# How did length of base polyline relate with length of Bézier curve?

Let cubic Bézier curve $C$ be based on points $p_0, p_1, p_2, p_3$. Suppose $L$ is the polyline through $p_0, p_1, p_2, p_3$. Is there some well known analytical relation between lengths of these two geometrical objects?

• I am almost sure that there is no equation linking the one to the other. Maybe an inequation ? Commented Dec 26, 2016 at 22:39
• @JeanMarie, I suppose that there is some relation. To check this I want to find relation of lengths of these objects. Unfortunately some additional work is needed (for example to find integral etc.) Commented Dec 27, 2016 at 7:42
• For the length of a Bezier curve, have you seen (math.stackexchange.com/q/12186), with many references in it ? Commented Dec 27, 2016 at 10:49
• Two questions: 1) Why don't you begin by a quadratic Bezier curve (i.e., an arc of parabola). Cubic Bezier curve are more complicated... 2) In your question, you say "a" polyline, why not "the" polyline ? Commented Dec 27, 2016 at 10:52
• I don't think you can find a relation between the two lengths. Take, as an example, two segments with a common vertex and the quadratic Bezier curve generated. If you open the angle between the two segments, the polyline will still have the same length, though the quadratic curve will be stretched and its length supposedly varies. So probably, that would also depend on the angles and not only on the length of the polyline. Commented Dec 27, 2016 at 11:34