Asymptotic combinatorial constructions of Geometries

04/10/2019 ∙ by Stanislaw Ambroszkiewicz, et al. ∙ 0

An approach to asymptotic finite approximations of geometrical spaces is presented.



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1 Introduction

Blackmore and Peters (2007) [3] presented the following problem in Open Problems in Topology II. Chapter 49 - Computational topology. Page 495:
Question 2.1. Is there a unifying topological abstraction covering manifolds, non–manifolds and other possible geometric models that might be useful to improve algorithmic design for geometric computations?

The geometric computations process finite combinatorial structures.

It seems that asymptotic finite approximations of geometrical spaces may answer the above question. That is, if a sequence of finite combinatorial structures (graphs) asymptotically and uniformly approximates a geometrical space, then all geometrical aspects of the space are encoded in the construction of the sequence.

Since the works of Alexandroff (1937) [1] and Freudenthal (1937) [11] it is known that any compact metric space can be approximated by a sequence of finite polyhedra, i.e. the space is the inverse limit of the sequence. Their work was continued by Mardevsic (1960) [16], Smyth (1994) [20], and Kopperman et al. (1997, 2003)[13, 14].

Asymptotic finite approximations of continuous mappings between such spaces are also important. They were investigated in Charalambous (1991) [7], and Mardevsic (1993) [17]. The recent papers by Debski and Tymchatyn (2017) [9] and (2018) [8] present a comprehensive (and up to date) approach to discrete approximations of complete metrizable topological spaces and continuous mappings.

This is the topological point of view, whereas we are interested in asymptotic and uniform finite approximations of geometrical spaces, as well as asymptotic finite approximations of continuous and smooth mappings between such spaces. The geometry is understood here in the spirit of Busemann’s (1955) book The geometry of geodesics [5]. His approach is based on axioms that characterize the geometry of a class of metric spaces (called G-spaces), and may be summarized as follows. The spaces are metric and finitely compact, i.e any bounded infinite sequence of points has at least one limit point; any two points can be connected by a geodesic.

The Busemann’s approach is general enough to include all complete Riemann and some Finsler spaces.

Summarizing the short review of the previous work, the mathematical framework for the constructive (and computational) approach to geometry has already existed for a long time, however, in the abstract topological setting. The idea explored in the paper is to adjust the abstract framework to geometry by explicitly introducing the geodesics. We propose a generic method for constructing finite approximations of geometrical spaces. In a sense, it may be viewed as a model of the axioms of the theory of G-spaces.

Hence, we follow the idea of Busemann that geodesics determine a geometry. However, in our approach, the most primitive notion is adjacency relation (finite graph) that is supposed to approximate a geometry. Geodesics in such a graph may be defined as minimal paths.

An inverse sequence of such graphs (inductively constructed according to a pattern) and its inverse limit (together with Hausdorff reflection) provide more than a topological space. If the pattern is sufficiently regular, then there is also a geometrical structure of the space with a pre-metric encoded in the pattern. Any geodesic in the space is the limit of corresponding geodesics of the finite graphs. A metric, consistent with the geodesics, may be introduced in many ways.

So that all topological features (invariants like homotopy type, covering dimension and embedding dimension), and the geometrical features (like flatness, elliptic or hyperbolic curvature) of the space can be deduced from its pattern. Such geometrical spaces are stand-alone constructions without any need of an ambient space, and present the intrinsic point of view in geometry, i.e. without the help of the notion of Euclidean spaces in particular.

We are going to show that the class of geometries that can be constructed by such patterns is quite large including also hyperbolic geometries. The crucial feature of the proposed approach is its computational aspect, i.e. the geometries are (by their constructions) asymptotically and uniformly approximated by finite graphs.

2 Working example

Figure 1: The duals to the first two quadrilateral tessellation graphs of torus

Let us consider flat torus and flat Klein bottle. Constructions of finite approximations for these geometries are extremely simple for any dimension . We will consider only 2-dimensional case that is sufficient to explain the basic idea. Also finite approximations of Euclidean spaces are simple and important. The unit interval and the unit square will serve as examples.

