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The center of a polytope is the point that is taken to be in its middle. They can be defined in several ways. Centers are generally assumed to be preserved by reflection, rotation and scaling.

By averages of points[edit | edit source]

By inscription and circumscription[edit | edit source]

If a polytope is circumscribable, the center of its circumsphere can be said to be the polytope's center. The same is true for polytopes that can have spheres inscribed into them, and the centers of said inspheres.

By symmetry[edit | edit source]

For any definition of center that is preserved by rotation, reflection and scaling, the center of a polytope must be preserved by the polytope's symmetries. If there is only one point that is preserved by a polytope's symmetries (as in all isogonal polytopes), that point is the polytope's center.

Center of incircle[edit | edit source]

In computer graphics and especially geographic information systems, a common problem concerns finding a "visual center" of a polygon that is neither regular nor convex (but has a defined interior). The centroid is unsuitable for this; a polygon with strong concave elements, such as one shaped like the letter "C," may not contain its own centroid, and a center outside the polygon is not considered visually salient.

One solution is to find a circle of largest possible radius that is completely contained in the polygon (incircle), and take the circle's center as the "visual center." Such a circle is not necessarily unique but often yields pleasing solutions.[1]