WillmoreEnergyFunctional - crowlogic/arb4j GitHub Wiki

The Willmore energy functional is a mathematical concept used in differential geometry to quantify the bending energy of a surface. It was introduced by British mathematician Thomas J. Willmore in the 1960s. The functional measures the amount of bending or deviation of a given surface from a minimal surface, such as a flat plane or a round sphere.

In mathematical terms, the Willmore energy W of a smooth, closed surface M is defined as the integral:

W(M) = 1/8 * ∫ (H^2 - k) dA

Here, H is the mean curvature, k is the Gaussian curvature, and dA represents the surface area element. The integral is taken over the entire surface M.

The Willmore energy functional has significant applications in various fields, including physics, computer graphics, and materials science. For example, it can be used to describe the shape and behavior of biological membranes, such as cell membranes, which tend to minimize their bending energy. Additionally, it is employed in computer graphics to create smooth surfaces and in the study of materials science to understand the formation and stability of various structures.