See [url=https://www.geogebra.org/m/yrzmwxqv]this applet[/url] for the transformation between conic metrics: elliptic, parabolic and hyperbolic.

The same scalar [i]s[/i] is used as circular angle, parabolic angle and hyperbolic angle:[br][list][*][i]θ[/i] = [i]s[/i] is the circular angle for R to rotate on the unit circle.[/*][*][i]s[/i] is the parabolic angle for S to "rotate" on the unit parabola.[br]This parabolic rotation is the [url=https://en.wikipedia.org/wiki/Shear_mapping]shear mapping[/url] with shear factor [i]s = [/i]tan([i]φ[/i]) where [i]φ[/i] is the shear angle.[/*][*][i]ψ[sub]s[/sub][/i] = [i]s[/i] is the hyperbolic angle for H to "rotate" on the unit hyperbola.[br]This hyperbolic rotation is the [url=https://en.wikipedia.org/wiki/Squeeze_mapping]squeeze mapping[/url] with squeeze factor [i]k[/i] = [i]e[i][sup]s[/sup][/i][/i] where [i]s[/i] = [i]ψ[sub]s[/sub][/i] is the hyperbolic angle relating to the squeeze angle [i]ψ[/i] by tan([i]ψ[/i]) = tanh([i]ψ[sub]s[/sub][/i]).[/*][/list]The scalar [i]s[/i], as the [b]conic angle[/b], is defined to be [b]double the [/b][b]area[/b] of the corresponding [b]conic sector[/b].[br][list][*]For circular and parabolic angle, they are also the arclength of the corresponding conic arc.[/*][*]For the hyperbolic case, using [url=https://en.wikipedia.org/wiki/Minkowski_space#Norm_and_reversed_Cauchy_inequality]Minkowski metric[/url], we also have the "arclength" of the hyperbolic arc equals hyperbolic angle.[/*][/list]