A new technology of 3D object description is realized via VRML 2.0 language (Virtual Reality Modeling Language). This tool allow us to develop high quality computer models of 3D objects. A high expansion the technology is in the field of Virtual Cities and Virtual Multiuser Chats development. It’s role for a scientific goals is not so important yet. At the same time one of problems of modern geometry and topology is analityc description and visual representation of geometric objects. 
        Before VRML a numerous tecnical difficulties didn’t allowed us to create a convenient tool for computer modeling so realistic geometric models. Therefore the object presentations which we can see [2][6][7] are drawing as usual by artist arm. 
        In VRML some geometric objects are creating as geometric primitives: Box, Sphere, Cone, Cylinder. More can be result of 3D Editor, form lofting and set-theoretic operations: union, intersection, substraction. On the whole tools give us the possibility to build very impressed and realistic models but we need to have a lot of non-automatic work of coordinates input even for simplest one-sheet hyperboloid. 
        General description of composed object/polyhedron we have from IdexedFaceSet of VRML node. Here are coordinates of vertexes, edges and faces/polygons of polyhedron and some appearance characteristics e.g. color, transparency, reflection normals etc. Input of parameters is unproductive, so they are importing from 3D editor as usual. 

        It is more rationally to get ones from user program by direct calculations. On the pic.1 you can see the presentation of VRML program result whish was obtained from user’s VBasic program (CosmoPlayer 2.0). 

 

        In principle any polyhedron building algorithm which approximate the surface, is a complex (cell-like or simplicial) of this surface with orientation. In this connection we have some features of construction, especially in the critical and singular points as well. On the pic.2 we have a complex of sphere. 

        In common rectangular cells attach one to other during parallels by the one choice algorithm. Such points and cells which cover them we shall name common. Close to a pole the algorithm of construction must be changed: either the pole is covered by polygon or in the pole we have set of triangular cells -simplexes. The variant of solveng depends of concrete algorithm realization. But in any case we have two points with neighborhoods for algorithm construction changing. 

        We shall name such points by singular points of algorithm construction. Here we have some questions: which sort are such points, how much them, can we decrease their number and how to change algorithm construction in their neighborhood. 

Some definitions: 
        Indexed complex of surface we shall name indexed sequence of cells a₁, a₂, … , a_N, where N- the number of points in polyhedron. 
        By the chain of indexed complex we shall name indexed sequence of adherent cells a₁, a₂, … , a_n, such that any two cells a_i, a_i+1 with adherent indexes have common edge. 
        By the band we shall name the chain where any cell has incidence no more than two other cells. We shall name one by regular band if all cells haven’t neighboring common edges with other cells. 
        By the loop we shall name the close loop chain, i.e. the chain where any cell has incidence exactly two other cells. We shall name one by regular loop if all cells haven’t neighboring common edges with other cells. 

 

         Theorem 1. Exist one-to-one correspondence between indexed complex of oriented surface Q and tangent vector fields continuous everywhere except finite number points. 

        In other words for any algorithm construction we can find the tangent vector field which definite it and back to front – for any tangent vector field we can find algorithm construction of surface this field submitting. 

        The Hopf theorem [3]. If on the closed oriented surface Q is defined non-zero tangent vector field continuous everywhere except finite singular points then sum of all indexes of singular points equal c (Q). 

        So, according to this theorem, if we confine to closed surfaces only, then non-singular vector fields are exist on the torus only. And, according to the theorem 1, the algorithm of construction without singularities is possible for a torus only. 

        Proof of theorem 1. Let A={ a_i}- complex of surface Q. With any cell a_i we shall linked unit tangent turbulent vector field which goes along the edges in positive direction of vertexes index. In view of the simple connectivity of cell, the field can be prolonged continuously inside cell with one simple pole – center of turbulence. Thus we can construct the tangent field with n poles and c (Q)-n saddle points, according Hopf theorem. 
        On other hand, let P is a tangent vector field with finite number of singular points { p_i}. For the field with simple singularities the points are sourses, drains, centers of turbulent and settle points. Cut these points from surface together with small open convex polygons. In addition we should shift a little the field in order to one flow around cutting sections close to settle points. Then the field on the surface will be looks as a laminar flux, along wich we can do the complex of surface by the open regular bands which are going from pole to pole and reguar loops arrounding the poles. Thus we can construct the indexes of cells, starting from each singular point · . 

