Working with Cubic Paths in a Quadratic World

This problem begins with Pierre Bézier, who developed a method for creating curves in the late 1960s while he was working for Renault SA (the French carmaker). He decided that a curve could be fairly accurately defined as two end points (called anchors) and two other points off the curve (called handles). Figure 3.10 shows these four points; notice that a line can be drawn from each anchor point and a handle.

The resulting curve must follow some rules. It must start at one of the end points, race off with a slope equal to the slope of the adjacent segment (which, by the way, is actually a tangent to the curve at that point), and then come home on the second anchor point, approaching with the slope of the second segment (another tangent). This typical configuration is called a Bézier curve of the third order, or a cubic Bézier. The name derives from the cubic equation used to very precisely define curves: y = Ax³ + Bx² + Cx + D.