Precise flattening of cubic Bézier segments

A Bézier curve segment is generally rendered by subdividing it into a series of disjoint curve subsegments, and then approximating each subsegment by joining its endpoints by a line segment (chord). The maximum transverse deviation of each curve subsegment from the corresponding chord (the achieved flatness) should be no greater than a minimum error value, , called the flatness. The standard technique for doing this is by a process called recursive subdivision [2], wherein the curve is recursively divided by two until the flatness criterion is met. The advantage of recursive subdivision is that the number of segments generated is variable— depending on the nature of the curve—rather than being fixed, as in the case of forward differencing [1]. The problem with recursive subdivision is that, if the flatness criterion is exceeded by even a small amount, the division is performed one more time, with each of the resulting segments having an achieved flatness of as little as 25% of . As a consequence, the number of segments in the resulting polyline is greater than necessary by as much as a factor of two. The described algorithm repeatedly reduces the front end of a curve by a segment whose flatness criterion is closely met, thus minimizing the number of generated segments in the approximating polyline.