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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?

bubba
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Alex Aparin
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  • I am almost sure that there is no equation linking the one to the other. Maybe an inequation ? – Jean Marie Dec 26 '16 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.) – Alex Aparin Dec 27 '16 at 07:42
  • For the length of a Bezier curve, have you seen (http://math.stackexchange.com/q/12186), with many references in it ? – Jean Marie Dec 27 '16 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 ? – Jean Marie Dec 27 '16 at 10:52
  • @JeanMarie, good mention. I did not see this post before. I prefer work with cubic bezier curves because it is the most popular type of bezier curve in computational geometry. I agree with your second question. I suppose that usage of "the" is preferable – Alex Aparin Dec 27 '16 at 11:15
  • Well, the closed-form arclength expression for a cubic Bézier is not terribly practical for you (it needs elliptic integrals), while getting the arclength of a polyline only needs judicious use of the Pythagorean theorem. – J. M. ain't a mathematician Dec 27 '16 at 11:33
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    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. – Harnak Dec 27 '16 at 11:34
  • Btw, I've just found this that may help you: http://math.stackexchange.com/questions/12186/arc-length-of-b%C3%A9zier-curves – Harnak Dec 27 '16 at 11:37
  • Yes, I agree this post clarifies answer. Note that it has showed earlier by JeanMarie. – Alex Aparin Dec 27 '16 at 11:40

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This paper has some material that might be relevant:

Specifying the arc length of Bézier curves
John A. Roulier
Computer Aided Geometric Design
Volume 10, Issue 1, February 1993, Pages 25-56

For example he shows that the length of the curve is less than the length of the polyline. The proof is fairly simple.

J. M. ain't a mathematician
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bubba
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