To draw a straight line with graphic-design software, it is usually sufficient to click once with the mouse to select the starting point and once again to select the ending point.
These two points define the line segment perfectly: it is a linear function—i.e., a first-degree polynomial—and is therefore defined by two points.
We would like to be able to do the same thing with curves: click to obtain the starting point and the ending point, and then use an intuitive means, accessible even to those with the most severe case of technophobia, to modify the curve and obtain the precise curve that we want from the infinite range of curves that can be drawn between two points.
That is precisely what Bézier curves do: they are third-degree polynomials and are therefore defined by four points. We already have two of those points: the starting point and the ending point; we need only define the two that remain. We call those two points the control points. In most software packages like Illustrator and CorelDraw, we click once to obtain the starting point, then drag the mouse to obtain the first control point, release the mouse button, move, click again to obtain the ending point, and drag the mouse again to obtain the second control point. How do we know where to place the last of these? Well, as soon as we have indicated the ending point, the software begins to draw a Bézier curve, which changes dynamically as we drag the mouse to set the second control point. Thus we ...