Classic torus as a subset of can be tessellated by convex quadrilaterals, however, not by equal squares because of the curvature. It is possible to tessellate torus with 4 quadrilaterals.

The next tessellation is done by dividing each of the 4 quadrilaterals into 4 smaller quadrilaterals. Then, each of the quadrilaterals may be divided into 4 smaller quadrilaterals, and so on.

Actually, any tessellation net (consisting of edges and vertices) is a graph. The dual (to this tessellation graph) is the graph where the vertices correspond to quadrilaterals, and edges correspond to the adjacency of the quadrilaterals.

The first two duals are shown in Fig. 1. The dual for the first tessellation is a multi-graph because some edges are duplicated.

We may construct the infinite sequence of finer and finer tessellations and corresponding dual graphs. For any of the graphs the geodesics are defined as the minimal paths. Let the length of the geodesics be normalized by the graph diameter. Then, the inverse limit of the graphs (with Hausdorff reflection) determines a flat metric space (known as flat torus) that locally looks like Euclidean space , i.e. for any point, there is an open neighborhood of the point that is isomorphic to an open subset of the Euclidean space . The space is homeomorphic to torus. By the famous Nash embedding theorem, a flat torus can be isometrically embedded in with a class 1 smooth mapping (see Borrelli et al. (2012) [4]) but not with a class 2 one.

Figure 2: The dual of the first quadrilateral tessellation graph of Klein-bottle
Figure 3: The dual of the second quadrilateral tessellation graph of Klein-bottle. It is composed of the two Möbius strips (where edges are solid lines) by joining (by edges shown as dashed red lines) the borders of the strips

Also Klein bottle can be tessellated by convex quadrilaterals. The simple argument is that the bottle can be constructed by joining the borders of two Möbius strips together, see Fig. 2 and Fig. 3. So that the infinite sequence of graphs (dual to finer and finer tessellations) can be constructed. The sequence determines the geometry known as flat Klein bottle. Although it cannot be embedded in (as a topological space), it is a concrete geometrical space that can be approximated by finite graphs. What is the specific structure of these finite graphs that determines the 4-embedding dimension of the Klein bottle?

Let denote the set of natural numbers without zero. For the flat torus, let us define explicitly the dual graphs denoted by . It is convenient to use the dot notation (ASN.1) to denote the sub-quadrilaterals resulting from consecutive divisions. The quadrilaterals (vertices) of , see left part of Fig. 1, are denoted by the letters and . The quadrilaterals (vertices) of are denoted by where and belong to the set .

The next finer tessellation results in quadrilaterals (vertices in ) denoted by labels of the form , where , and belong to the set .

In general case, the vertices in the graph are denoted by finite sequences of the form . Let denote the set of all such of length . So that, is defined as the set of vertices in the graph . The edges of are determined by the adjacency relation between the quadrilaterals.

For our purpose it is convenient to consider the adjacency relation (denoted by ) instead of the set of edges of the graph .

For less or equal to the length of , let denote the prefix (initial segment) of of length . Note that for any of length , the sequence () may be interpreted as a sequence of nested quadrilaterals converging to a point on the torus if goes to infinity.

Note that is symmetric, i.e. for any and in , if , then . It is also convenient to assume that is reflexive, i.e. any is adjacent to itself, formally .

Let denote the union of the sets for . The relations (for ) can be extended to , i.e., to the relation in the following way. For any and of different length (say and respectively, and ), relations and ) hold if there is of length such that is a prefix (initial segment) of , and holds, i.e. is of the same length as and is adjacent to .

3 Topology and geometry

We continue to use the flat torus as the working example.

Consider the infinite sequences of the form . Let denote the set of such sequences. By the construction of () any such infinite sequence corresponds to a sequence of nested quadrilaterals (on the 2D-torus) converging to a point. However, there may be four such different sequences that converge to the same point of the torus.

