Talk:Circumscribed circle

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Contents 1 Discussion: Merge with circumcircle 2 Triangles: Circumscribed circle is a circumcircle 3 Incorrect Formula? 4 Frustrating Merges

[edit] Discussion: Merge with circumcircle

I assert that circumscribed circle and circumcircle not be merged. They are fundamentally different concepts with different definitions, and to merge them would show a profound lack of geometric sophistication. For example, if you draw a five-point star, this star has no circumscribed circle (since the star has 10 vertices), but it does indeed have a circumcircle. The circumcircle does not have to touch every vertex; instead, it merely has to be the unique circle with the smallest radius such that the entirety of the star's interior is bounded within it. User:KyleGoetz

[edit] Triangles: Circumscribed circle is a circumcircle

A circumscribed circle for any triangle is the same circle as the circumcenter. Whoever wrote otherwise is absolutely wrong: http://mathworld.wolfram.com/Circumcircle.html User:KyleGoetz Just to clarify this with my comment above concerning the differences between a circumcircle and a circumscribed circle: for a triangle, they are the same; however, for other polygons this may not be true. User:KyleGoetz

I think it is very wrong to have a definition for triangles that is incompatible with the definition for polygons in general, but I also don't think the circumscribed circle and circumcircle are ever different. At best note the "smallest enclosing circle" version in passing as a variant usage rather than making it the main definition of the word. As source I cite this paper — it's my own work, so I'm not going to edit into the actual article, but it's a well respected survey from a number of years ago and clearly states (top of page 9, and theorem 1 page 13) that circumcircle = circumscribing circle for triangles, and that the smallest enclosing circle is something different. The only single word synonym I've seen in professional usage for smallest enclosing circle is minidisk or minidisc (or miniball in higher dimensions). —David Eppstein 04:49, 15 October 2006 (UTC)

Yeah, there is no difference between circumscribed circle and a circumcircle. Any polygon has a circumcircle, and when polygon's vertices happen to be all on that circumcircle, it is called a circumscribed circle. Oleg Alexandrov (talk) 05:05, 15 October 2006 (UTC)

Er. There are polygons which have all vertices on a circle, but for which that circle is not the smallest enclosing circle. Polygons other than triangles, even. So I'm not sure which circle your comment is referring to. To me, "circumscribed circle" and "circumcircle" can only mean a circle passing through all vertices, even if that circle is not the smallest one enclosing the polygon. Anything else needs a different name. —David Eppstein 05:17, 15 October 2006 (UTC)

OK, then my changes to the article were wrong. I still think that my merger of circumcircle into circumscribed circle was a good thing. I will try to fix my errors tomorrow. Oleg Alexandrov (talk) 05:34, 15 October 2006 (UTC)

Thanks for fixing my mistake! Oleg Alexandrov (talk) 16:03, 15 October 2006 (UTC)

David, I agree with you, and I did not mean to say anything different than what you did. It's now been so long since I edited this article, that I cannot remember exactly what brought about my mentioning of triangles specifically in the talk page, but I think it's because the article itself explicitly said that circumcenter and circumscribed circle were the same thing, and used a triangle as proof of the statement. I merely pointed out this was an erroneous proof that the two concepts are the same. KyleGoetz 23:25, 17 December 2006 (UTC)

[edit] Incorrect Formula?

is the formula under: "Coordinates of circumcenter" right, i'm not sure philb 14:53, 24 March 2007 (UTC)
The formula is correct. Note that these barycentric coordinates are not "normalized" (as in areal coordinates, where λ₁+λ₂+λ₃=1). If you want the barycentric coordinates that are most commonly used, divide the entire thing by the sum of the components 128.237.234.118 (talk) 19:37, 13 April 2008 (UTC)

[edit] Frustrating Merges

I agree with KyleGoetz. I spent time separating circumsphere and circumscribed sphere from eachother just to have someone come along and merge them. One must "touch 3 points and enclose the polygon" and the other "touch all the vertices of the polygon". As a software engineer who has worked on rigid body dynamics, there is a need for the former (circumscribed sphere) to define the collision area of ANY arbitrary polyhedron. Circumspheres (and circles) cannot be applied to any arbitrary polyhedra (or polygons) make the need for distincition clear. Coder0xff 19:24, 29 April 2007 (UTC)

The article as it stands correctly distinguishes between two concepts, (1) touching all vertices, and (2) enclosing and smallest. As I understand it you are trying to introduce a distinction between these two and a third concept, (3) touching three vertices and enclosing. Circles satisfying this third definition always exist but are uniquely defined only when the convex hull of the polygon is cyclic. It's a reasonable definition, though I think not as useful as the other two. But can you provide reliable sources in the mathematical literature clearly stating that definition (3) is associated with one of the phrases "circumcircle" or "circumscribed circle" and that definition (1) is associated with the other phrase? Because I don't think I've ever seen such a definition in the papers I've read. This is not the place for making up distinctions that do not represent what is already in the literature. —David Eppstein 19:33, 29 April 2007 (UTC)