# Applications of circumcircles?

Are there examples where the circumcircle of a triangle is useful in everyday life (not unrealistic or far-fetched ones)?

Are there other interesting applications (physics, technology, mathematics)?

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The circumcircles and their radii become important when you study triangulation for the purposes of meshing. Creating a mesh representation of an object or space is a task frequently encountered in applied mathematics, and it is a richer subject than you might think.

If you have a bunch of sample points in the plane and you want to connect them together into triangles, how would you do it? A natural way to define a neighbour relation is to give each sample point $p$ its own territory, $V(p)$, defined as that portion of the plane that can call $p$ its closest sample point. The resulting tesselation of the plane is called a Voronoi diagram, and $V(p)$ is the Voronoi cell of $p$.

So we say $p$ and $q$ are neighbours if their Voronoi cells share a boundary edge. Connecting the Voronoi neighbours like this generally gives you a triangulation, the Delaunay triangulation (if you think about this you might realize that this isn't quite true).

Now consider three Voronoi neighbours $p$, $q$, and $u$ that form a Delaunay triangle. Their Voronoi cells all mutually intersect, and in fact there is one point, $c$, that is shared by $V(p)$, $V(q)$, and $V(u)$: it is a corner in each of them. This point $c$ says that $p$, $q$, and $u$ all qualify as the closest sample point. Thus it is equidistant to all of them. This means that $c$ is the circumcentre of triangle $[p,q,u]$. Notice that there can be no other sample point within the circumcircle of $[p,q,u]$, since all other sample points are further away.

The Delaunay triangulation can be defined directly by this characteristic that all the circumcircles are empty: it is the only triangulation that enjoys this property.

Now imagine that your points do not lie on the flat plane, but on some smooth curved surface that you want to model. In computer graphics, smooth surfaces are often represented by triangle meshes. In order to make the triangle mesh look smooth, it is important that the triangles lie flat against the surface. It turns out that one way to ensure that the triangles lie flat is to enforce that the circumradii of the triangles are small. A triangle with a large circumradius can lie at a bad angle to the surface even if all of its edges are small. The magic of the Delaunay triangulation is that if you put enough points on the surface, you are guaranteed that the triangles will all have small circumradii. There are choices and subtleties involved with defining the Delaunay triangulation on the surface though.

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