The following pictures will provide the basic idea.
Bezier control helps a lot, but can be far too clumsy for practical modeling.
When many points are close together, the tiny overlapping handles get very confusing. We need a way to simply grab single control points to get the result we want. For reference, here's the cardinal spline example, with the funny twitches that resulted from moving only the single point in the middle. If we convert the very same curve to a B-spline, the twitches disappear, and the curve bulges out only where the point was moved. This makes for much more organic modeling.