The Surface Evolver

Kenneth A. BrakkeMathematics DepartmentSusquehanna UniversitySelinsgrove, PA 17870

email: brakke@susqu.edu

May 13, 1992

This version is close but not identical to the published version in Experimental Mathematics, 1 (1992) 141–165. Some figures have been regenerated, and the bibliography updated. Pagination is different, and thetext may not be exactly identical.

Abstract

The Surface Evolver is a computer program that minimizes the energy of a surface subject to con-straints. The surface is represented as a simplicial complex. The energy can include surface tension,gravity, and other forms. Constraints can be geometrical constraints on vertex positions or constraintson integrated quantities such as body volumes. The minimization is done by evolving the surface downthe energy gradient. This paper describes the mathematical model used and the operations available tointeractively modify the surface.

AMS classification (1990):Primary:49-04 Calculus of variations explicit machine computation and programs.Secondary:35-04 PDE explicit machine computation and programs.49Q05 Minimal surfaces.49Q20 Variational problems in a geometric measure-theoretic setting.

Keywords: minimal surfaces, calculus of variations.

Introduction

One of the fundamental problems in the calcu-lus of variations is to find a surface minimizing someenergy subject to constraints. A soap film on a wireframe minimizes its area subject to its boundary stay-ing on the frame. A cluster of soap bubbles minimizesthe total soap film area subject to enclosing fixed vol-umes in each bubble. A capillary surface minimizesthe gravitational energy of a fluid in a vessel plusthe surface energy of its free surface and its contactenergy with the vessel walls. Other examples of sur-faces are grain boundaries in metals, crystal facets,fluid interfaces, and cell membranes [Almgren 1982;Almgren and Taylor 1976]. Note that these surfacesneed not be simply connected, need not be orientable(as in a Mobius band soap film), and need not bemanifolds (as in bubble clusters).

The Surface Evolver is an interactive program forthe study of surfaces shaped by surface tension and

other energies. The user specifies an initial surface,the constraints that the surface should satisfy through-out the evolution, and an energy function that de-pends on the surface. The Evolver then modifiesthe surface, subject to the given constraints, so asto minimize the energy. The user can intervene dur-ing the evolution, changing the surface’s propertiesor applying certain operations to keep the evolutionwell-behaved.

The action of the Evolver is meant to model theprocess of evolution by mean curvature, which wasstudied in [Brakke, 1977] for surface tension energy inthe context of varifolds and geometric measure the-ory. The energy in the Evolver can be a combina-tion of surface tension, gravitational energy, squaredmean curvature or user-defined surface integrals. TheEvolver can handle complicated topology (as seen inreal soap bubble clustes), volume constraints, bound-ary constraints, boundary contact angles, prescribed

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mean curvature,crystalline integrands, gravity, andconstraints expressed as surface integrals.

The main focus of the Surface Evolver is on two-dimensional surfaces in three-dimensional space, theso-called soap-film model. The principal data struc-tures are set up with this in mind, but they are de-signed in such a way that the dimension of the am-bient space and the ”surfaces” of interest can be ar-bitrary. Thus, one-dimensional strings and higher-dimensional surfaces can also be handled. Moreover,the ambient space can be endowed with an arbitraryRiemannian metric, and even be a quotient space un-der a group action.

The Evolver has a graphical interface that allowsthe user to follow the evolution of the surface on thescreen. The graphics can also be output to files inseveral formats, including PostScript.

The Surface Evolver program is freely available(see ”Software Availability” at the end of this article)and is in use by a number of researchers. Some of theapplications of the Evolver so far include modellingthe shape of fuel in rocket tanks in low gravity [Tegart1991], calculating areas for the Opaque Cube Prob-lem [Brakke 1991b], computing capillary surfaces incubes [Mittelmann] and in exotic containers [Calla-han et al. 1991], simulating grain growth, study-ing grain boundaries pinned by inclusions, searchingfor partitions of space more efficient than Kelvin’stetrakaidecahedra, modelling the shape of molten sol-der on microcircuits [Racz et al.], studying polymerchain packing, and classifying minimal-surface singu-larities. Section 9 of this paper gives a proof, basedon the use of the Evolver, that a conjectured area-minimizing cone in R4 is not area-minimizing.

The strength of the Surface Evolver program is inthe breadth of problems it handles, rather than opti-mal treatment of some specific problem. It is undercontinuing development, and not every feature is de-scribed in this paper. Users are invited to suggestnew features.

This paper describes the capabilities of the Sur-face Evolver so that readers can evaluate its useful-ness in their research and so that users have a pub-lished description to cite. It is not a substitute for theSurface Evolver Manual [Brakke 1991a], although itdoes mention certain operational details so that userscan find the corresponding features in the program ormanual.

Acknowledgments.

The Surface Evolver is a result of my participa-tion in The Geometry Center at the University ofMinnesota (formally The National Science and Tech-nology Research Center for Computation and Visual-ization of Geometric Structures, formerly the Geom-

etry Supercomputer Project), I would like to thankFred Almgren, Jean Taylor, and Al Marden for mak-ing my participation in The Geometry Center pos-sible and for their enthusiasm. I would also like tothank Charlie Gunn, Toby Orloff, Stuart Levy, andthe rest of the staff of The Geometry Center for tech-nical assistance, especially in graphics. Thanks alsoto John Sullivan, who has used the Evolver since itsinception and has made many suggestions and foundmany bugs.

1 Surface models

1.1 Representing surfaces

Surfaces have been represented mathematically asgraphs of functions, level sets of functions, imagesof maps, measures, simplicial complexes, polyhedralcomplexes, spline patches, etc. Each way has itsstrengths and weaknesses. The Surface Evolver uses asimplicial complex representation. This permits therepresentation of surfaces like soap films and bub-bles, which may have arbitrary topologies and maynot be orientable. It permits the exact specificationof a surface in terms of a finite number of parameters,the vertex coordinates. See the figures accompanyingthis paper for some simple examples.

The main focus of the Surface Evolver is ontwo-dimensional surfaces in three-dimensional space,which is called the soapfilm model. The principaldata structures are set up with this in mind, but theyare so designed that the dimension of the ambientspace can be arbitrary and one-dimensional “strings”can also be represented. Higher dimensional surfacescan also be handled to a limited degree.

The units of measurement are dimensionless. Ifthe user wishes to model a specific physical problem,then all values should be in one consistent set of unitssuch as cgs or MKS.

Section 3 describes what is available in the Evol-ver in terms of alternatives or elaborations to the ba-sic soap-film model: ”surface” of arbitrary dimension,ambient spaces with an arbitrary Riemannian metric,and so on.

1.2 Energies

Broadly speaking, the energies that the Evolver min-imizes are any quantities that may be expressed asintegrals over the surface. Foremost among them issurface tension. Soap films and interfaces betweendifferent fluids have an energy proportional to theirarea, which can also be regarded as a surface ten-sion, or force per unit length. That is, across any linein the surface, there is a tension whose value is the

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same as the surface energy density. In the Evolver,the user can specify a value of the surface tension foreach facet.

Another common energy is gravitational potentialenergy, which can be written as a surface integral bymeans of the divergence theorem. Capillary surfacesmay be modeled by including both surface tensionand gravitational energy in the total energy. For moredetails, see section 4.

1.3 Constraints

Several types of constraints are available in the Evol-ver. Vertices may be fixed in place, reflecting the factthat the edge of a soap film should be attached to awire. Vertices may be constrained to lie on smoothmanifolds, reflecting the case when the edge of a soapfilm should lie on a wall, but is free to move on thewall. Edges and facets may likewise be constrained,which simply means that any vertices generated onthem will inherit those constraints. Bodies may beconstrained to have fixed volumes, as in the case ofthe volume of the column of liquid underneath a cap-illary surface. Section 5 explains the exact mathe-matical form of the various types of constraints.

1.4 Basic Operations

The fundamental operation of the Evolver is the it-eration step, which reduces energy while obeying anyconstraints. A gradient descent method is used. Theforce at each vertex is the gradient of the total energyas a function of the position of that vertex. Everyvertex is moved simultaneously by a global multilpleof its force. This multiple, called the scale factor,can either be fixed by the user or be the factor thatoptimizes the decrease in energy. Details of all theoptions available for the iteration step are given inSection 6.

Several operations are available for manipulatingthe triangulation. Refinement is the subdivision ofeach facet into four similar facets, for better approx-imation of curved surfaces. Equiangulation readjuststhe triangulation of a surface to make the facets asnearly equilateral as possible. Vertex averaging moveseach vertex to the average position of its neighboringvertices. These and other operations, including somethat change the topology of the surface, are more fullydescribed in section 7.

2 Three examples of the Evol-ver in action

2.1 The Catenoid

The catenoid is the minimal surface whose boundaryis two parallel rings not too far apart. It is an ex-tremely simple surface, yet it illustrates some of thesubtleties of evolving triangulated surfaces. Stages inits evolution are shown in Figure 1. The surface inthe initial data file consists of six rectangles forminga cylinder between the two rings. In general, a datafile contains only the minimum amount of informa-tion needed to correctly define the topology of thesurface. When initially read in, the rectangles areautomatically triangulated into facets (top left). Thevertices and edges on the rings are fixed. The ringsthemselves are not shown. With so few facets, theinitial surface cannot shrink, so the user refines thesurface (top middle). Here, only one refinement isdone to keep the facets large enough to be seen eas-ily. Normally, there would be alternating stages ofrefinement and iteration. Note that the vertices cre-ated by subdividing the edges on the rings are them-selves fixed on the rings. Equiangulation gives themuch nicer triangulation shown top right, by switch-ing the diagonals of some quadrilaterals to make thefacets more equiangular. Fifty iterations with opti-mizing scale factor result in an area of 6.458483 (bot-tom left). At this point, each iteration is reducingthe area by only .0000001, the triangles are all nearlyequilateral, everything looks nice, and the innocentuser might conclude the surface is very near its min-imum. But this is really a saddle point of energy.Further iteration shows that the area change per it-eration bottoms out about iteration 70, and by iter-ation 300 the area is down to 6.4336 (bottom right),near the true local minimum. We know it is a mini-mum because iteration produces no change, and theEvolver can calculate the Hessian matrix to be posi-tive definite. One can see that the triangulation reallywants to be twisted around so that there are edgesfollowing the lines of curvature.

