*panes --*focusing on McCLIM, for such application, namely as with regards to the MetaCommunity.info (MCi) fork of the original McCLIM codebase. Although this initial study would be non-trivial, namely to my own limited academic knowledge of mathematical systems, but it's my hope that I may be able to trudge my way through this study, to develop something of an idea for a functional application of existing theory about hyperbolic graphs -- if not the broader algebraic topology -- namely towards a design of a new graphing component for McCLIM.

Reading [KobourovS2005], this evening, the authors denote two popular approaches for mapping a graph in the hyperbolic space onto the Euclidean plane:

- Poincaré Disk Model
- Beltrami-Klein Projections.

There are two additional approaches denoted by [ParksH]

- Poincaré Half-Plane Model
- Hyperboloid

This article, as a short collection of notes, will presently focus on the presentation in [ParksH]

- Beltrami-Klein Projections
- Named after the work of Eugenio Beltrami and Felix Klein (ca. 1870)
- Euclidean space is constrained by a
*disk*, for purpose of the projection *Points*are defined within the circumference of the*disk**Lines*are presented as*chords*between*points*- For points A, B not on the circumference of the disk, the geometric extension of each of A and B to the circumference of the disk -- respectively, to points P, Q -- may be required for some calculations -- such as to calculate the projected
*distance*between points A and B: - d(A,B) = 1/2 |log (AP*BQ) / (BP * AQ)|
- For a A situated on the circumference of the disk, A is denoted as an
*ideal point* - The
*pole*C for two points A, B, is the point on the Euclidean plane at which the lines defined as*tangent*to each of A and B meet in an intersection. C, of course, is not within the space of the*disk* - Concept of parallelism:
*Lines*are*parallel*if they do not intersect within the area of the*disk* - Concept of perpendicularity:
- For line M being a
*diameter*of the disk, N is perpendicular to M -- in the hyperbolic projection -- if N is perpendicular to M in a a sense of the Euclidean coordinate space - For lines M and N not being a
*diameter*of the disk: N is perpendicular to M, only if the*geometric extension*of N beyond the area of the*disk*would intersect the*pole*of M. - Angular measure: TBD
- Poincaré Disk Model
- Named after the work of Henri Poincaré (ca. 1880)
- Likewise, Euclidean space is constrained by a
*disk*, for purpose of the projection - Likewise,
*points*are defined within the*circumference*of the disk -- excluding the circumference or*boundary*of the disk. See also: [TernesM] - For a circle -- as a geometric object -- defined within the hyperbolic projection, the circle has essentially two functional
*center points --*one for the circle in the Euclidean space, and one for the circle in the hyperbolic space, as projected onto the Euclidean space. See also: [TernesM], which the following sub-points are in reference to - The midpoint of a line likewise has two coordinates -- one in the Euclidean space, and one in the projected hyperbolic space
- The definition of an angular
*bisector*must also be calculated for the characteristics of the projection - Two types of
*line,*for points A, B: - A d
*iameter*of the*disk*, passing through points A, B - An
*arc*passing through points A, B and ending as*orthogonal*to the circumference of the*disk*(see also: [VenemaG] p. 97, ch. 14) - Concept of parallelism (arcs) : That the arcs M, N do not intersect at any point within the disk
- Concept of perpendicularity: That two lines M, N meet
*orthogonally i..e*at a*right angle*(See also: Angular measure) - A relatively easy matter to calculate, computationally, for two lines that are
*diameters*in the Poincaré Disk projection - Less relatively easy to calculate, if either or both of M and N is an
*arc* - to calculate the projected
*distance*between points A and B, with ideal points P and Q: - d(A,B) = |log (AP*BQ) / (BP * AQ)|
- This is expressed in [VenemaG] (p. 97, ch. 14) as follows:
- d(A,B) = |ln(AB, PQ)|
- Angular measure:
*"To measure the angle between two lines in a**Poincaré Disk, you measure the Euclidean angle between their tangent lines."* - TBD: Some ambiguity as to what the term
*tangent line*may mean, in that definition. In any interpretation, that definition would be constrained only ti*arc*type lines within the*disk* - If the term refers to a line draw tangent to a point on the circumference of the
*disk,*and external to the circumference, but for any line AB, there would be two*tangent lines*in that interpretation. Those lines, of course, would be*parallel*if AB describes a*diameter*of the disk. - If the term refers to a line drawn tangent to each endpoint of an
*arc --*namely, at the points where the*arc*meets the*circumference*of the disk, ortogonally -- then for any line AB, there would be only two such*tangent lines*, both meeting on the Euclidean plane, at a point within the*circumference*of the*disk* - In either interpretation, there would be -- respectively --
*four*or*two*individual*tangent lines*for each*arc.*It would then be effectively impossible to measure the angle between tangent lines for two intersecting arcs - In a third interpretation: The term
*tangent line*, in that application -- and namely, for an*arc*type line -- might refer to a line drawn tangent to the arc, at the point of the intersection. For a non-arc type line, the line itself might be used for the determination of angle, at the intersection.*This is probably the correct interpretation of the term.*See also: [WolframP] *Limit Rays*: Arcs that meet at a common point on the circumference of the disk [WolframP]- The author describes a method for projecting a Beltrami-Klein Projection onto the Poincaré Disk Model

