Talk:Convex curve
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Book of Girko
Book "Treatise of Avalysis" Vol. IV DIEUDONNE has nothing common with book of Girkin "Spectral Theory of Random Matrices"/ It look like error link of Google. Jumpow (talk) 15:04, 23 February 2015 (UTC)Jumpow
Necessarily a closed curve?
Different passages in the article either require or don't require a convex curve to be closed.
From the lead:
- A convex curve is a curve ... which lies on one side of each of its tangent lines.
From "Definition by supporting lines":
- A plane curve is called convex if it lies on one side of each of its tangent lines.
From "Definition by convex sets":
- A convex curve may be defined as the boundary of a convex set....[or] a subset of the boundary of a convex set.
From "Properties":
- Every convex curve has a well-defined finite length.
The first two quotes imply that a parabola is a convex curve, while the last two imply that it is not. If standard terminology requires it to be a closed curve (or subset thereof), the first two quotes should be modified to reflect that. On the other hand, if the term is used both ways, with and without a restriction that the curve be closed, then this should be explicitly mentioned. Thanks. Loraof (talk) 16:14, 28 May 2015 (UTC)
- The third quote does not necessarily contradict the first two. E.g, a parabola can also be seen as a boundary of a convex (unbounded) set. I am not sure about the 4th quote. --Erel Segal (talk) 19:04, 28 May 2015 (UTC)
- But the parabola is not a closed curve, so the claim that the "boundary of convex set" definition implies that the curve is closed appears to be incorrect. As another example: an open semicircle (i.e. one that is missing its two endpoints) would seem to satisfy the definition by supporting lines, but is not the boundary of a convex set (instead it obeys the "subset of the boundary" definition). The statement of the four-vertex theorem is also incorrect; it requires smoothness. —David Eppstein (talk) 19:34, 28 May 2015 (UTC)
- Right, Erel, the third quote above permits parabolas; unfortunately I left out a key part of the passage. The complete version of the third quote is
- A convex curve may be defined as the boundary of a convex set in the Euclidean plane. This means that a convex curve is always closed (i.e. has no endpoints). Sometimes, a looser definition is used, in which a convex curve is a curve that forms a subset of the boundary of a convex set. For this variation, a convex curve may have endpoints.
- As David points out, the second sentence here does not logically follow from the first one.
I disagree with David about his example the open semicircle--I think it is the boundary of a convex set, namely an open half-disk.Loraof (talk) 16:20, 29 May 2015 (UTC) Strike that-- of course it's a subset of the boundary. Loraof (talk) 16:37, 29 May 2015 (UTC)
Also, the article four-vertex theorem defines a convex curve as one with strictly positive curvature. Modifying this to say non-negative curvature (to allow for the non-strict case) would seem to me to be another good definition (equivalent I think to the one about tangent lines) which does not appear in this article.Loraof (talk) 16:32, 29 May 2015 (UTC) Strike that too--it's in there toward the bottom. Loraof (talk) 16:55, 29 May 2015 (UTC)
Determining convexity
I would suggest that the following two related issues be discussed in this article:
1. Given the equation of an algebraic plane curve (or perhaps more specifically a closed one), how does one determine whether it is convex?
2. Given the vertex coordinates of a polygon, what is the most efficient way to determine if it is convex?