If the two rings are too far apart, the neck of thecatenoid will shrink down to a point, as shown inFigure 1. Upon iteration, the neck forms a ring ofvery short edges (top middle). These edges can be re-moved by identifying their endpoints, in a step (takenby the user) known as tiny edge weeding (§7.5). Thisproduces a single vertex at the neck (top right). TheEvolver can recognize the topology around the neckvertex as improper for a soap film and split the ver-tex into two (bottom left), when instructed to do soby the user. This called vertex popping (§7.7) Thetwo parts of the surface then quickly collapse to disks(bottom right).

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Figure 1: The evolution of a stable catenoid. Top left: The initial surface. The boundary wire circles arenot shown. The rectangular faces of the datafile have been automatically triangulated. Top middle: Afterone refinement. Note how vertices on top and bottom edges follow the boundary wire circles. Top right:After equiangulation. Note the edges that have switched direction. Bottom left: After iterating fifty times.This is a saddle point in area. Bottom right: Ultimate endpoint of iteration, with edges following the linesof curvature, which are horizontal and vertical.

Figure 2: The evolution of an unstable catenoid. The neck pinches down to a point, and the surface splitsinto two pieces.

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Figure 3: Capillary surface making contact with a wall. Left: the initial surface. The facets on the verticalwall (darker shading) have surface tension 0.5, while the ones on the horizontal free surface have surfacetension 1.0. Thus the equilibrium contact angle is 60◦. The free surface is bounded by a fixed wire in frontand plane constraints on either side. Center left: After some evolution, the free edge has crept up the wall.Note that several wall facets are getting very narrow, which would soon cause problems in the evolution.Center right: After vertex averaging, the free edge has more room to migrate without overrunning vertices.Right: the final equilibrium surface.

2.2 Capillary Surface Meeting a Wall

Figure 3 shows the evolution of an example where weuse constraints and varying surface tension to modela surface meeting a wall at a given contact angle –here 60 degrees. This is the situation for a capillarysurface, but this example does not include volumeconstraints or gravity.

The initial configuration is shown on the left. Thefluid surface is the light gray horizontal surface, whichhas one edge movably attached to the dark gray verti-cal wall. The junction of the surface and wall consistsof edges belonging to both surface and wall facets.We give the wall facets half the surface tension of thesurface facets, so the equilibrium contact angle is 60degrees (0.5 = cos60◦). All vertices on the wall arecnostrainted to stay in a vertical plane. The threevertices at the top of the wall are fixed in place, asare the vertices of the surface on the edge oppositethe wall. The two lateral sides of the surface and wallare constrained to lie in vertical planes (not shown)perpendicular to the wall. These planes contain nofacets and contribute no energy, so the equilibriumcontact angle is 90 degrees.

After several iteations (center left), the contactline has moved up the wall, seeking the equilibriumcontact angle. The shrinking of the wall facets morethan offsets the stretching fo the surface facets. Theinterior vertices of the wall do not move, since thereare no net forces on them. This can give rise to prob-lems if the contact line overruns interior wall vertices,as is about to happen. At this point, the user mustintervene to adjust the triangulation, using vertex av-eraging, after which (center right) the contact line cancontinue to move up the wall. The final equilibriumstate consists fo a plane surface (far right).

The wall facets sereve two purposes: to generatethe correct contact angle and to help visualize the

surface. But they also are the source of problems asthe surface moves up the wall, requiring repeated ver-tex averaging. They occupy more computer memoryand calculation time than necessary. It is possibleto omit them. In §4.5 I show how to use edge inte-grals to compute the equivalent energy of the facets.The visualization function could be served by havingfixed facets on the wall that do not participate in thecalculations and do not refine.

2.3 Grain Growth

When liquid metal solidifies, crystallization gener-ally starts at many nuclei, with random orientations.The crystal lattices are mismatched where the grainsmeet, and the atoms along the grain boundaries are ina higher energy state than interior atoms. To a goodapproximation, the energy is independent of the ori-entation of the boundary. In the process of annealing,the metal is warmed enough for boundary atoms toswitch from one lattice to the other through thermalmotions, and the boundaries migrate at a rate pro-portional to its curvature, assuming that impuritiesor other obstacles do not interfere.

Figure 4 shows this process for a two-dimensionalmetal in a unit flat torus that initially crystallizesfrom 100 random nuclei, resulting in an initial grainconfiguration of 100 Voronoi cells (top left). The ul-timate product of evolution consists of four unequalhexagonal grains. A video of the evolution is availablein [Brakke 1992b].

Modeling the dynamics of the evolution requiresusing a fixed scale factor (the time step) much smallerthan the optimizing scale factor. Here, a time stepof 5 × 10−6 was used. The Evolver has several fea-tures used to automate the evolution, including au-tomatic topology changes in the string model. Witha feature called autopopping, any edge whose length

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Figure 4: The evolution by mean curvature of 100 Voronoi cells in a two-dimensional flat torus. The timestep is 5 × 10−6 From left to right: The initial configuration; configuration after 200 steps, still with 100cells; after 1600 steps, with 66 cells left; and after 5000 steps, with 35 cells left.

is projected to become less than the critical lengthis deleted and any improper vertices resulting arepopped. The critical length is automatically set tothe critical length for stability described in §6.4, whichworks out to 0.003 here. Only edges whose lengthsare decreasing are tested in order not to eliminate thevery short edges generated by vertex popping. Also,with autochopping, edges that become longer than achosen cutoff length of 0.015 are automatically sub-divided.

3 Surface Models

This section describes several variations on the basicsoap-film model discussed in §1.1.

3.1 One-Dimensional ”Surfaces”: The StringModel

The “surface tension” can be declared to reside inedges instead of facets. The surface then becomesa network of elastic strings, hence the term stringmodel is applied to this mode of operation. Thestrings may reside in any dimensional space, but ifthe domain is two-dimensional, then the strings maybound regions. In this case, a region is defined witha facet structure and a body structure. The facetmay have any number of sides. The body has justone facet on its boundary. The effect is to stretch thestring network into a cylindrical surface of height 1,whence the mechanisms for surfaces can be appliedwith minimal changes. The grain growth example in§2.3 uses the string model.

3.2 Quadratic model

In an attempt to approximate curved surfaces bet-ter than by flat facets, there is a mode in whicheach facet is a quadratic spline patch. A midpointis added to each edge, giving a total of six control

points. Each coordinate is then quadratically inter-polated from these six points to form the surface.An edge becomes a curve that depends only on thecontrol points on the edge, thus guaranteeing thatneighboring facets meet without a gap. The disad-vantages of the quadratic mode are that it is slower,surface area is calculated approximately by numericalintegration, facets are displayed as if flat, and someEvolver features are not implemented.

3.3 Higher-Dimensional Surfaces

Higher-dimensional surfaces cannot be represented bythe basic vertex-edge-facet scheme. There is a modeof operation that permits the facets making up thesurface to be represented directly as simplices (vertexlists). This permits an arbitrary dimensional surfacein an arbitrary dimensional space. However, manyfeatures are not implemented yet for this mode.

3.4 Quotient spaces

The ambient space can be a quotient space of Rn un-der some symmetry group. The vertex coordinatesare taken to be in a fundamental region, and eachedge is marked with a group element to tell how itshead vertex should be transformed (wrapped) relativeto its tail. The user invents an integer representationfor the group elements to be used in marking edges.The flat torus quotient space is built-in with somespecial notation in the datafile for specifying the fun-damental parallelepiped and the wraps of the edges.Using some other quotient space requires the user towrite C functions that handle group transformationsand compositions.

The display of a surface in a quotient space posessome interesting problems, since the display space isEuclidean. The Evolver offers three options, two ofwhich are illustrated in Figure 5, which shows a sur-face in a flat three-dimensional torus whose funda-mental unit is a unit cube. The three options are:

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Raw facets. Each facet is displayed as it is located inthe fundamental domain. Location is based onthe first vertex in the facet, with other verticesunwrapped as needed.

Connected bodies. For each body, all facets on theboundary of that body are gathered in a list.One point is chosen as a base point, and un-wrapping of vertices spreads out from neighborto neighbor facet until the entire body surfaceis done. This nicely displays each logical bodyas a single visual body. An example is shownin Figure 5 (left).

Clipping. For a torus domain, each facet is clippedto the fundamental parallelepiped. Pieces lyingoutside are wrapped around so they lie inside.This option clearly shows the fundamental do-main and how it wraps around. An example isshown in Figure 5 (right).

The surface shown in Figure 5 bounds two tetra-kaidecaheda. Lord Kelvin [Thompson 1887] conjec-tured that the optimal way to partition space intoequal volume cells with least area is a packing of veryslightly curved tetrakaidecahedra. Several people (in-cluding the author) have used the Evolver in attemptsto beat Kelvin’s partition, but nobody has succeededyet.

3.5 Background metric

The ambient space can be endowed with a Rieman-nian metric. Only one coordinate patch is allowed,but quotient spaces are possible. Basically, the Evol-ver operates as usual in Euclidean coordinates, ex-cept that the metric is used for the calculation ofedge lengths and facet areas. Edges are not taken tobe geodesics, nor are facets geodesic surfaces; ratherthey keep their Euclidean shape. Surfaces are dis-played as if their coordinates were in Euclidean space.

An example using a metric is discussed in sec-tion 9 below. Other possible ways to use a metricare: modeling a cylindrically symmetric surface bymeans of the string model, and implementing a spa-tially varying scalar surface energy, as in a surfacewhose own weight is not negligible.

The metric need not be positive definite. Onecan do minimal surfaces in a Minkowski metric, butit takes a little care.