For an infinite sequence, denoted by , let denote its initial finite sequence (prefix) of length .

The set may be considered as a topological space (Cantor space) with topology determined by the family of clopen sets such that where is the length . Note that the adjacency relation is not used in the definition.

We are going to introduce another topological and geometrical structure on the set determined by .

Two infinite sequences and are defined as adjacent if for any , the prefixes and are adjacent, i.e. holds. Let this adjacency relation, defined on the infinite sequences, be denoted by . Note that any infinite sequence is adjacent to itself. Two different adjacent infinite sequences may have a common prefix.

The transitive closure of is an equivalence relation denoted by . Let the quotient set be denoted by , whereas its elements, i.e. the equivalence classes be denoted by and . Actually, is the inverse limit of the graph sequence, whereas is the Hausdorff reflection of the limit relatively to the equivalence relation .

There are rational and irrational points (equivalence classes) in . For the flat torus, each rational equivalence class has four elements, whereas the irrational classes are singletons. Each rational class corresponds to a vertex of a tessellation graph, and equivalently to a square face in .

By the construction of , any point of the initial 2D-torus corresponds exactly to one equivalence class (a point in ) and vice versa.

The set of rational classes is dense in .

The relation determines natural topology on the quotient set .

Usually, a topology on a set is defined by a family of open subsets that is closed under finite intersections and arbitrary unions. The set and the empty set belong to that family. Equivalently (in the Kuratowski style), the topology is determined by a family of closed sets; where finite unions and arbitrary intersections belong to this family.

Let us define the base for the closed sets as the collection of the following neighborhoods of the points of . For any equivalence class , i.e. a point in , the neighborhoods (indexed by ) are defined as the sets of equivalence classes such that there are and such that is adjacent to , i.e. holds. Note that may be equal to .

A sequence of equivalence classes of (elements of the set ) converges to , if for any there is such that for all ,    .

The topological space is a Hausdorff compact space. The sequence of graphs may be seen as finite approximations of the space , more exact if is grater. By the construction, the space is homeomorphic to Euclidean 2-torus. Actually, the graph sequence contains the geometric structure (specific to the flat torus) that is much more rich than the topology. The structure allows to define geodesics in many ways. The geodesics determine a pre-metric that is the basis to define concrete metrics on .

The normal metric and corresponding geodesics are defined in the following way. Let the k-geodesics are defined as the shortest paths in . The length of k-geodesic is the length of the corresponding shortest path. The distance between two vertices and of is denoted by , and defined as the length of a geodesic between and divided by the diameter of graph .

Metrics on is defined by

Let the metric be called normal.

Geodesic metric space means that for any its two points, there is a geodesic path between them whose length equals the distance between the points.

Hence, is a geodesic metric space known as flat torus.

Different metrics may be also defined for .

3.1 Summary of the example

Figure 4: The first four consecutive graphs of the unit interval
Figure 5: The first four consecutive graphs of the flat circle

The presentation of the example was a bit informal. In the next Section 4 a general and rigorous framework will be introduced.

Borelli et al. (2012) [4] constructed class 1 isometric embedding mapping of flat -torus into Euclidean space . Although, the embedding is interesting by itself, it does not explain (in a simple way) the nature of intrinsic geometry of the flat torus. The pattern of inductive construction of the finite graphs approximating the geometry of the flat -torus is extremely simple. It is even simpler than the pattern of Euclidean -cube. The flat torus of dimension may be taken as the basis, i.e. as the ambient space instead of the Euclidean -dimensional space.

The essence, of the asymptotic combinatorial construction of the flat 2-torus presented above, is the sequence of graphs (). Actually the topology and the geometry were defined on the basis of the sequence alone without reference to an Euclidean -torus that serves only as a helpful intuition. The same method can be applied to the unit interval (Fig. 4), circle (Fig. 5), the unit square (Fig. 6) as well as to -cube for arbitrary . Also dimensional torus and Klein bottle can be approximated by such sequences of finite graphs.

