[color=#ff7700][i][b][size=50][right]october 2021[/right][/size][/b][/i][/color][br][size=85][b]Poincaré[/b] disk model for the [i][b]hyperbolic plane[/b][/i] [math]\mathbb{H}_2[/math][br]A hyperbolic [i][b]LINE[/b][/i] is the intersection of the inner of [b]c[sub]abs[/sub][/b] ([math]\(\mathbb{H}_2\)[/math]) with a [color=#ff0000][i][b]circle[/b][/i][/color] in the extended complex plane [math]\mathbb{C}\cup\{\infty\}[/math] [br]perpendicular to [/size][size=85][size=85][b]c[sub]abs[/sub][/b][/size].[br]For two points [b]a[/b],[b]b[/b] in [math]\mathbb{H}_2[/math] there is an unique hyperbolic [i][b]LINE[/b][/i] through [/size][size=85][b]a[/b],[b]b[/b][/size][size=85].[br]The [i][b]hyperbolic distance[/b][/i]: |[b]a[/b] , [b]b[/b]|[b][sub]hyp[/sub][/b] = | [b]ln[/b](|[b]dv[/b]([b]a[/b], [b]b[/b], [b]s[sub]1[/sub][/b], [b]s[sub]2[/sub][/b])|) |, where [math]\mathbf{dv}\left(z_1,z_2,z_3,z_4\right)=\frac{z_1-z_3}{z_2-z_3}\cdot\frac{z_2-z_4}{z_1-z_4}[/math]: [color=#0000ff][i][b]complex cross-ratio[/b][/i][/color],[br]and [/size][size=85][size=85][b]{ s[sub]1[/sub][/b], [b]s[sub]2 [/sub][/b][/size][b]}[/b] = [/size][size=85][size=85][size=85][b]c[sub]abs [math]\cap[/math][/sub][/b][/size][/size][color=#ff0000][b] c[/b][/color] , [color=#ff0000][b]c[/b][/color] : the [/size][size=85][size=85]perpendicular [color=#ff0000][i][b]circle[/b][/i][/color] to [/size][size=85][size=85][b]c[sub]abs[/sub][/b][/size][/size] through [/size][size=85][size=85][b]a[/b],[b]b.[/b][/size][br][br]In the [b]GAUSS[i]ian plane [/i][/b][math]\mathbb{C}\cup\{\infty\}[/math] is the [color=#ff7700][i][b]hyperbolic parabola[/b][/i][/color] a [color=#ff7700][i][b]bicircular quartic[/b][/i][/color], with double-point on [/size][size=85][size=85][size=85][size=85][b]c[sub]abs[/sub][/b][/size][/size][/size], [br]symmetric to [/size][size=85][size=85][size=85][b]c[sub]abs[/sub][/b][/size][/size]. The [color=#0000ff][i][b]directrix[/b][/i][/color] isn't a [i]hyperbolic[/i] [i][b]LINE[/b][/i]![br][br][math]\hookrightarrow[/math] [color=#980000][i][b]geogebra-book[/b][/i][/color] [url=https://www.geogebra.org/m/fzq79drp][color=#0000ff][u][i][b]Brennpunkte und Leitlinien = foci & directrices[/b][/i][/u][/color][/url][br][math]\hookrightarrow[/math] [url=https://www.geogebra.org/m/xrdwb6nj][color=#0000ff][u][i][b]hyperbolic hyperbola[/b][/i][/u][/color][/url][br][math]\hookrightarrow[/math] [color=#0000ff][u][i][b][url=https://www.geogebra.org/m/fubg3ema]elliptic ellipse[/url][/b][/i][/u][/color][/size]