3.6 Internal Representation

Each geometric element (vertex, edge, facet, body)is implemented as one data structure. An elementis stored as an oriented entity, but may be referredto with this orientation or the inverse one. There is

a data type element id that contains a pointer to anelement structure and a relative orientation (normalor inverted). This type is used for all references toelements. The connectivity of the surface is speci-fied by having links in each element structure to thenext higher- or lower-dimensional elements it inter-sects. One design principle followed is that each ele-ment should contain links to a fixed number of otherelements. Thus the body structure does not recordbounding facets itself; rather, the facet structure hastwo slots to record which body (if any) is on each ofits sides.

The surface connectivity is completed by intro-ducing facet-edge structures, which are a simplifica-tion of the scheme described in [Dobkins and Laszlo1987]. A facet-edge structure contains links to a facetand an edge on the facet’s perimeter (with properboundary orientation), plus links to the previous andnext facet-edges around the facet and to the previousand next facet-edges around the edge. This permitsa quick run-through of all facets containing a givenedge, and also the representation of facets with an ar-bitrary number of sides, which is necessary in certainsituations.

4 Energies

The Surface Evolver tries to minimize the total en-ergy of a surface. This energy may have several com-ponents.

4.1 Surface tension

Soap films and interfaces between different fluids havean energy proportional to their area. In the Evol-ver, each facet has a surface tension of 1 unless oth-erwise specified. Different facets may have differentsurface tensions. It is possible to endow both facetsand chosen edges with tension in order to model sur-faces wehre singular curves have energy, as in [Mor-gan 1994].

Contact angles between free surfaces and walls, asin capillary problems, can be specified by introducingfacets that are confined to the wall and have a dif-ferent surface tension, as in the wall example in §2.2.Negative tensions are allowed, so all contact anglesare possible. However, this method has the drawbackthat a moving free boundary on the wall can overrunwall facets, as noted in Figure 3. Another methodof prescribing contact angles, described in §4.5, usesedge energy integrals.

There is no general mechanism yet to include theintegral over the surface of a general scalar integrandwhich may depend on position and tangent plane ori-entation. However, the use of a Riemannian metric

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Figure 5: Two of Kelvin’s tetrakaidecahedra in a flat torus. The fundamental region is a unit cube. A. Thesurface plotted as the boundaries of connected bodies. B. The surface clipped to the fundamental region, aunit cube.

on the ambient space can often acheive the same ef-fect.

4.2 Crystalline integrands

The Evolver can model energies of crystalline sur-faces. A crystalline surface energy density dependson the direction of the surface normal vector. Sucha quantity is known as a crystalline integrand [Tay-lor 1983; 1988]. The energy density is given by thelargest dot product of the surface normal with a set ofvectors known as the Wulff vectors. Surface area canbe regarded as a crystalline integrand whose Wulffvectors coinciding with the unit sphere. In the Evol-ver, a finite set of Wulff vectors may be specified, andthe corresponding crystalline energy computed.

4.3 Surface integrals

A facet may contribute an energy resulting from in-tegrating a vectorfield over the facet as a surface in-tegral. Multiple vectorfields may be defined in thedatafile as functions of the coordinates, and any facetmay use any number of them. Besides surface ener-gies, these can be used to do volume energies, such asgravitational energy, by using the Divergence Theo-rem to convert a volume integral into a surface inte-gral. Integrals are done numerically using Gaussianquadrature.

4.4 Gravity

A body B having a density ρ contributes its gravita-tional energy to the total. The acceleration of gravityG is under user control. Letting ρ be the body den-sity, the energy is defined as

E = Gρ

∫ ∫ ∫

B

z dV,

but is calculated by the Divergence Theorem as

E = Gρ

∫ ∫

∂B

z2

2~k · ~dS.

The integral is taken over each facet that bounds abody. If a facet bounds two bodies of different den-sity, then the appropriate difference in density is used.Vertical facets or facets lying in the z = 0 plane makeno contribution, and may be omitted if they are oth-erwise unneeded. Facets lying in constraints may beomitted if their contributions to the gravitational en-ergy are contained in edge energy integrals.

Gravity is a special case of a surface integrand,but it is implemented internally for several reasons:it can be evaluated exactly without numerical inte-gration; it is common enough to be worth saving theuser the trouble of setting it up; and, in general, eval-uation of user-defined nitegrand expressions is slowerthan that of compiled-in energies.

The built-in gravity does not apply to Riemannianmetrics or quotient spaces; users must define theirown gravitational energy integrands in such cases.

Gravity applies to bodies, not surfaces. Surfacesare weightless. If one did want heavy surfaces, onecould use the metric mechanism to simulate a scalarsurface integrand.

4.5 Edge integrals

An edge may contribute an energy resulting from in-tegrating a vectorfield over the edge as a line integral.The objective of this is to let the free edges of a sur-face have energy. This is useful to control the contactangle of a surface on a wall. As mentioned earlier insection 3.1, the contact angle can be specified by giv-ing an energy density to the wall on one side of thesurface edge. Alternatively, an edge integral can bedefined to give an energy equivalent to the wall en-ergy by means of Stokes’ Theorem. This eliminatesthe need to coat the wall with facets. Likewise, edgeintegrals can be used to replace other facet energies,such as gravitational energy, for facets on constraints.

An edge integral is evaluated once for each edge– regardless of how many facets it is on – using theorientation given in the datafile.

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4.6 Prescribed mean curvature and pressure

For an equilibrium surface with a constant surfacetension, the mean curvature can be interpreted asproportional to the pressure difference across the sur-face. Therefore, prescribing the mean curvature isequivalent to prescribing the pressure difference. TheEvolver permits the user to prescribe pressures inbodies. Pressure can be defined as the rate of changeof energy with respect to volume, so the pressure fea-ture is implemented by having each body with a pre-scribed pressure P contribute energy

E = −PV,

where V is the actual volume of the body. The energyis actually calculated by a surface integral

E = −P

∫ ∫

∂B

z ~k · ~dS.

The desired surface need not really be the entire bound-ary of a body; it may have fixed edges. The energycontributed by the omitted boundary is constant andso does not affect the shape of the surface.

This method will only work if the desired sur-face is stable at the prescribed pressure. If curvaturedecreases as volume increases, then the surface willeither blow up or implode. For example, if a roundsoap bubble of surface tension T and initial radius R0

is prescribed to have a pressure of P and P > 2T/R0,then the pressure force will cause the bubble to ex-pand, which reduces the curvature 1/R, so the pres-sure can never be balanced by the curvature, and thebubble expands indefinitely.

4.7 Squared mean curvature

There are circumstances under which one wants theenergy to include the integral over the surface of thesquared mean curvature. For example, surfaces whichminimize this integral are by definition Willmore sur-faces. This presents a slight problem for a piecewiselinear surface, as the mean curvature (in the form offirst variation measure) is singular and concentratedon the edges. Its square integral is therefore alwaysinfinite. However, it is possible to come up with ausable approximation. An average mean curvaturearound each vertex can be calculated, and the inte-gral of the square of this average counted as energy.

The definition of mean curvature used here is avariational one, and corresponds to the average ofthe sectional curvatures rather than their sum. Theintegral of squared mean curvature in the soapfilmmodel is calculated as follows: Each vertex v has astar of facets around it of total area Av. The forceon the vertex is

~Fv = −∂Av

∂v.

Since each facet has three vertices, the area associatedwith v is Av/3. Hence the average mean curvature atv is

~hv =12

~Fv

Av/3,

and this vertex’s contribution to the total integral is

Ev = h2vAv/3 =

34

F 2v

Av.

Ev can be written as an exact function of the vertexcooordinates, so the gradient of Ev can be fed intothe total force calculation.

The alternative to locating curvature at verticesis to locate it on the edges, where it really is, andaverage it over the neighboring facets. But this hasthe problem that a least area triangulated surfacewould have nonzero squared curvature, whereas inthe vertex formulation it would have zero squaredcurvature.

Squared mean curvature is also implemented forthe string model, but not for quadratic models. Inthe string model, let Lv be the sum of the lengths ofthe edges adjacent to v, so the force on a vertex is

~Fv = −∂Lv

∂v.

Each edge has two endpoints, so the length associatedwith v is Lv/2, so the curvature is

~hv =~Fv

Lv/2,

and this vertex’s contribution to the total integral is

Ev = h2vLv/2 =

2F 2v

Lv.

4.8 Gaps

Figure 6: Surface in a ring, showing the gap problem.A. The initial surface. The vertices are free to movealong the boundary wire. B. After one iteration, thegaps have grown and the surface area shrunk.

Consider a soap film spanning a circular wire. TheEvolver must approximate this surface with a collec-tion of facets, as shown in Figure 6. The straight

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edges of these facets cannot conform to the curvedwire, and hence the computed area of the surfaceleaves out the gaps between the outer edges and thewire. If the vertices are free to move along the wire,then the Evolver will naturally try to minimize areaby moving the outer vertices around so the gaps in-crease, as in Figure 6 right, ultimately resulting in asurface collapsed to a line. This is not good. Some-times the vertices can be fixed on the wire, but othertimes this is not possible, for example when the sur-face is spanning the inside of a cylinder. Thereforethere is provision for a “gap energy” to discouragethis. A constraint of the type defined in §5.1 may bedeclared convex in the datafile. For an edge on sucha constraint, an energy is calculated

E =k

6

∥∥∥~S × ~Q∥∥∥

where ~S is the edge vector and ~Q is the projection ofthe edge tangent to the constraint at the tail vertex ofthe edge. The constant k is a global constant calledthe gap constant. A gap constant of 1 gives the bestapproximation to the actual area of the gap. A largervalue of k minimizes gaps and gets vertices nicelyspread out along the wire.

Another way to handle gaps is to define an edgeintegral (see §4.5) that is zero on the constraint andpositive on the convex side of the constraint. Edgeintegrals are evaluated on interior points of the edge,so the bigger the gap, the bigger the integral. Thisencourages the edges to be equal length.

Of course, any addition of energy changes theproblem slightly. But this energy decreases quadrat-ically with the fineness of the triangulation, and soshould not change the solution significantly. By chang-ing the constant associated with this energy, one cansee whether the problem is being significantly altered.

In actual practice, gap energy would seldom beused with a wire boundary, since it would be muchsimpler just to declare the vertices on the wire fixed.The real use for gap energy comes with surface edgeson walls, where the vertices cannot be fixed.

