Hyperbolic Angle - Comparison With Circular Angle

Comparison With Circular Angle

In terms of area, one can consider a circle of radius √2 for which the area of a circular sector of θ radians is θ. (The area of the whole circle is 2π.) As the hyperbola xy = 1, associated with the hyperbolic angle, has shortest diameter between (−1, −1) and (1, 1), it too has semidiameter √2.

There is also a projective resolution between circular and hyperbolic cases: both curves are conic sections, and hence are treated as projective ranges in projective geometry. Given an origin point on one of these ranges, other points correspond to angles. The idea of addition of angles, basic to science, corresponds to addition of points on one of these ranges as follows:

Circular angles can be characterised geometrically by the property that the if two chords P₀P₁ and P₀P₂ subtend angles L₁ and L₂ at the centre of a circle, their sum L₁ + L₂ is the angle subtended by a chord PQ, where PQ is required to be parallel to P₁P₂.

The same construction can also be applied to the hyperbola. If P₀ is taken to be the point (1,1), P₁ the point (x₁,1/x₁), and P₂ the point (x₂,1/x₂), then the parallel condition requires that Q be the point (x₁x₂,1/x₁1/x₂). It thus makes sense to define the hyperbolic angle from P₀ to an arbitrary point on the curve as a logarithmic function of the point's value of x.

Whereas in Euclidean geometry moving steadily in an orthogonal direction to a ray from the origin traces out a circle, in a pseudo-Euclidean plane steadily moving orthogonal to a ray from the origin traces out a hyperbola. In Euclidean space, the multiple of a given angle traces equal distances around a circle while it traces exponential distances upon the hyperbolic line.