Of course, the hyperbolic trigonometric functions may have some applications in a graphing model for information visualization. See also: [KobourovS2005][KobourovS2012]

For a more in-depth analysis of the geometry of the hyperbolic space in the Poincaré Disk projection, see also: [SchutzA]

The qualities of the abstract geometric spaces of these projections would certainly make for a lengthy study, to the student of mathematics. Concerning a question of how these projections may be applied for information visualization, there are some many illustrations provided in [KobourovS2005] and [KobourovS2012]. Perhaps it may be well to consider some additional models, furthermore -- not limiting the discussion to the set of known hyperbolic projections.

Certainly, there is a significant variety of information visualization models defined in the information sciences. [HermanI] Particularly, either of the Cone Tree / Relative Disk Tree (RDT) projection models proposed in [JeongC] or the model proposed in [MelanconG] might serve to define a visually appealing graph with a minimum of spatial compaction, in a visual model for a directed acyclic graph (DAG).

Towards development of an ontology model, such as may be suitable for construction of a DAG view: In an application of SKOS, as within an OWL ontology model, it may be advisable to ensure that every

Of course, although the graph display procedures might be the most time-consuming features to design for an ontology visualization system -- as to display any single structure of any single graph of

*object property*defined in the ontology will be a subclass --*whether directly or indirectly -- of exactly one of the**object properties*, skos:broader or skos:narrower, With that level of consistency in an ontology, and given any single*individual*instance M1 within the ontology, it would be possible to define a simple algorithm for computing the*descendants*of M1 by following the graph of skos:narrower relations -- directly, or as the compliment of skos:broader - for all property expressions within a specific instance of applying the graphing algorithm. to the ontology, selecting each property of which M1 would be a*subject*, then subsequently taking each*object*of the relation as the*subject*for a subtree, for any number of*object properties*selected in any one instance of applying the graphing algorithm -- the graphing instance being constrained, for purpose of brevity, constrained to any number of generational iterations of the algorithm.Of course, although the graph display procedures might be the most time-consuming features to design for an ontology visualization system -- as to display any single structure of any single graph of

*individual*instances and their*object property*relations, within an OWL ontology -- however, a graph itself may not serve to convey all of the information presented in an ontology.
In at least, a complimentary table/page view model might be applied, as to create a focused view for display of ontologically structured information for any single

*individual*instance within the ontology. The table/page view might be applied, moreover, as to allow for a convenient editing onto the set of ontology properties defined to any single*individual*instance.
Considering the effective depth of complexity that this article has now arrived to, this article will conclude at the simple remark: "TO DO: Study"

Some additional works were discovered, during the research for this article, namely as with regards to hyperbolic projections onto the Euclidean space. [ChepoiV][BowditchB].

Some additional works were discovered, during the research for this article, namely as with regards to hyperbolic projections onto the Euclidean space. [ChepoiV][BowditchB].

*Non-Euclidean Spring Embedders*(2005)

[ParksH] Parks, Hal.

*The Poincare Disk Model and The Klein—Beltrami Model*.

[VenemaG] Venema, Gerard A.

*Exploring Advanced Euclidean Geometry with Geometer's Sketchpad*. Archived

[WolframP] Wolfram MathWorld.

*Poincaré Hyperbolic Disk*

[SchutzA] Schutz, Amy.

*Hyperbolic Functions*.

[TernesM] Ternes, Megan.

*Tangent Circles in the Hyperbolic Disk*

[KobourovS2012] Kobourov, Stephen G.

*Spring Embedders and Force Directed Graph Drawing Algorithms*(2012)

[HermanI] Herman, Ivan , Guy Melançon , and M. Scott Marshall. Graph Visualization and Navigation in Information Visualization: a Survey

[JeongC] Jeong, Chang-Sung and Alex Pang.

*Reconfigurable disc trees for visualizing large hierarchical information space.*See also: [Related]

[MelaonconG] Melançon G and I. Herman.

*Circular Drawings of Rooted Trees*

[ChepoiV] Chepoi, Victor

*et al. Diameters, Centers, and Approximating Trees of δ-Hyperbolic Geodesic Spaces and Graphs*

[BowditchB] Bowditch, B. H.

*Relatively Hyperbolic Groups*