5 Constraints

Energy minimization takes place subject to con-straints of two types: constraints on the motion ofvertices and constraints on the value of surface in-tegrals. Vertices can be individually constrained bydeclaring them fixed, by confining them to level setsof functions (which is Evolver’s narrow meaning ofthe term constraint), or by defining their positionin terms of parameters (Evolver boundaries). Sur-face integrals in general are called quantities, and the

particular case of body volumes is implemented in-ternally.

5.1 Level-Set Constraints

A vertex may be confined to the zero level set ofone or more functions. Such a function is called aconstraint in Evolver terminology. It should be clearfrom context when the term constraint is used in thisnarrow sense or in the broader mathematical sensein this paper. The default is the narrow sense. Con-straint functions are defined by the user in the initialdatafile and numbered for reference. Vertices may bedeclared to be on one or more constraints simulta-neously, but it is the user’s responsibility to ensurethat the constraint function gradients at a vertex arelinearly independent. There are also one-sided con-straints, which means a vertex is restricted to theregion where the constraint function has nonnegative(or nonpositive) values.

When a vertex is moved for whatever reason, New-ton’s method is used to project it back to its con-straints. There is a global constraint tolerance pa-rameter the user can set to control the accuracy ofconstraint satisfaction. The coordinates of a vertexin the initial datafile do not have to exactly satisfy itsconstraints; it will be automatically projected. A ver-tex on constraints may also be declared fixed, whichmeans that it will not move after its initial projection.If a constraint is modified during runtime by chang-ing an adjustable parameter, all vertices are projectedagain so as to satisfy the new constraint.

Edges and facets may be declared to be on con-straints, which means that all vertices generated bysubdividing them will be on the same constraints.

A single constraint is the best way to attach a freeedge of a surface to a wall. Two constraints confinea vertex to a curve; but if a one-dimensional wire isdesired instead of a two-dimensional wall, then it maybe better to use the parameterized boundary featuredescribed below.

5.2 Boundaries

Evolver boundaries are one or two dimensional pa-rameterized manifolds; they are an alternate wayto constrain the position of vertices. A vertex ona boundary cannot also have constraints. Vertices,edges, and facets may be deemed to lie in a bound-ary. For a vertex, this means that the fundamen-tal parameters of the vertex are the parameters ofthe boundary, and its coordinates are calculated fromthese. When a vertex on a boundary moves, the mo-tion is projected back to parameter space and appliedto the parameters. Edges and facets on a boundarybequeath the boundary to descendant vertices.

10

A delicate question is how to handle wrap-aroundson a boundary such as a circle or cylinder. Sub-dividing a boundary edge requires a midpoint, buttaking the average parameters of the endpoints cangive nonsense. Therefore the average coordinates arecalculated, and that point projected on the bound-ary parameters as continued from one endpoint. Therings in the catenoid example in §2.1 are representedas one-parameter circles and show a case where theendpoint extrapolation is nexessary in the refiningoperation.

A general guideline is to use constraints for two-dimensional walls and boundaries for one-dimensionalwires. If one uses a boundary wire, the vertices andedges on the boundary can probably be declared tobe fixed. Then the boundary becomes just a guidefor refining the boundary edges.

5.3 Quantities

A quantity in the Evolver is a value that can be ex-pressed as the sum of integrals of vectorfields oversurfaces and edges. Quantities can be evaluated forinformational purposes, or they can be used as con-straints (in the mathematical sense). Body volume isan examples of a built-in quantity. Surface area is nota quantity in this sense, since it cannot be written asa vector integral.

Uuser-defined quantities are supported – for ex-ample, one might use as quantities the center of mass,the moment of inertia, the magnetic flux, and so no.The key to many of these is the Divergence Theo-rem, which permits volume integrals to be writtenas surface integrals. One can also use a quantity inplace of a body volume that is awkward to do withthe built-in volume mechanisms, for example, whena background metric is used.

Each quantity is defined in the datafile by givingthe surface integrand and which facets it is to be in-tegrated over, and an edge integral and the edges itis to be integrated over. A quantity acts as a mathe-matical constraint when it is declared fixed and givena specified value. Its actual value may be displayedand its target value changed interactively. Quantitieshave the same mathematical form as surface and edgeenergy integrals discussed in section 3, but are usedfor constraints or information rather than as part ofthe objective function.

5.4 Volumes

The term volume is used for the highest dimensionalmeasure of a region in n-space: area in R2, volumein R3, etc. A body may have a volume specified inthe datafile, which then becomes a volume constraint.

The volume of a body B can be written as

V =∫ ∫ ∫

B

1 dV,

which by the Divergence Theorem can be written asurface integral:

V =∫ ∫

∂B

z~k · ~dS.

This integral is evaluated over all the boundary facetsof a body.

The part of the boundary of a body lying on aconstraint need not be given as facets. In that case,Stokes’ Theorem can be used to convert the part ofthe surface integral on the constraint to a line integralover the edges where the body surface meets the con-straint. The line integral integrands are given as partof the constraint definition in the datafile. These edgevolume integranls can also be used to overcome thevolume calculation problems caused by gaps betweencurved constraints and flat facets.

Volumes are a special case of quantities, and areimplemented internally for Euclidean space and flattorus domains only. In general quotient spaces or inRiemannian metrics, it is up to the user to define vol-ume constraints using the general quantity constraintmechanism.

5.5 Volumes in a torus domain

The volume of a body can be automatically calcu-lated in a torus domain, but the wrapping of the edgesacross the faces of the fundamental region makes thecalculation a bit tricky. Ideally, we would like to ad-just the vertices by multiples of the fundamental re-gion basis vectors to get a body whose volume wecould find with regular Euclidean methods. Unfor-tunately, all we know are the edge wraps, i.e. thedifferences in the adjustments to endpoints of edges.But this turns out to be enough, if we are a littlecareful with the initial volumes in the datafile.

Let the facets of a particular body be indexed bym, and let vmi, i = 0, 1, 2, be the vertices of facetm. Let ~Ami be the (unknown) vertex adjustment forvertex vmi, and ~Tmi be the (known) wrap vector (dif-ference in endpoint adjustments) for edge i of facetm. Then

V = 16

∑facets m(~vm0 + ~Am0)

·(~vm1 + ~Am1)× (~vm2 + ~Am2)

= 16 (S1 + S2 + S3 + S4),

(5.1)

11

where

S1 =∑

facets m

~vm0 · ~vm1 × ~vm2,

S2 =∑′

facets m

~vm0 · ~vm1 × ~Am2,

S3 =∑′

facets m

~vm0 · ~Am1 × ~Am2,

S4 =∑

facets m

~Am0 · ~Am1 × ~Am2.

In S2 and S3, the∑′ notation means that each facet

is included three times in cyclic permutation, oncewith each vertex as base point.

The first of these sums is straightforward. Thesecond sum can be regrouped with one term for eachedge j, pairing the two facets j and j′ for each edgetogether:

S2 =∑

edges j

~vj0 · ~vj1 × ~Aj2 + ~vj1 · ~vj0 × ~Aj′2

=∑

edges j

~vj0 · ~vj1 × ( ~Aj2 − ~Aj′2)

=∑

edges j

~vj0 · ~vj1 × 12(~Tj′2 + ~Tj1 − ~Tj2 − ~Tj′1).

We can regroup this into a sum which can be donefacet by facet, again including each facet three times:

S2 =12

∑′

facets m

~vm0 · ~vm1 × (~Tm1 − ~Tm2).

In S3, we group terms with a common vertex to-gether, with the inner facet sum over facets with ver-tex k as base vertex:

S3 =∑

vertices k

~vk ·

∑

facets i

~Ai1 × ~Ai2

=∑

vertices k

~vk ·

∑

facets i

( ~Ai1 − ~Ak)× ( ~Ai2 − ~Ak)

=∑

vertices k

~vk ·

∑

facets i

~Ti0 ×−~Ti2

=∑′

facets m

~vm0 · ~Tm2 × ~Tm0,

which again can be done facet by facet. The stepintroducing Ak is valid since the Ai1 are just a rela-beling of the Ai2 and so

∑i Ai1 =

∑i Ai2.

The sum S4 is a constant, and so only needs to befigured once. Also, it is a multiple of the fundamental

region volume Vc, so its contribution to the body vol-ume is a multiple of 1

6Vc by equation (5.1). Therefore,if we assume the volume prescribed in the datafile iswithin 1

12Vc of the actual volume, we can calculatethe other sums and figure out what the fourth sumshould be.

The body volume gradient at vertex v can be eas-ily found from the above sums, since the base vertexis dotted with other terms. The gradient is a sumover all facets with base vertex v:

∂V

∂v=

16

∑

facets m on v

~vm1 × ~vm2

+12~vm1 × (~Tm1 − ~Tm2) + ~Tm2 × ~Tm0

).

6 Iteration

The heart of the Evolver is the iteration step that re-duces energy while obeying any constraints. The sur-face is changed by moving the vertices. No changesin topology or triangulation are made. The idea is tocalculate the force at each vertex and move the ver-tex in that direction, thus using a gradient descentmethod of minimization.

6.1 Force Calculation

The first step in an iteration is the calculation of theforces on the vertices. The total energy of the sur-face is viewed as a function of the coordinates of thevertices. The negative gradient of the energy as afunction of the position of a single vertex gives theforce on that vertex. Collectively, all the forces on allthe vertices make up the negative of the total gradi-ent of energy. No new approximations are introducedby the force calculation; the energy may be an ap-proximate energy due to numerical integrations, butthe gradient is the exact gradient of the approximateenergy.

Vertices on constraints have their forces projectedto the tangent spaces of the constraints. Vertices onboundaries have their forces mapped back to forces onthe boundary parameters. Fixed vertices have theirforces set to zero.