Actually, without correspondence to a metric space, we may construct a graph sequence () according to a regular and geodesic pattern, see the next Section 4. Then, the graph may be considered as an approximation of a geometry (may be a novel one), more exact if is approaching infinity. Hence, the method presented above gives rise to construct new intrinsic and stand-alone geometries that are geodesic metric spaces.

Figure 6: The duals of the first three quadrilateral tessellation of the unit square

4 Abstraction and generalization

We are going to generalize the notions introduced in the previous section.

Let () be a sequence of graphs and inverse mappings generated by an arbitrary pattern of inductive construction (denoted by ) such that:

  • denotes the set of vertices of . For , the sets and are disjoint.

  • Inverse mappings , for , are surjective, i.e. the image and the codomain are equal.

The sequence is called an inverse sequence.

Let denote the union .

Note that the inverse mappings determine a tree-like partial order on the set .

Inverse sequence is a special case of inverse systems (see [14], [20], and [8]) where topological spaces are substituted for finite graphs, and directed set is substituted for linear order.

For and , let denote the “prefix” of of length such that ,   , and so on. It is convenient to define the tree-like partial order (i.e. being a prefix denoted by ) determined by the inverse mappings. That is, if is a prefix of or . Let denote that is a prefix of .

Note that by the definition of the mappings , for any , its consecutive preceding elements are uniquely determined by the mappings. So that, these elements may be considered as “prefixes”. Although it is a slightly abuse of the notion of prefix as an initial segment of a sequence, it is a convenient notation that operates on the tree-like partial order instead of the inverse mappings, and is similar to the notion of prefix from the preceding section.

We say that and are of the same length if they belong to the same . Let the adjacency relation be defined in the very similar way as in the previous section.

Let denote an infinite sequence such that its k-th element belongs to .

A thread is defined as an infinite sequence such that for any ,   -th element (say ) of belongs to , and is the -th element of .

Let denote the set of all threads, whereas denote its Hausdorff reflection.

Since for any its preceding elements are determined by the inverse mappings, let (as an element of ) denote also the initial segment (“prefix”) of thread of length .

For these new (and more abstract and general) notions, to have the intended sense, some restrictions are necessary on the inductive construction of the sequence of graphs along with the inverse mappings.

4.1 Regular patterns

Pattern is regular if it satisfies the following two conditions.

  1. Connectedness: If and (belonging to , for some ) are adjacent, then there are two different elements of (say and ) such that and , and and are adjacent.

  2. Consistency: If of length , and of length are adjacent, then for any and any :    and are adjacent.

For any regular pattern , the adjacency relation for threads (elements of ) can be defined. Its transitive closure is an equivalence relation. So that, the Hausdorff reflection of relatively to , i.e. , is a compact Hausdorff space.

Since regular pattern determines the space , from now on, the space will be denoted by .

5 Finite approximations and continuous functions

Let regular patterns and be arbitrary.

A function is defined as monotonic if for any and , if , then .

Monotonic function is strictly monotonic if for any thread and any there is such that and .

A strictly monotonic function is defined as continuous if for any two adjacent (according to ) elements and of , the elements and are – adjacent.

This continuity may be seen as more intuitive if and are of the same length. Actually, it is the same; it follows from the definition of adjacency relation for objects of different length (see the end of Section 1), and from the strict monotonicity.

5.1 Extension to geometrical spaces

The extension of a strictly monotonic function to the function is natural. That is, is defined as element in such that for any there is such that is a prefix of .

The strict monotonicity of is essential here. An equivalent definition of the extension is that is the limit of the sequence . By the monotonicity, for any ,  . By the strict monotonicity, the length of consecutive elements of the sequence is unbounded.

The transformation of to the function is not always possible. Recall that denotes , and denotes .

