Content

• A brief history of hyperbolic geometry
• Early hyperbolic browsing systems
• Available UI components and modules 2018

Euclid's Axioms 300 BC

• Any two points lie on a unique line.
• Any straight line can be continued indefinitely in either direction.
• You can draw a circle of any centre and any radius.
• All right angles are equal.
• If a straight line, crossing another two straight lines L, L',
  makes angles α, β with L, L' on one side,
  and if α + β < π, then L, L' if extended sufficiently far meet on that same side.

Playfair's Axiom 1795

• Any two points lie on a unique line.
• Any straight line can be continued indefinitely in either direction.
• You can draw a circle of any centre and any radius.
• All right angles are equal.
• If a straight line, crossing another two straight lines L, L',
  makes angles α, β with L, L' on one side,
  and if α + β < π, then L, L' if extended sufficiently far meet on that same side.
• In a plane, through a point not on a given straight line,
  at most one line can be drawn that never meets the given line.

Nikolai Lobachevsky 1823 (Janos Bolyai 1823 too)

• Any two points lie on a unique line.
• Any straight line can be continued indefinitely in either direction.
• You can draw a circle of any centre and any radius.
• All right angles are equal.
• In a plane, through a point not on a given straight line, at most one line can be drawn that never meets the given line.
• For any given line R and point P not on R, in the plane containing both line R and point P there are no/at least two distinct lines through P that do not intersect R.

What could this axioms be applyed to?

Spherical

Hyperbolic

Embedding

Projection

Only Parts of ℍ² fit into ℝ³

A Theorem of Hilbert states that it is not possible to place a full hyperbolc plane in ℝ³.

Negative Curved Plane Construction

Parts of ℍ² in Reality

Parts of ℍ² in ℝ³

Curved Space

Spherical

Euclidean

Hyperbolic

Embedding:

Projection:

Model:

𝕊

𝔼

𝔻 or ℍ

Lines:

arcs of great circles

euclidean lines

arcs orthogonal to boundary

Parallel lines:

through P ∊ L there is
no line not meeting L

through any P ∊ L there is
a unique line not meeting L

through P ∊ L there are
infinitely many lines not meeting L

Curvature:

>0

0

<0

Angle sum of triangle:

>π

π

<π

Circumference of circle:

2π sin r

2πr

2π sinh r

Area of circle:

4π sin² r/2

πr²

4π sinh² r/2

Poincaré Disk Model 1873

$$z_h \mapsto z_e \qquad z_h, z_e \in ℂ$$

$$translation:\ P \in ℂ$$

$$rotaion:\ \theta \in ℂ$$

$$transformation:\ z_e = {\frac{\theta z_h + P}{1 + \overline{P} \theta z_h}}$$

$$inverse:\ z_h = {\frac{\overline{\theta} z_e + -\overline{\theta}P}{1 + \overline{-\overline{\theta}P} \overline{\theta} z_e}}$$

$$composition:\ \Big(P_1,\theta_1\Big)\circ\Big(P_2,\theta_2\Big) = \Big(\frac{\theta_2 P_1 + P_2}{\theta_2 P_1 \overline{P_2} + 1}, \frac{\theta_1\theta_2+\theta_1\overline{P_1}P_2}{\theta_2 P_1 \overline{P_2}+1}\Big)$$

$$identity\ transformation:\ \Big(P=0, \theta=1\Big)$$

$$focusing\ point\ \alpha_h:\ \Big(P=-\alpha_h, \theta=1\Big)$$

Poincaré Disk Model Animation

lserf.shinyapps.io

Models

Length

Area

Straightness

Angle

Circle

1871

Beltrami–Klein model

❌

❌

✔️

❌

❌

1873

Poincaré disk model

❌

❌

❌️

✔️

✔️

1873

Poincaré half-plane model

❌

❌

❌️

✔️

✔️

Band model

〰️

❌️

❌️

✔️️️️️️

✔️

Poincaré Disk Model of Schläfli {7,3}, {∞,3} Tiling and Complete Tree

Early Systems

Pipeline

Known Implementations and Applications

Walrus

h3py

Inxight Applications

HyperTree

HyperProf

Roget2000

HVS

Ontology4us

JavaScript InfoVis Toolkit (JIT)

Treebolic

RougeVis

Hyperbolic Tree of Life 2018

Thank You For Your Attention

• Slides: glouwa.github.io/d3-hypertree-examples/hyperbolictree-slides
• Twitter: @GlatzhoferM
• Code: github.com/glouwa/d3-hypertree
• Demo: hyperbolic-tree-of-life.github.io