6.2 Volume and Quantity Constraints

The second step in an iteration is to enforce con-straints on body volumes and other integrated quan-tities. This has two parts. The first part consists ofcorrecting for any errors in the current values of thequantities. The second part consists of projecting thevertex forces to be orthogonal to the quantity gradi-ents. Both parts use the gradients of the quantities

12

as functions of the vertices. Let ~Wvk be the gradientof quantity k as a function of the position of vertexv. These gradients are projected on constraint levelsets or boundaries or set to zero in the same manneras forces.

The value correction consists of applying a sin-gle step of Newton’s method. Let the current excessvalue of quantity k be δk (which may be negative,of course). We assume the correcting motion ~Rv atvertex v is of the form

~Rv =∑

k

ck~Wvk

such that∑

v

~Rv · ~Wvk = −δk for each k.

This leads to the following linear system for the ck:∑

k

ck

∑v

~Wvk · ~Wvk′ = −δk′

for each quantity k′. This system is solved for theck, and hence for the motions ~Rv. This motion is notdone immediately, but is done as part of the over-all motion described below. The quantity values arenot perfectly corrected by this step, but over severaliterations they should converge to the target values.

The second part is the projection of the forces.Let ~Fv be the total force at vertex v. We want aprojected force ~Fv,proj of the form

~Fv,proj = ~Fv −∑

k

ak~Wkv

such that∑

v

~Fv,proj · ~Wvk = 0 for each quantity k,

which leads to the linear system for ak

∑

k

ak

∑v

~Wvk · ~Wvk′ =∑

v

~Fv · ~Wvk′

for each quantity k′. The coefficients ak are the La-grange multipliers for the quantity constraints, andcan be interpreted as pressures for body volume con-straints. Whenever the user asks for body volumesto be displayed, these pressure values are also shown.

6.3 Motion

Each vertex is moved by the quantity correction mo-tion plus a scale factor times the force at the vertex.The scale factor is a global constant, the same forall vertices. The user may set the scale factor ex-plicitly, or let the Evolver seek the optimal value. In

the latter mode, Evolver will successively double orhalve the scale factor until a minimum in energy isbracketed. Then quadratic interpolation is used toestimate the optimum scale factor, and that value isused in the final motion.

Each time a motion is done, all vertices on con-straints are projected back to their constraints by re-peatedly applying Newton’s method until the con-straint function value is smaller than the constrainttolerance factor, which the user may set. If verticessubject to one-sided constraints are on the wrong sideof the constraint, they are projected to the constraint.If such a vertex wants to move to the proper side ofthe constraint, it is freed from the constraint.

From experience, it seems that for two-dimensionalsurfaces driven by surface tension that the optimumscale factor is around 0.2 independent of the finenessof the triangulation as long as the surface is evolvingwithout problems. The universality the scale factor isexpected in this case since the scale factor is unitlessin length for two-dimensional area only. When thescale factor dives toward zero, it is usually a sign ofproblems like an edge length becoming zero, or a facetarea becoming zero. Unfortunately, the 0.2 value ap-plies best to minimizing area; other models, such asthe string model and squared mean curvature, have“normal” scale factors that vary with triangulationsize and other factors, so it is hard to tell when thesurface is evolving properly and when it is gettinginto trouble. In these circumstances, other methodshave to be used, such as watching the surface visuallyand checking edge length and facet area histograms.

6.4 Motion by mean curvature

The mean curvature vectorfield ~h of a surface S isdefined to be the gradient of the area of S, in thesense that if S is deformed with an instantaneousvelocity ~u, the rate of change of surface area is

dA

dt=

∫ ∫

surface

~u · ~h dA.

By definition, ~u = −~h under motion by mean curva-ture, so that

dA

dt= −

∫ ∫

surface

h2 dA. (6.1)

By default, the gradients of quantities like energyand volume are calculated simply as the gradient ofthe quantity as a function of vertex position. THisgives the force on a vertex, for example. But to sim-ulate motion by mean curvature, it is necessary tohave force per unit area instead. In the triangulationformulation, let Av be the area of the star of facetsaround vertex v and let

~Fv = −∂Av

∂v

13

be the force on vertex v. ThendA

dt=

∑v

~u(v) · −~Fv.

Take the area associated with a vertex to be one-thirdof the total areas of the facets surrounding the vertex,dA = 1

3Av. Since each facet has three vertices, thisallocates all area. Hence as approximation to ~h, wetake

~h = −3~Fv

Av.

If the Evolver is operated in “area normalization”mode, then the vertex motions are calculated usingthis formula. In this mode, the user should set thescale factor, which is the timestep for the evolution,to a constant small enough for the iteration to bea good approximation to the continuous evolution.Using an optimizing scale factor makes the time steptoo large.

The string model is more amenable to closescrutiny of how the Evolver does motion by meancurvature. In §2.3 an example was given of grainboundaries evolving in two dimensions. In approxi-mating motion by mean curvature, there are two dis-cretizations that must be made, in space and in time.The space discretization here consists of represent-ing the grain boundaries as a set of line segments,and the time discretization consists of the iterationsteps. There is an intermediate stage of discretiza-tion, that of the continuous time evolution of the dis-crete boundaries. The equation of motion for thisproblem has been chosen to approximate the contin-uous problem in a certain respect. Two-dimensionalgrain evolution has the property that the rate ofchange of area of a grain depends only on the numberof its sides. If ~h is the mean curvature of a grain G, sothe boundary velocity is −~h, then the rate of changeof area of a grain is

dA

dt=

∫

∂G

−hds = −∫

∂G

dθ

dsds = −

∫

∂G

dθ.

The total turning angle around the boundary of agrain is 2π, but each triple vertex contributes a turn-ing angle of π/3, so for a grain of N sides, so

dA

dt= (N − 6)

π

3.

The space-disrete problem preserves this property.The motion of each vertex is such that the areachange of the grain due to the motion of that vertexis proportional to the turning angle at that vertex (orto the excess turning angle at triple vertices).

A property that is not exactly preserved is thatthe rate work is done sweeping out area is propor-tional to the rate of length loss,

dW

dt=

∫

∂G

~v · ~F ds =∫

∂G

h2ds = −dL

dt.

The last equality is the string version of equation(6.1). To preserve this property would require solv-ing a system of linear equations linking together themotions of all the vertices.

The space-discretized evolution is guaranteed dis-sipative, but the time-discretized evolution can suf-fer from instabilities. Consider a boundary made upof segments of length L zigzagging about a straightmidline with small amplitude y. The velocity of avertex will be 4y/L2, and if the time step is ∆t, thenthe amplitude will grow if 4∆ty/L2 > 2y. Hencethe maximum timestep allowable is ∆t = L2/2, or,conversely, the minimum edge length is L =

√2∆t.

Exact damping of the zigzag occurs when ∆t = L2/4.

6.5 Conjugate gradient

For minimizing a quadratic function, there is a tech-nique called the conjugate gradient method [Press etal. 1988 §10.6] that can minimize much faster thangradient descent. With exact arithmetic, it minimizesan n-dimensional quadratic function in at most n it-erations. This method does not follow the gradientdownhill, but makes an adjustment using the pasthistory of the minimization.

The Evolver uses the Fletcher-Reeves variant ofthe conjugate gradient method. At iteration step i,let Si be the surface, Ei its energy, ~Fi(v) the force atvertex v as described above, and ~hi(v) the “historyvector” of v. Then

~hi(v) = ~Fi(v) + γ~hi−1(v)

where

γ =∑

v~Fi(v) · ~Fi(v)∑

v~Fi−1(v) · ~Fi−1(v)

.

For the actual motion, a one-dimensional minimiza-tion is performed in the direction of ~hi, using thebracketing method described in §6.3. It is importantthat all volumes and constraints be enforced duringthe one-dimensional minimization, or else the methodcan go crazy.

the energy function of a surface is not exactlyquadratic, but the method can stil be applied, andsometimes it yields very good results. But some-thimes it’s worse than regular iteration. The saddlepoint of energy in the catenoid example of §2.1 seemsto confuse the conjugate gradient method. With con-jugate gradient in effect, the saddle point is passed atiteration 17 and the area decreases again until itera-tion 30, when it reaches 6.4486. But at this point fur-ther iteration produces no change, and the conjugategradient mode has to be turned off and on to erase thehistory vector. Once restarted, another 20 iterationswill get the area down to 6.4334. This shows that theconjugate gradient mode can work much better thanordinary mode, but it can also have problems.

14

6.6 Hessian minimization

Minimization by gradient descent or even conjugategradient can take many iterations. A more directway to try to minimize is to calculate the Hessianmatrix of second derivatives of energy and solve forthe motion that gives zero gradient. This method iscurrently implemented only for the case where sur-face tension is the only energy and there are no con-straints of any kind except fixed vertices. It assumesthat the surface is close enough to a local minimumfor the Hessian to be positive definite, which is notalways true. When the method works, it can find theminimum energy to 15 decimal places in three or fouriterations. But if the Hessian is not positive definite,the method blows up. It is easy to find exampleswith a saddle point of energy; see the catenoid exam-ple below. There are checks in place to ensure thatthe calculated motion does indeed reduce energy.

6.7 Diffusion

In real soap-bubble clusters, air can diffuse across thesoap films, driven by pressure differences. Smallerbubbles tend to have higher curvature and hencehigher pressure, so tend to shrink. One can watcha foam evolve over the course of minutes, changingits topology as bubbles disappear. The Evolver cansimulate diffusion. If the diffusion mode is on, then atthe start of an iteration target volume is transferredacross each facet, in the amount equal to the area ofthe facet times the difference in body pressures timesthe global diffusion constant. The iteration step thencorrects the actual volumes to the target volumes anddoes its normal energy minimization step. Topologychanges are not done automatically yet; those are upto the user to do using the operations described inSection 7.

7 Surface Operation

This section describes the main commands availableto the user, aside from the iteration step described inthe previous section.

7.1 Refining

To refine a triangulation is to subdivide each facet tocreate a finer triangulation. THe Evolver does thisby creating new vertices at the midpoints of edges,which it then uses to subdivide each facet into fournew facets, each similar to the original.