5.2 The continuity of approximation functions is necessary

If is not continuous, then can not be a function. That is, there are two adjacent threads sequences and (i.e. ) such that . Since , the value can not be defined.

For and where the both patterns and are (i.e Euclidean pattern of dimension 1 corresponding to the unit interval, see Fig. 4), the proof is simple. Suppose that is strictly monotonic and not continuous. Then, there are and in (of the same length, because is a regular pattern) that are adjacent, and and are not adjacent. So that, either and are adjacent, or and are adjacent. That is, either and are equivalent (adjacent), or and are equivalent, where (resp. ) denotes the infinite sequence consisting only of (resp. of ). Let and ) be the infinite sequences from the above ones that are equivalent, i.e. they represent the same real number. By the assumption that and are not adjacent, and by the strict monotonicity of and the regularity of pattern ,    and ) are not equivalent and correspond to different real numbers.
Note that in the proof above, only the regularity of the pattern was essential. That is, the construction of and can be carried out for any regular pattern. So that we may generalize the proof and get the following conclusion.

5.3 Conclusion

For arbitrary regular patterns and , and strictly monotonic function :

  1. If is a function, then is continuous.

  2. If is continuous, then is also continuous in the classical sense, i.e. as the continuity of a function from topological space into topological space .

  3. For any continuous mapping (in the classic sense), there is a strictly monotonic and continuous function , such that .

It follows from (1) i (2) that if is a function, then is a continuous function.

The proof of (2) is simple and is as follows. If is not continuous, then there is sequence converging to such that the sequence converges to . Let , and , and for any . Then, there are and such that for all and all , the finite sequences and are not adjacent. Since is strictly monotonic, there is such that is a prefix of , and there is such that is a prefix of . By the regularity of the pattern , also and are not adjacent. Since the sequence (, ) converges to   (here the regularity of is essential), for sufficiently large , the finite sequences and are adjacent. Hence, function is not continuous, and the proof is completed.

In order to prove (3), let us suppose for awhile that is a continuous (in the classic sense) mapping on the unit interval . We are going to define a strictly monotonic and continuous function , such that and are the same up to the homeomorphism between and the unit interval.

For any , let denote the sub-interval of corresponding to . For any , the let be defined as such that is the smallest (with regard to inclusion) interval such that for any ,   . If such smallest interval does not exits, then by the continuity of , the function is constant on . Then, let be defined as the first of length greater than the length of , such that for any ,   . The function defined in this way is strictly monotonic and continuous. The classic continuity of is essential here. The extension of , i.e is the same (modulo the homomorphism) as .

In the general case, the proof is almost the same as for the Euclidean pattern . The only difference consists in definition of the set . Now, it is defined that , if for some .

5.4 Discussion

Is any mapping the extension of a strictly monotonic and continuous function ? If it is, then it is continuous.

The extension (defined at the beginning of Section 5) implies that if such that , then for any there is such that is a prefix of . In order to compute an initial segment (prefix) of the value , the function needs only an initial segment of elemnts of the equivalence class . So that, the computations done by the function are only potentially infinite. It is one of the postulates (axioms) of the Brouwer’s intuitionistic analysis.

In the classic Analysis, there are mappings that contradict the postulate.

As an example, let us define  (where the space is homeomorphic to the unit interval ) in the following way. If corresponds to a rational number, then let , where is the thread consisting of elements that all are equal to . Otherwise, let .

Another simple example is the mapping such that for any that corresponds to a real number less or equal , . Otherwise, . For the thread such that , the determination of the value must take the whole infinite sequence . Note that the mappings and are not continuous.

Strictly monotonic and continuous functions of type (where and are arbitrary regular patterns) are of particular interest in finite element method (numerical method for solving problems of engineering and mathematical physics), as well as in computer graphics where 2-dimensional grid of pixels, and 3-dimensional grid of voxels, correspond to , where determines the (2D and 3D) image resolution, finer if is bigger.

