The geometric intuition is important but missing. Calculus and algebra is not everything.<p>Short version:<p>Imagine a line connecting A to B.  Now imagine a point sliding across that line, starting at A and going to B.<p>Now consider a line A to B and a second line B to C. A point slides across each line: one from A to B, another from B to C. Now draw a line connecting these two points, and imagine a point sliding across that line.<p>This moving line starts as A to B, ends as B to C, so the point sliding along it goes from A to C along a curved path. This is a quadratic Bézier curve.<p>Add another layer and you get cubic.