The first stage of refining is to subdivide all edgesby inserting a midpoint. Hence all facets temporar-ily have six sides. For an edge on constraints, the

midpoint gets the same set of constraints, and is pro-jected to them. For an edge on a boundary, theparameters of the midpoint are calculated by pro-jecting the vector from the edge tail to the midpointback into the parameter space and adding that to thetail parameters. This avoids averaging parameters ofendpoints, which gives bad results when done withboundaries that wrap around themselves, such as cir-cles. In the second stage, each facet is subdivided intofour facets by connecting the new midpoints.

Certain attributes of new elements are inheritedfrom the old elements from which they were created.The new facets inherit the surface tension of theirparent facets. Fixity, constraints, and boundaries arealways inherited by offspring of all dimensions. In aquotient space, some, but not all, new edges inheritsymmetry group wrapping, so that the surface is cor-rectly embedded.

Refinement can change surface area, energy, andvolumes if there are curved constraints or boundaries.

In seeking the minimum energy, it is best to evolvewith a coarse triangulation as far as possible. Eachiteration can propagate a position adjustment onlyone edge at a time, so the finer the triangulation, thelonger adjustments take to travel across the surface.

7.2 Equiangulation

b c

a

d e

θ1

θ2

Figure 7: The two adjacent facets on the left violatethe equiangulaton criterion, since we have θ1 + θ2 >π. Equiangulation flips the quadrilateral diagonal,making the triangles more nearly equilateral.

Triangulations work best when the facets are asclose to equilateral (that is, equiangular) as possiblefor a given set of vertices. Given a set of vertices,how does one make a triangulation for those verticesthat has triangles as nearly as possible equilateral? Inthe plane, the answer is the Delaunay triangulation,in which the circumcircle of each triangle containsno other vertex [Sibson 1978]. It is almost alwaysunique. It can be constructed by local operations be-ginning with any triangulation. Consider any edge as

15

the diagonal of the quadrilateral formed by its adja-cent triangles. If the angles of the two vertices off ofthe diagonal add to more than π, then the circum-circle criterion is violated, and the diagonal shouldbe switched to form a replacement pair of triangles(Figure 7. When no more switches can be done, wehave a Delauney triangulation.

Now suppose now we have a triangulation of acurved surface in space. For any edge with two adja-cent facets, we switch the edge to the other diagonalof the skew quadrilateral if the sum of the angles atthe off vertices is more than π. As in Figure 7, leta be the length of the common edge, b and c thelengths of the other sides of one triangle, and d ande the lengths of the other sides of the other triangle.Let θ1 and θ2 be the off angles. Then by the law ofcosines,

a2 = b2 + c2 − 2bc cos θ1, a2 = d2 + e2 − 2de cos θ2.

The condition θ1 + θ2 > π is equivalent to cos θ1 +cos θ2 < 0. So we switch if

b2 + c2 − a2

bc+

d2 + e2 − a2

de< 0.

The equiangulation procedure over the whole surfacemay have to be repeated several times to get completeequiangulation, but almost never more than three orfour times. The process is guaranteed to terminatesince a switch reduces the radii of the circumcirclesand a finite number of vertices has a finite number oftriangulations.

Equiangulation can have an almost magical effectin improving a triangulation, and I highly recommendits regular use. It may temporarily increase area andchange volumes, but the magnitudes of these effectsare within the approximation error of using flat facetsfor a curved surface. A use of equiangulation is shownin Figure 1.

7.3 Vertex averaging

An evolving surface can get into trouble if some ofthe vertices of the triangulation get too scrunchedtogether, as shown in Figure 3 B. To get vertices tospread out, there is an operation called vertex averag-ing. For each vertex, this operation computes a newposition as the area-weighted average of the centroidsof the facets adjoining the vertex. Fixed vertices orvertices on boundaries are not moved, nor are ver-tices on triple or higher edges. Also, to keep the newsurface as close as possible to the old one, volumes onboth sides of the surface are preserved. If vertex v ison facets fi with centroids ~xi, then the new positionis calculated as

~vavg =∑

i area(fi) · ~xi∑i area(fi)

.

The volume on one side of all the facets around thevertex calculated as a cone from the vertex is

V =∑

facets f

~v · ~Nf ,

where ~Nf is the facet normal representing its area.The total normal ~N is

~N =∑

facets f

~Nf .

To preserve volume, we subtract a multiple λ of thetotal normal from the average position:

(~vavg − λ ~N) · ~N = ~v · ~N,

so

λ =~vavg · ~N − ~v · ~N

~N · ~N.

Then the new vertex position is

~vnew = (~vavg − λ ~N).

Constrained vertices are then projected to their con-straints.

Vertex averaging may slightly increase area, butthis is usually offset by its benefits. It is useful in get-ting the vertices spread out evenly. Evolution can beawkward when facets are of very different sizes, sincethe same scale factor applies to the whole surface.

7.4 Notching edges

A surface can be locally highly curved, resulting infacets forming pronounced ridges along edges. Oneway to selectively refine the surface is to just refinearound those edges whose adjacent facets are too farfrom parallel, putting a notch in the edge to make itmore saddle-shaped. There is a command that letsthe user do this with a cutoff angle of his choosing.The refinement is actually done by subdividing eachadjacent facet by putting a new vertex in the cen-ter. Equiangulation then completes the process. For-merly, the Evolver did notching by just subdividingthe offending edges, but that tended to create lotsof long skinny triangles and not always help matters.The new method seems to work better.

7.5 Edge and facet operations

One way to improve a triangulation is to simply elim-inate all edges that have become too short. This op-eration is known as tiny edge weeding. Every edgeshorter than a user-set cutoff length that can legiti-mately be removed is deleted by identifying its end-points.

16

Sometimes there are very skinny triangles thatshould be eliminated, but which don’t have a shortedge to be found by tiny edge weeding. Thereforethere is an operation called area weeding that removestriangles whose area is smaller than some cutoff. Thisprocedure finds the shortest edge of a triangle andeliminates it by the same process as the regular tinyedge removal.

There is a command that will bisect all edgeslonger than a user-chosen length. All facets adjoiningthe edges are also subdivided into pairs of facets. Ifthe new edges are still longer than the cutoff length,they are not further subdivided. It is suggested thatthis step be followed by equiangulation.

Histograms of edge lengths and facet areas can bedisplayed in conjunction with any of these commands.

7.6 Annealing, or jiggling

Sometimes it may be desirable to perturb the sur-face to get it off a metastable position. There areboth random and user-definable perturbations possi-ble. Because of its similarity to the thermal pertur-bations responsible for annealing in metals, the char-acteristic magnitude of the perturbation is called thetemperature.

Under a random perturbation, or jiggle, each coor-dinate of each non-fixed vertex is moved by δx = TLgwhere g is a random value from the standard gaussiandistribution (calculated from the sum of five randomvalues from the uniform distribution on [0,1]), T is thecurrent temperature, and L is a characteristic lengththat starts as the diameter of the surface and is cutin half at each refinement.

A long jiggle is a sinusoidal displacement of eachvertex v by ~A sin(~v · ~w + ψ). The amplitude ~A, thewavevector ~w, and the phase ψ may be specified bythe user or chosen at random.

7.7 Popping Edges or Vertices

The Evolver does not change the topology of a sur-face on its own. However, there are many times whena naturally evolving surface will need to change itstopology. For example, a neck might pinch out in acatenoid whose boundary rings are too far apart, ortwo growing metal grains might meet. Fortunately,the types of singularities possible in soapfilm-like sur-faces in three-dimensional space were classified in [Tay-lor 1976] for uniform surface tension. Three surfacesmay meet along a curve, or four triple curves maymeet at a point. The Evolver has procedures knownas edge popping and vertex popping to detect impropersingularities and reduce them to proper types. Theseroutines are designed only for surfaces with uniform

surface tension. There are many more types of sin-gularities possible if the different component surfacesmeeting at a singularity have different surface ten-sions.

Edge popping looks for edges with more that threefacets that are not fixed and are not on boundariesor constraints. When found, such an edge is splitlongitudinally with a new facet in between. The twoold facets with the smallest dihedral angle betweenthem are attached to the new edge. This is repeateduntil only three facets are on the original edge. Eachsplit is propagated along the multiple junction lineas far as possible. If it is impossible to propagate thesplit beyond either endpoint, the edge is subdividedto provide a vertex which can be split.

Vertex popping assumes that all edges have atmost three facets adjoining, so edge popping shouldbe done first. The facet and edge structure aroundeach vertex is analyzed to find which have the wrongtopology. The analysis is done by looking at the net-work formed by the intersection of the facets andedges containing the vertex with the surface of a smallsphere around the vertex. The numbers of sides ofthe cells of the network are counted. A simple planevertex has two cells of one side each. A triple edgevertex has three cells of two sides each. A tetrahe-dral point has four cells with three sides each. Anyother configuration is popped. The popping is doneby replacing the vertex with a hollow formed by trun-cating each cell-cone except the cell with the largestsolid angle. In case the network is disconnected, thesolid angles of all the cells will add up to over 4π.Then the vertex is duplicated and the different com-ponents are assigned to different vertices. This letsnecks shrink to zero thickness and pull apart.

In the string model, vertices with more than threeedges are popped by finding the pair of edges makingthe least angle, pulling them out a short distance witha new vertex, and joining the new and old vertex witha short edge. This is repeated until the original vertexhas only three edges.

Improper vertices may exist in the original datafile,or they may be introduced by short edge elimina-tion. For example, the pinching neck in a catenoidmust have all the short edges around its waist elim-inated to pinch the waist down to one vertex, whichcan then be popped. The other operations describedin this section (refining,vertex averaging, equiangu-lation, notching) do not change the global topologyand so do not introduce improper vertices. Impropervertices are not automatically detected unless the au-topop feature is on.

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7.8 Zooming

Sometimes the detail of a surface may require a closerlook. The graphics display can magnify a surface,but that doesn’t change the triangulation to followthe detail. There can be cases, as when a bound-ary wire passes through a soapfilm, where the detailaround a point is on a scale 100,000 times smallerthan the whole surface [Brakke 1992a]. Hence theEvolver contains a mechanism to zoom in on a vertex,throwing away the rest of the surface to save mem-ory and time and to keep all triangle sizes reasonablyclose together.