6 Geodesics and metrics

For regular pattern , let us consider the graph for a fixed . A path between vertices and in the graph is defined in the usual way as a finite sequence of vertices such that and , and for any , there is edge between and . Since the adjacency relation is supposed to be reflexive, there is loop (edge from vertex to itself) for each vertex in .

For any path its contraction is defined as the longest sub-path such that for any , the consecutive vertices and are different.

A -geodesic between two vertices in is defined as a shortest path between these two vertices.

Note that the phrase “shortest path” means that each edge has the same unit length. For a different meaning of “shortest path” see Section 10.

A -geodesic is called an extension of -geodesic (where ), if:

  • the contraction of path is the same as the path where is the prefix of of length .

Let a sequence of geodesics , where is a -geodesic, satisfies the following conditions:

  • for any ,   is a -geodesic and is its extension.

If such minimal number exists, then let it be denoted . Then, the limit of the sequence denoted , is defined as a -geodesic in the space .

Geodesics will be denoted by symbol (possibly with indexes), and identified with the corresponding sequence .

6.1 Geodesic patterns

A regular pattern is called geodesic if for any two points in there is a -geodesic between them.

The above definition of geodesics captures the geometrical intuition of shortest path, and is a bit different than the classic definition of geodesics in Riemannian manifolds and the theory of general relativity.

6.2 Pseudo–metric

Note that the length of a -geodesic (as the length of the corresponding path) can be normalized by the diameter of the graph . So that any –geodesic (as the limit of expanding -geodesics) may have finite length less or equal to . Usually, the normalization determines a metric (let it be called normal metric) on the space . In general case it is a pseudo–metric.

Pseudo–metric means that if the distance between two points, say and , is zero, then it is not necessary that . An example is presented in subsection 7.2.2.

The definition of -geodesics may depend on a priori metric where lengths of the edges of graph are not equal. As a good example, see the construction of a space similar to Euclidean 2-sphere in Section 10.

7 Examples

Figure 7: Circles. From the top: flat, elliptic at one point, finite hyperbolic at one point, and infinite hyperbolic at one point

Our intuition fails to imagine a flat circle, a flat torus, and a flat Klein bottle.

Circle is one of the simplest geometrical spaces. What is a flat circle? An Euclidean circle is elliptic. Is it possible that there is also a hyperbolic circle? The terms: flat, elliptic and hyperbolic refer to the notion of curvature.

For a regular and geodesic pattern , the curvature at a point may be defined on the base of the following simple idea.

Elliptic curvature at a point means that the normalized length of a -geodesic ending at this point is getting shorter if goes to infinity. This may be interpreted as gravitation force attracting to this point. The idea is consistent with the Einstein’s theory identifying gravity with space curvature.

Hyperbolic curvature at a point means that the normalized length of a -geodesic ending at this point is getting longer if goes to infinity. This, in turn, may be interpreted as repulsive force from this point, and interpreted as anti-gravity or the electromagnetic repulsion.

In order to explain the idea, let us consider the geometrical space corresponding to the flat circle, see Fig. 7, the first picture from the top. Its pattern is denoted by . Let the sequence () denote the corresponding inverse sequence. The graph consists of two vertices and . The set of vertices of graph is . The inverse mappings are defined as follows. For vertex of graph ,   .

The space (also denoted ) is homeomorphic and isometric to an Euclidean circle. However, it is not isomorphic because it is flat like the flat torus presented in Section 1.

Let us choose the point that corresponds to the two threads , where each element is , denoted by , and , where each element is , denoted by . Note that and are adjacent, and the equivalence classes and are equal.

7.1 Elliptic one-point curvature

Let us transforms the inverse sequence () into the inverse sequence denoted by (), see Fig. 7, the second picture from the top, in the following inductive way.

Let denote the length of , i.e. the number such that . Since the sets and for are mutually disjoint, the length is well defined.

For any of length , let be the set of all , such that .

