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My goal is to find a way to calculate the convex hull of the union of some parameterized curves.

For instance, I had to calculate the convex hull of $A=\{(-4k,k^2+2,2k^2-2k)|k\ge 2\}\cup \{(k,k^2/4-1,-k^2/4-k)|k\le -2\}$. I am aware with the Fenchel-Bunt theorem, so I just have to consider every (closed) triangles made by $a,b,c\in A$. I wanted to do this by hand, but it was very complicated to do. My question is : is there a way to calculate this convex hull?

I do not believe that this is a research-level question, but I didn't get any comments or answers in math.stackexchange.

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  • $\begingroup$ I gave a partial answer at MSE. Here is the original question: math.stackexchange.com/q/1166518/166535 $\endgroup$
    – Joonas Ilmavirta
    Commented Mar 1, 2015 at 8:15
  • $\begingroup$ What do you mean by "calculate"? In what terms do you want the result? $\endgroup$
    – Alex Degtyarev
    Commented Mar 1, 2015 at 9:31
  • $\begingroup$ Anything geometrically special about the two curves? What kind of an answer do you expect (qualitatively)? $\endgroup$
    – Włodzimierz Holsztyński
    Commented Mar 1, 2015 at 9:54
  • $\begingroup$ I want to calculate the minimum value of $z$ when $x,y$ is given, and while $(x,y,z)\in\mathrm{con}(A)$. $\endgroup$
    – Gheehyun Nahm
    Commented Mar 1, 2015 at 10:39
  • $\begingroup$ I am working on specific families of inequalities, and one of my problems state that every inequality, understood geoemetrically, are in the convex hull of two specific curves. One of the simplest case is this. These curves are algebraically special : they are not random, but I cannot find a significant geometrical meaning. $\endgroup$
    – Gheehyun Nahm
    Commented Mar 1, 2015 at 10:43

1 Answer 1

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You might see if this paper helps:

Ranestad, Kristian, and Bernd Sturmfels. "On the convex hull of a space curve." arXiv:0912.2986 (2009). (arXiv abstract link)

      CurveHull
      The yellow surface is $z - 4x^3 + 3x = 0$. The green surface has degree $16$. The pink triangle is planar.
      (Image due to Frank Sottile, Philip Rostalski.)
If an approximate hull would suffice, it is "easy" to compute the 3D convex hull of many points along your curves. Here is a crude attempt on your two curves $A$:
      CurveHull

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  • $\begingroup$ @Houtarou: Could you describe your "different way"? $\endgroup$
    – Joseph O'Rourke
    Commented Mar 1, 2015 at 22:37
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    $\begingroup$ I viewed this set in R3 as a set of inequalities.(By some calculation I have done before) In this perspective, finding out the minimum of z was quite easy to do. $\endgroup$
    – Gheehyun Nahm
    Commented Mar 2, 2015 at 13:08

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