The user specifies which vertex to zoom in on andthe radius around it to keep. All vertices beyondthe cutoff distance from the given vertex are deleted.Then all edges and facets containing any deleted ver-tices are deleted. Any remaining edge that had afacet deleted from it is made fixed to anchor the cutedges of the surfaces.

8 User Interface Details

This section describes the Evolver’s user interface, in-cluding the initial data file, the command mechanism,and the graphics interface.

8.1 The Initial Data File

The initial configuration of a surface is read froma text file referred to as the datafile. The datafilehas five sections: general definitions, vertices, edges,faces, and bodies. The catenoid datafile cat.fe ispresented here to give the flavor. This file is slightlyatypical in that none of the vertices are given directlyby their coordinates. The .fe filename extension isa relic of early versions of the Evolver in which facet-edges had to be explicitly listed in the datafile; I con-tinue to use it out of habit to identify datafiles.

// cat.fe// Evolver datafile for catenoid.

// ring radius and height// adjustable at runtimePARAMETER radius = 1PARAMETER height = 0.55

// upper ring, parameterized by p1boundary 1 parameters 1x1: radius * cos(p1)x2: radius * sin(p1)x3: height

boundary 2 parameters 1x1: radius * cos(p1)x2: radius * sin(p1)x3: -height

vertices // second column = value of p11 0*pi/3 boundary 1 fixed2 1*pi/3 boundary 1 fixed3 2*pi/3 boundary 1 fixed4 3*pi/3 boundary 1 fixed5 4*pi/3 boundary 1 fixed6 5*pi/3 boundary 1 fixed7 0*pi/3 boundary 2 fixed8 1*pi/3 boundary 2 fixed9 2*pi/3 boundary 2 fixed10 3*pi/3 boundary 2 fixed11 4*pi/3 boundary 2 fixed12 5*pi/3 boundary 2 fixed

edges // given by endpoint vertices1 1 2 boundary 1 fixed2 2 3 boundary 1 fixed3 3 4 boundary 1 fixed4 4 5 boundary 1 fixed5 5 6 boundary 1 fixed6 6 1 boundary 1 fixed7 7 8 boundary 2 fixed8 8 9 boundary 2 fixed9 9 10 boundary 2 fixed10 10 11 boundary 2 fixed11 11 12 boundary 2 fixed12 12 7 boundary 2 fixed13 1 714 2 815 3 916 4 1017 5 1118 6 12

faces // given by oriented edge list1 1 14 -7 -132 2 15 -8 -143 3 16 -9 -154 4 17 -10 -165 5 18 -11 -176 6 13 -12 -18

The datafile syntax provides several features forflexibility and ease of use. Simple macros can be de-fined to do text substitution. Compound expressionscan be used anywhere a real number or a formula isexpected. Normal arithmetic and standard functionsare available. For functions that are evaluated duringruntime (constraints, quantities, etc.) expressions arestored as syntax trees that are interpreted when theexpression needs evaluation. If interpretation is tooslow, user-defined functions may be written in C andcompiled into the Evolver. Named variables may bedeclared and used in runtime expressions. Such vari-ables (called adjustable parameters) can be changedinteractively during runtime, and are useful for mov-ing constraints and boundaries around, modifying themetric, changing contact angles, and so forth.

The definitions section describes everything notpertaining to particular geometric elements, such as

• declarations and initial values for adjustable pa-rameters;

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• the dimension of the surface and the dimensionof the ambient space containing the surface;

• the specification of a quotient space, or, for aflat torus domain, the vectors defining the fun-damental parallelpiped;

• Riemannian metric tensor components;

• for a crystalline surface energy, the name of theWulff vector file;

• constraint function formulas, together with en-ergy and volume integrands for edges on con-straints;

• boundary definitions via formulas of coordinatesin term of parameters;

• quantity integrands, with target values for con-strained quantities;

• initial values for the gravitational constant, thediffusion constant, the weighting factor for thesquared mean curvature in the energy, the Gaus-sian integration order, and linear or quadraticmode.

None of these items are required. If missing, the fea-ture is not used or has a natural default value.

The vertices section lists the vertices, one per line.Each vertex is numbered for later reference, and isdefined by its coordinates (or boundary parameters),which constraints or boundaries it is on, and whetherit is fixed.

The edges section lists the edges, one per line,also numbered for reference. Each edge is defined byits tail and head vertex numbers, the constraints orboundaries it is on, the quantities it contributes to,and whether it is fixed. If a quotient space is beingused, the group element for wrapping the head vertexto the proper place with respect to the tail vertex isalso given.

The face section lists polygons forming the initialsurface. Each polygon is given by its edge numbersin order around its circumference. Edges traversedin opposite direction from that given in the edgessection are given as negative numbers. The poly-gons need not be planar, and they need not be tri-angles (which is why the section is “faces” instead of“facets”). The Evolver will immediately triangulatenon-triangular faces by putting a new vertex at theaverage position of the original vertices and puttingin edges from the new vertex to each original vertex.Each face may be on constraints, boundaries, or befixed. It may be given a specific surface tension; thedefault is 1.0. It may be deemed to contribute to cer-tain quantity integrals, and to have certain surfaceintegrands contribute to the total energy.

In the bodies section, each body is defined by list-ing its bounding faces by number, negative numbersif the orientation of the face in the face list has aninward normal. There may be any number of faces inany order. Faces do not have to completely enclosea body; they are used to compute volume and otherintegrals, and if certain faces are not needed for that,they may be omitted. A body may be declared tohave a fixed volume of a certain value. The actualinitial volume need not be that exact value; the vol-ume will be adjusted during the iteration process. Abody may also be given a density, which will causethe total energy to include the gravitational potentialenergy of the body with that density.

8.2 Command Interface

The user command interface is built on a simpleterminal-type model for maximum portability. Themain prompt is “Enter command: ”. There are twotypes of commands. The first consists of one let-ter occasionally followed by a number; the second isan embryonic query language. Currently, queries aresupported that list, display, refine, or delete elementsby various criteria. Commands may be read from afile with the command “read filename”. The out-put of any command can be piped to a system com-mand. Commands that change the surface or changethe model will cause energies and volumes to be recal-culated. Commands can be logged to a file for laterrepetition with the read command.

8.3 Graphics

It is possible to run the Surface Evolver without anygraphics. The program uses the standard teletype-style interface for maximum portability. But it’s al-ways nice to see the surface, and often essential to un-derstanding what is happening. Unfortunately, everycomputer system has its own way of displaying graph-ics, and there is no universal standard. The Evol-ver isolates the system-dependent graphics to draw-ing two-dimensional triangles. This cuts down theeffort in porting the Evolver to a new system to areasonable level. The Evolver’s main graphics rou-tine calculates the triangles to display, and calls thedevice-dependent subroutine to do the display. Thedevice could be a screen display or a graphics outputfile writer.

Two classes of graphics devices are provided for:those which can do their own viewing transforma-tions and hidden surface removal, and those whichcan’t. The former are simply provided a list of tri-angles with vertex coordinates in three dimensions.For the latter, the Evolver keeps an internal viewingtransformation matrix, sorts the transformed trian-

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gles from back to front, and feeds them to the displayroutine (painter’s algorithm). The sorting algorithmwill not subdivide intersecting triangles. If two trian-gles overlap, it will just find one point in the overlapand compare depths there. This can lead to somestrange-looking displays for strange surfaces, but itworks well for the types of surfaces the Evolver ismeant for.

My favorite graphics system is the GeomView pro-gram on Iris workstations. GeomView is an interac-tive viewer that lets the user rotate, translate, zoom,etc. the surface by dragging a mouse cursor overthe window in a trackball metaphor. A high-endIris workstation can light, shade, and smoothly ro-tate a surface consisting of several thousand trian-gles. The Evolver uses a shared memory interfacewith GeomView, so that the display is automaticallyupdated whenever the surface changes. GeomViewwas written at The Geometry Center, and is freelyavailable (see “Software Availability” at the end ofthis article).

Other types of screen displays do not have suchfancy view control as GeomView. Instead, there isa teletype-style graphics command interface that letsthe user control the viewing angle and size of thedisplay. This viewing transformation is also used forthe graphics output files.

There are several graphics output file formats,most notably PostScript. There are also formats thatlist transformed or untransformed triangles as text,suitable for input to other programs.

8.4 Other Commands

It is possible to reset the values of many parametersduring runtime, including the gravitational constant,body volumes, constrained quantity targets, the dif-fusion constant and the user-defined variables used informulas for constraints, boundaries, quantities andmetrics.

The current surface can be dumped to a text file inthe same format as the datafile. This is the only wayto save a file; there is no binary save format. Thetext format has the advantages that it is portable,editable and not too much larger than a binary filewould be.

After minimizing energy at several levels of re-finement, it is possible to extrapolate the energy toan infinitely fine refinement. The extrapolation usesthe final energies of three successive refinements andassumes a power law approach to the ultimate mini-mum.

9 Application: The Hopf ConeConjecture

In this sectin I present an example of a conjecturethat was settled (negatively) through the use of theEvolver.

In constrast to the situation in R3, the classifi-cation of area mininizing hypersurface cones in R4

is unknown. Frank Morgan once conjectured thata certain cone in R4 is absolutely area minimizing[Morgan 1986, p. 1278]. This example shows that itis not by having the Evolver generate a comparisonsurface with less area for the same boundary. It il-lustrates the use of a Riemannian metric to permit athree-dimension surface to represent a four-dimensionsurface by projecting out a symmetry.

Morgan’s cone is based on the Hopf fibration ofthe 3-sphere S3. Let S3 be parametrized by

0 ≤ α ≤ π/2, 0 ≤ β ≤ 2π, 0 ≤ γ ≤ 2π

so that Euclidean coordinates are

x1 = cos α cos β, x2 = cos α sin β,

x3 = sin α cos γ, x4 = sin α sin γ.

Then the boundary of Morgan’s Hopf cone consistsof the three surfaces

β − γ = 0 (mod 2π),β − γ = 2π/3 (mod 2π),β − γ = 4π/3 (mod 2π).

