Surface Evolver is an interactive program for the study of

surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...

s shaped by

surface tension Surface tension is the tendency of liquid surfaces at rest to shrink into the minimum surface area possible. Surface tension is what allows objects with a higher density than water such as razor blades and insects (e.g. water striders) to f ...

and other energies, and subject to various constraints. A surface is implemented as a

simplicial complex In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their ''n''-dimensional counterparts (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial set ...

. The user defines an initial surface in a datafile. The Evolver evolves the surface toward minimal energy by a

gradient descent In mathematics, gradient descent (also often called steepest descent) is a first-order iterative optimization algorithm for finding a local minimum of a differentiable function. The idea is to take repeated steps in the opposite direction of the ...

method. The aim can be to find a minimal energy surface, or to model the process of evolution by

mean curvature In mathematics, the mean curvature H of a surface S is an ''extrinsic'' measure of curvature that comes from differential geometry and that locally describes the curvature of an embedded surface in some ambient space such as Euclidean space. The ...

. The energy in the Evolver can be a combination of surface tension,

gravitational energy Gravitational energy or gravitational potential energy is the potential energy a massive object has in relation to another massive object due to gravity. It is the potential energy associated with the gravitational field, which is released (conver ...

, squared mean curvature, user-defined

surface integral In mathematics, particularly multivariable calculus, a surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analogue of the line integral. Given a surface, one may ...

s, or knot energies. The Evolver can handle arbitrary

topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...

, volume constraints, boundary constraints, boundary

contact angle The contact angle is the angle, conventionally measured through the liquid, where a liquid–vapor interface meets a solid surface. It quantifies the wettability of a solid surface by a liquid via the Young equation. A given system of solid, liq ...

s, prescribed mean curvature, crystalline integrands, gravity, and constraints expressed as surface integrals. The surface can be in an

ambient space An ambient space or ambient configuration space is the space surrounding an object. While the ambient space and hodological space are both considered ways of perceiving penetrable space, the former perceives space as ''navigable'', while the latt ...

of arbitrary

dimension In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...

, which can have a

Riemannian metric In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ''T ...

, and the ambient space can be a quotient space under a

group action In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism ...

. Evolver was written at

The Geometry Center The Geometry Center was a mathematics research and education center at the University of Minnesota. It was established by the National Science Foundation in the late 1980s and closed in 1998. The focus of the center's work was the use of computer ...

, sponsored by the

National Science Foundation The National Science Foundation (NSF) is an independent agency of the United States government that supports fundamental research and education in all the non-medical fields of science and engineering. Its medical counterpart is the National I ...

, the

Department of Energy A Ministry of Energy or Department of Energy is a government department in some countries that typically oversees the production of fuel and electricity; in the United States, however, it manages nuclear weapons development and conducts energy-rel ...

, Enterprise Minnesota, and the

University of Minnesota The University of Minnesota, formally the University of Minnesota, Twin Cities, (UMN Twin Cities, the U of M, or Minnesota) is a public university, public Land-grant university, land-grant research university in the Minneapolis–Saint Paul, Tw ...

References

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