Editing Hyperbolic angle (section)

===Relation To The Minkowski Line Element===
There is also a curious relation to a hyperbolic angle and the metric defined on [[Minkowski space]]. Just as two dimensional Euclidean geometry defines its [[line element]] as
:<math>ds_{e}^2 = dx^2 + dy^2,</math>
the line element on Minkowski space is<ref>{{cite web |last1=Weisstein |first1=Eric W. |title=Minkowski Metric |url=https://mathworld.wolfram.com/MinkowskiMetric.html |website=mathworld.wolfram.com |language=en}}</ref>
:<math>ds_{m}^2 = dx^2 - dy^2.</math>

Consider a curve embedded in two dimensional Euclidean space,
:<math>x = f(t), y=g(t).</math>
Where the parameter <math>t</math> is a real number that runs between <math> a </math> and <math> b </math> (<math> a\leqslant t<b </math>). The arclength of this curve in Euclidean space is computed as:
:<math>S = \int_{a}^{b}ds_{e} =  \int_{a}^{b} \sqrt{\left (\frac{dx}{dt}\right )^2 + \left (\frac{dy}{dt}\right )^2 }dt.</math>

If <math> x^2 + y^2 = 1 </math> defines a unit circle, a single parameterized solution set to this equation is <math> x = \cos t </math> and <math> y = \sin t </math>. Letting <math> 0\leqslant t < \theta </math>, computing the arclength <math> S </math> gives <math> S = \theta </math>. Now doing the same procedure, except replacing the Euclidean element with the Minkowski line element,
:<math>S = \int_{a}^{b}ds_{m} =  \int_{a}^{b} \sqrt{\left (\frac{dx}{dt}\right )^2 - \left (\frac{dy}{dt}\right )^2 }dt,</math>
and defining a unit hyperbola as <math> y^2 - x^2 = 1 </math> with its corresponding parameterized solution set <math> y = \cosh t </math> and <math> x = \sinh t </math>, and by letting <math> 0\leqslant t < \eta </math> (the hyperbolic angle), we arrive at the result of <math> S = \eta </math>. Just as the circular angle is the length of a circular arc using the Euclidean metric, the hyperbolic angle is the length of a hyperbolic arc using the Minkowski metric.