These three surfaces have zero mean curvature andmeet at 120◦ angles along the two orthogonal circlesx2

1 + x22 = 1 and x2

3 + x24 = 1.

Theorem 9.1. Morgan’s Hopf cone is not absolutelyarea minimizing.

Proof. The idea of the proof is to take the quotientspace of R4 modulo the Hopf fibers S1, giving R3

with a metric such that the area of a surface in R3

is the same as the 3-area of the lift of the surfaceback into R4. The metric turns out to have a naturalinterpretation as a cone space, leading to a simplecounterexample which can be verified by the Evolver.

The metric on R4 in Hopf spherical coordinates(r, α, β, γ) is

ds2 = dr2 + r2dα2 + r2 cos2 α dβ2 + r2 sin2 α dγ2.

The coordinates of the quotient space R3 will be(r, α, θ), where θ = β − γ. The orthogonally pro-jected metric is

ds2 = dr2 + r2dα2 + r2 sin2 α cos2 α dθ2.

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The lift of a point in R3 is a circle of circumference2πr, Hence we can make the 3-area of the lift of asurface have the 2π times the 2-area in R3 by multi-plying the linear metric ds by

√r, giving an effective

metric of

ds2 = r dr2 + r3dα2 + r3 sin2 α cos2 α dθ2.

This can be made to look more like the ordinaryspherical coordinate metric by using coordinates ρ =r3/2/2 and φ = 2α. Then

ds2 =169

dρ2 + ρ2dφ2 + ρ2 sin2 φdθ2.

This is the Euclidean spherical coordinate metric ex-cept for the factor 16/9, which is bigger than 1, mak-ing R3 into a cone space.

Figure 8: This surface is a counterexample to Mor-gan’s Hopf cone conjecture. The three outer edges areequally spaced half-great-circles on the unit sphere.Note how the surface avoids the center of the sphere.

Morgan’s Hopf cone projects to three planes meet-ing at 120◦, which is known to be absolutely area min-imizing in the standard metric of R3. Its area insidethe unit sphere (in the Hopf metric) is 2π. However,the cone factor makes it more expensive for a sur-face to go radially inward rather than go sideways.In a two-dimensional cone, it is easily seen (by un-rolling the cone) that geodesics avoid the origin. Asimilar phenomenon happens here. The comparisonsurface was made by deforming the three planes bypushing the point at the origin out toward one of theboundaries. This configuration was fed into the Sur-face Evolver, which evolved the surface and producedthe surface of Figure 8 with an area of 6.14, which iseasily less than that of Morgan’s Hopf cone. The nu-merical errors in this area are due to the numericalintegration used to calculate the area of the facetsand the gap between the curved boundary and thefacets. Both of these can easily be estimated to beless than the improvement of 8 over the Hopf cone.

The Hopf cone that projects to the tetrahedralcone is also not minimizing. The Evolver gives 7.44

Figure 9: Comparison surface for the tetrahedralHopf cone. The three outer edges lie on the unitsphere. Again, the surface avoids teh center of thesphere.

for the area of the comparison surface Figure 9, whilethe area of the cone is 4 cos−1(−1/

√3) = 7.6425. The

only other minimizing cone in standard R3, the flatplane, also deforms to avoid the origin. Thus noneof the area-minimizing cones in standard R3 lifts viathe Hopf fibration to a minimizing cone in R4.

10 Future directions

The Surface Evolver is under continual development.I welcome suggestions from users for new features.If they are reasonable, they will be added as timepermits.

Some of the major items on my current list ofthings to do are:

• How close in various senses is an Evolver mini-mal surface to the true smooth minimal surfacefor a given problem?

• The Evolver gives an upper bound for the areaof a minimal surface. The technique of cali-brations, a generalization of the network the-ory min-cut max-flow duality, can give lowerbounds. A near-minimal surface should be ableto generate a near-maximal calibration. Hencea goal is to have the Evolver generate such cal-ibrations.

• How close is an Evolver evolution by mean cur-vature to an ideal smooth evolution? Given aninitial smooth surface, is it possible to constructan Evolver approximation that stays close tothe ideal evolution? A more permissive notionof approximation would say that for each Evol-ver evolution there is an ideal smooth evolutionthat stays near it.

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• The current method of approximation to mo-tion by mean curvature needs further investiga-tion. The gradient of energy is a covector, andmotion is a vector. The conversion from covec-tor to vector requires a metric or inner product.The inner product used by the Evolver is theEuclidean inner product at vertices, weightedby the verex star area. Other inner productsare possible and may have desirable properties.

• Instabilities of the type described in §6.4 oftenlimit the size of the time step in an evolution.These instabilities need to be understood andmethods have to be developed to speed evolu-tion.

• Automatic triangulation management. Cur-rently, users have to closely monitor the surfacetriangulation and intervene manually when itgets fouled up. I hope to be able to have any ini-tial surface evolve for any length of time with-out any user intervention, as is now the case forstring evolution as described in §2.3.

• An interactive graphical interface, which willlet users select with a mouse the geometric ele-ments (vertices, edges, etc.) they wish to workwith, and it could be an interactive tool for thedesign of initial surfaces and datafiles.

11 Bibliography

[Almgren 1982] F. Almgren, “Minimal surfaceforms,” The Mathematical Intelligencer 4, no. 4(1982), 164–172.

[Almgren and Taylor 1976] F. Almgren and J.Taylor,“The geometry of soap films and soap bubbles,”Scientific American, July 1976, 82–93.

[Brakke 1977] K. Brakke, The motion of a surfaceby its mean curvature, Princeton University Press,Princeton, NJ (1977).

[Brakke 1991a] K. Brakke, Surface Evolver Manual,Research Report GCG 31, The Geometry Center.See “Software Availability” below for informationon how to obtain it.

[Brakke 1991b] K. Brakke, “The opaque cube prob-lem,” in Computing Optimal Geometries (videoand booklet edited by J. E. Taylor), AmericanMathematical Society, Providence RI, 1991.

[Brakke 1992a] K. Brakke, “Minimal surfaces, cor-ners, and wires,” Journal of Geometric Analysis2 (1992), 11–36.

[Brakke 1992b] K. A. Brakke, “Grain growth withthe Surface Evolver,” in Video Proceedings ofthe Workshop on Computational Crystal Growing(edited by J. E. Taylor), American MathematicalSociety, Providence, RI, 1992.

[Callahan et al. 1992] M. Callahan, P. Concus andR. Finn, “Energy minimizing capillary surfaces forexotic containers,” in Computing Optimal Geome-tries (video and booklet edited by J. E. Taylor),American Mathematical Society, Providence RI,1991.

[Dobkin and Laszlo 1987] D. Dobkin and M. Las-zlo, “Primitives for the manipulation of three-dimensional subdivisions,” technical report CS-TR-089-87, Department of Computer Science,Princeton University, Princeton, NJ, 1987.

[Mittelman] H. Mittelmann, “Symmetric capillarysurfaces in a cube” Math. Comp. Simulation 35(1993) 139–152.

[Morgan 1986] F. Morgan, “Harnack-type massbounds and Bernstein theorems for area-minimizing flat chains modulo ν,” Comm.Part. Diff. Eq. 11, (1986) 1257–1283.

[Morgan 1994] F. Morgan, ”Surfaces minimizing areaplus length of singular curves”, Proc. AMS 120(1994) 1153-1161.

[Press et al. 1988] W. Press, B. Flannery, S. Teukol-sky, and W. Vetterling, Numerical Recipes in C,Cambridge University Press, New York, 1988.

[Racz et al.] L. M. Racz, J. Szekely and K. A. Brakke,“A General Statement of the Problem and Descrip-tion of a Proposed Method of Calculation for SomeMeniscus Problems in Materials Processing”, ISIJInternational 33 No. 2, (February 1993) 328–335.

[Sibson 1978] R. Sibson, “Locally equiangular trian-gulations”, Comput. J. 21 (1978) 243–245.

[Taylor 1976] J. E. Taylor, “The structure of singu-larities in soap-bubble-like and soap-film-like min-imal surfaces,” Ann. Math. 103 (1976), 489–539.

[Taylor 1983] J. E. Taylor, “Constructing crystallineminimal surfaces”, pp. 271–288, in Seminar onMinimal Submanifolds (edited by E. Bombieri),Annals of Math. Studies 105 Princeton Univer-sity Press, Princeton, NJ, 1983.

[Taylor 1988] J. E. Taylor, “Constructions and con-jectures in crystalline nondifferentiable geometry,”Proceedings of the Conference in Differential Ge-ometry, Rio de Janiero, 1988, Pittman PublishingLtd.

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[Tegart 1991] J. Tegart, “Three-dimensional fluid in-terfaces in cylindrical containers,” AIAA paperAIAA-91-2174, 27th Jount Propulsion Conference,Sacramento, CA, June 1991.

[Thompson 1887] W. Thompson (Lord Kelvin), “Onthe division of space with minimum possible error,”Acta Math. 11(1887), 121–134.

12 Availability

The Surface Evolver program is in the public do-main, and there is no charge for it. It is writtento be portable between systems. The only require-ment is a C compiler. So far, it has been ported toSun, Iris, NeXT, Xenix, and MS-DOS systems. Themajor effort in porting to a new system is writing ascreen graphics interface. However, this is fairly sim-ple, since the system-dependent routines need onlydisplay triangles. The program can also be run with-out any screen graphics, which makes it possible torun remotely.

A package containing source code, manual, andsample data files is available by anonymous ftp fromgeom.umn.edu (128.101.25.31) as pub/evolver.tar.Z.For those with NeXT computers, there is also a sepa-rate ftp archivepub/evolver.next.tar.Z which contains a NeXTexecutable Evolver and Interface Builder files. TheEvolver is also available on floppy disk from the au-thor. The manual in TEX dvi format is included in theftp archive. A hardcopy version of the manual can berequested separately from the author, or directly fromThe Geometry Center, 1300 South Second Street,Minneapolis, MN 54554. (Later note: the previousparagraph is obsolete. The Evolver home web pageis now http://www.susqu.edu/evolver. Geomview isnow available from http://www.geomview.org.)

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