In a previous page we discussed quadratic and cubic Bézier curves. As you can see above, the cubic form has 2 control points, while the quadratic form only has 1 control point.
The main difference between them is that the quadratic form can only allow a curve in one direction, whereas the cubic form allows curves of many directions.
If you think of the control points as magnets acting on a metal ball moving from the start to the end point, you can see how this would be possible.
Here we are attempting to create a cycloid curve using cubic Bézier curves, and it appears we are very close - the red curve is a true cycloid, while the yellow curve is produced by the following path using a cubic Bézier:
The control points are indicated by the 2 blue dots. They are crucial to the outcome of the final curve.