For any the set of verices of is defined inductively as follows.

  • Let be equal to , and be equal to .

  • For

  • For

  • For

  • For

Edges of graph are determined by adjacency relation restricted to the set of the vertices of the graph.

Note that the above construction is generic, i.e. it can be done for any regular pattern. Also the number in (interpreted as the attraction factor) may be bigger. Then, if approaching the point, the distance to the point is getting much more shorter.

The finite elliptic curvature at the point seems to be equal . The formal definition of curvature is in the Section 8.2.

7.2 One-point hyperbolic curvature

Most models of hyperbolic space are constructed on the basis of Euclidean spaces, see the excellent exposition by Cannon et al. (1997) [6]. We are going to present a generic method for constructing intrinsic and stand-alone hyperbolic spaces.

On th base of the inverse sequence of the flat circle , we are going to construct two inverse sequence where the corresponding spaces have a point with finite hyperbolic curvature, and infinite hyperbolic curvature.

7.2.1 Finite hyperbolic one-point curvature

For a natural number , let us define function such that for any argument different than any prefix of and any prefix of ,  ; and for any natural number :
and .

Function may be viewed as -weak repulsion at point of the circle .

For the sake of presentation let us consider the case .

The function transforms the sequence
) into the sequence (, see Fig. 7, the second picture from the bottom, in the following way.

For any such that , let , i.e. be defined as the set of all , such that . Otherwise, let be the set of of length such that , i.e. is a prefix of .

  1. Let be equal to , and be equal to .

  2. Suppose that and are already defined. Then, let

    and for any , let be defined as such that .

Note that the above construction is generic, i.e. it can be done for any regular pattern and any parameter .

The finite hyperbolic curvature at the point seems to be equal . The formal definition of curvature is in the Section 8.2.

7.2.2 Infinite hyperbolic one-point curvature

For a natural number , let us define function such that for any argument different than any prefix of and any prefix of ,  ; and for any natural number : and .

The function is injective (on–to–one) and strictly monotonic, and may be viewed as -strong repulsion at point of the circle .

Let us consider the case .

The function transforms the inverse sequence () into the inverse sequence denoted by (), see Fig. 7, the first picture from the bottom, in the following inductive way.

For any such that , let . Otherwise, let be the set of of length such that , i.e. is a prefix of .

  1. Let be equal to , and be equal to .

  2. Suppose that and are already defined. Then, let

    and for any , let be defined as such that .

Edges of graph are determined by adjacency relation restricted to the set of the vertices of the graph.

Intuitively, the hyperbolic curvature at point of the space (interpreted as the velocity of a particle approaching the point) is infinite.

If in the definition of function we substitute for , then the distance between any two points (different than ) is zero. Hence, the distance is a pseudo–metric.

7.3 Inverse sequences for circles

Let the notion of circle be understood here as a closed, linearly ordered, and 1 dimensional geometrical space with a constant curvature.

There are three kinds of intrinsic stand-alone geometries that correspond to this notion.

The first one is the well known flat circle where the corresponding pattern is . The second kind consists of elliptic circles. The third kind is for hyperbolic circles. The Euclidean circles seem to be of the second kind.

7.3.1 Hyperbolic circles

Constructions of hyperbolic circles may use the method used in Subsection 7.2.1. The base is the flat circle and the pattern . Let points with a fixed finite hyperbolic curvature (like ) be called hyperbolic points. They can be introduced uniformly in the consecutive graphs in the following way.

  1. Starting with the second graph , and (corresponding to one point) are hyperbolic, as well as and (also corresponding to one point) are hyperbolic.

  2. Starting with the graph , and (corresponding to one point) are hyperbolic, as well as and (also corresponding to one point) are hyperbolic.

  3. Let us introduce the lexicographical order (denoted ) on the set assuming that . For graph , if and are adjacent and if , then and (corresponding to one point) are hyperbolic; otherwise and are hyperbolic.

