Hyperbolic Browsing

Scalable Hierarchy Browsing in Hyperbolic Space

Michael Glatzhofer

Wed 31 Oct 2018 Updates at GitHub

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

Peter Mercator^{Public Domain}
Glatzhofer Michael^Observable
Trevorgoodchild^{Public Domain}

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

Frank Sottile^CC3
Daina Taimina^Web

Parts of ℍ² in Reality

Frank C. Müller^CC4
Toby Hudson^CC3
Glatzhofer Michael^CC4

Parts of ℍ² in ℝ³

1+2: C.T.J. Dodson^Web
3: David Dumas^Web
4+5: Ermishin Fedor^CC3

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

Ag2gaeh^CC4
Tomruen^{Public Domain}
Lasunncty^CC4
Loren Serfass^Web
Michael Glatzhofer^Observable
Parcly Taxel^{Public Domain}

Poincaré Disk Model 1873

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

$$translation:\ P \in ℂ $$

$$rotaion:\ \theta \in ℂ $$

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

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

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

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

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