Analytic geometry

Isoperimetric ratio

In analytic geometry, the isoperimetric ratio of a simple closed curve in the Euclidean plane is the ratio L2/A, where L is the length of the curve and A is its area. It is a dimensionless quantity that is invariant under similarity transformations of the curve. According to the isoperimetric inequality, the isoperimetric ratio has its minimum value, 4π, for a circle; any other curve has a larger value. Thus, the isoperimetric ratio can be used to measure how far from circular a shape is. The curve-shortening flow decreases the isoperimetric ratio of any smooth convex curve so that, in the limit as the curve shrinks to a point, the ratio becomes 4π. For higher-dimensional bodies of dimension d, the isoperimetric ratio can similarly be defined as Bd/Vd − 1 where B is the surface area of the body (the measure of its boundary) and V is its volume (the measure of its interior). Other related quantities include the Cheeger constant of a Riemannian manifold and the (differently defined) Cheeger constant of a graph. (Wikipedia).

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Related pages

Convex curve | Dimensionless quantity | Cheeger constant | Isoperimetric inequality | Jordan curve theorem | Area | Curve-shortening flow | Cheeger constant (graph theory) | Surface area | Invariant (mathematics) | Riemannian manifold | Euclidean plane | Volume | Analytic geometry | Circle | Similarity (geometry) | Length