  4. And so on. It resembles the construction of an Euclidean circle from regular -polygons inscribed in the circle. The limit of the polygons, as goes to infinity, is the circle.

The hyperbolic points may have different repulsion factors. So that, the resulting spaces may have hyperboloidal shape.

7.3.2 Elliptic circles

Constructions of elliptic circles may be based on the method used in Subsection 7.1 where one-point elliptic curvature was introduced into flat circle resulting in the space . In the very similar way as for hyperbolic cycles, one-point curvatures can be introduced in an uniform way. The points with such fixed finite elliptic curvature form a dense set in the circle. The elliptic metrics on the dense set can be extended to the whole circle.

When embedded into 2D Euclidean space, the radius of the corresponding circle is determined by the equation .

The elliptic points may have different attraction factors. So that, the resulting spaces may have ellipsoidal shape.

Also hybrid construction that use hyperbolic points and elliptic points may be interesting.

7.4 Summary of the examples

Let us summarize the examples of this Section. Three kinds of circles were presented: flat, elliptic, and hyperbolic.

The simple examples of elliptic and hyperbolic spaces presented in the section seem to be instructive, and can be easily generalized in order to construct intrinsic sophisticated hyperbolic and elliptic spaces on the base of flat -torus, or flat -Klein bottle.

It is interesting to construct an inverse sequence that gives intrinsic geometry isomorphic to an Euclidean 2-sphere, i.e the set of such that for a fixed radius . See Section 7.3 for a discussion. Any Euclidean 2-sphere is elliptic. Is it possible to construct inverse sequences that give flat 2-sphere or hyperbolic 2-sphere?

8 Parametrization of a geodesic

Let be a geodesic pattern, and () be the corresponding inverse sequence.

Note that any geodesic is homeomorphic to the unit interval .

Figure 8: Pre-parametrization of the geodesic of length from to

Let the space be given the normal metrics , so that the length of geodesics is defined.

Any geodesic determines uniquely a bijective mapping , where is the length of the geodesic, and is an interval, a subset of .

The mapping is called normal pre-parametrization of .

Note that is the normalization factor, because the maximal geodesic length in is according to the normal metric .

The first derivative of the mapping may be interpreted as an approximation of the velocity of a particle moving along the geodesic from its beginning at time to the end of the geodesic at time .

The idea is explained in Fig. 8. Here, a sequence of partitions of the unit interval is constructed from the one-point hyperbolic circle (see Fig. 7) by making the vertices and not adjacent for any . The graph determines partition of the unit interval, such that each vertex of has its own element of the partition. In the next partition , the element is divided into sub-intervals of the same size. The number of the sub-intervals is equal to the number of successors of in .

Let be -geodesic, see Section 6 for definition. It is the limit of the sequence of -geodesics. Let .

Note that is a linear subgraph (denoted by ) of . Since the pattern is fixed, let be denoted by .

The length of is equal to the number of vertices of minus . Let denote the set of vertices of graph , for .

Let .

Inverse mapping is constructed in the following way. For any vertex in let

such that .

Hence, the inverse sequence () has been constructed and determines the intrinsic stand-alone geometric space that corresponds to , and is also denoted by . Although, the space is homeomorphic to the unit interval, its geometry may be hyperbolic or elliptic, see the examples in Section 7.

Below we present the construction of the mapping .

For any in , let denote the maximal sub-path of such that for any in :  , i.e. .

Let us construct sequence of nested partitions of the interval .

  1. For . Number denotes the length of the sequence . If , then consists of one element . If , then partition of is defined as the set of the following sub–intervals (of length denoted by ) of the interval .

    The mapping from elements of (actually, ) onto is defined by the correspondence , for . If , then the mapping is . Let the mapping be denoted by .

  2. Suppose that for a fixed () the partition and the mapping have already been constructed.

    For any in , is the sub-interval denoted by . Let denote the length of the sub-interval.

    Let denote the length of the sequence . So that