        From these theorems we have the minimal number of points are defined on the surface Q by Euler characteristic c (Q). It is well known that closed oriented non-singular manifold has homeomorphism to sphere with handles set. Adding one handle decrease Euler’s characteristic by 2. Therefore beginning with second handle we can construct the tangent vector field with settle points only. And it was minimal their number m=2k-2, where k is number of handles. 
 

Pic. 3. Sphere with handles

Contrary to expectations the number of points minimization of singular points for algorithm construction don’t advisable always. Consider the example of handle pasting in a sphere. In simplest case the handle pasting in a pole of sphere. In the case from poles we cut their polygons and past the cylinder with the same number of sides. Here is no problem with field prolongation evidently.

        But by this way we can’t past more than one handle. In general case we should choice a rectangle with laminar flux and replace it by other one with two cutting polygons and some reconstructions in their neighboring as it is shown on the pic.4 and pic.5. 

 

        We suppose the indexed complex existence on the sphere wich is induced by vector field with two pole outside selected square. 
        In other hand the handle is topological equivalence to cylinder and its vector field is done by topological multiplication S x I. In points of pasting we should provide the vector fields pasting too. On the pic. 4 we can see that the constructing field has two pole in cutting and four “half-settled” points with half-integer indexes –1/2 (simplexes N₁, N₂, N₃, N₄). It is clear by some transformation we can reduce four “half-settled” points to two real settled points X₁X₂, how it is shown on the pic. 5. 
        But in this way we can have evident bad appearance of the model. “Half-settled” point can be pasting by individual cell don’t incoming in any band, and settled point can be pasting by individual simplex. 

        Classical way of vector field construction with finite number of singular points is to definite the Morse function [5] on the surface, wich uniquely define the gradient vector field (vectors tangent to steepest descent). By the Morse theorem the non-singular critical points of Morse function unique associate with singular points of induced vector field and Euler characteristic of surface c (Q). Thus the effective Morse function definition on the surface we can take for most perspective first stage for algorithm of surface construction develop. 

        A torus is topological multiplication of two circles S x S. His homological group Z + Z has two generating homological loops C₁ and C₂. Their linear combination define new loop wich generates vector field on the torus. So the loop 2C₁+3C₂ is knotted on the torus. The closed knot is well known as a torical knot or a three-leaf. 
        That way shows us the principal possibility for VRML-modeling of topological knots using vector fields torsion on the handles of the sphere[8][9]. 
        We should mark as well numerous questions about linked and knotted surfaces, multicomponent compact and non-compact algebraic surfaces and surfaces with singularities. 

Pic.6. Linked toruses

Pic.7. 3D model of Mebius band.

        Among surfaces with edge we can select as most interest non-oriented manifolds. A simplest example Mebius band you can see on the pic 7. 
        In view that it is one-sided surface we should to do the band as double-sided surface or to use its its double covering. However first way is evidently simplest because we should do 180° sheet rotation per full cycle. In line of pasting together we shall have the algorithm construction changing as far as sheet changes his orientation here. 
 
        On the pic.8 we have 3D presentation of simplest non-oriented compact surface well known as “Klein bottle”. We carried on construction by horizontal bands with plain construction rotation. In the last step we had inverse pasting together first and last bands. Number of cells in the band must be even sertanly. 
        We should mark that transversal self-intersection of surface is a result of 3D inclusion impossibility. But the feature hasn’t any reflection in the algorithm construction and we didn’t cut the hole in the side of bottle. 

 

        We don’t consider theories of non-oriented surfaces in the paper. But for our goals here is one usefull feature – double covered r : Y® X of such non-oriented surface X. Under the covered manifold Y is oriented and Y hasn’t any difference with X by appearance. Set of Y vertexes are congruent to set of X vertexes, but faces in the VRML-description of Y we have done twice – with direct and inversed index of vertexes. Thus we have no troubles about inverse pasting edges of surface together in the pointline of the contact. Instead we can prolong our construction up to next contact. With a “Klein bottle” case the covering manifold is a topological equivalence for a torus wich twice envelop one. 

        In the conclusion we can mark that for a smooth geometric surfaces construction it doesn’t necessary indefinite subdivision of model by a set of small polygons. In last versions of VRML browsers is realized the Normal node which allow us to define the smooth field of surface’s reflection normals on the set of polyhedron’s faces. Thus we can have a good appearance with a real number of polygons in the model.