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.
Read more about this topic: Hyperbolic Angle
Famous quotes containing the words comparison with, comparison, circular and/or angle:
“[Girls] study under the paralyzing idea that their acquirements cannot be brought into practical use. They may subserve the purposes of promoting individual domestic pleasure and social enjoyment in conversation, but what are they in comparison with the grand stimulation of independence and self- reliance, of the capability of contributing to the comfort and happiness of those whom they love as their own souls?”
—Sarah M. Grimke (1792–1873)
“But the best read naturalist who lends an entire and devout attention to truth, will see that there remains much to learn of his relation to the world, and that it is not to be learned by any addition or subtraction or other comparison of known quantities, but is arrived at by untaught sallies of the spirit, by a continual self-recovery, and by entire humility.”
—Ralph Waldo Emerson (1803–1882)
“‘A thing is called by a certain name because it instantiates a certain universal’ is obviously circular when particularized, but it looks imposing when left in this general form. And it looks imposing in this general form largely because of the inveterate philosophical habit of treating the shadows cast by words and sentences as if they were separately identifiable. Universals, like facts and propositions, are such shadows.”
—David Pears (b. 1921)
“The good lawyer is not the man who has an eye to every side and angle of contingency, and qualifies all his qualifications, but who throws himself on your part so heartily, that he can get you out of a scrape.”
—Ralph Waldo Emerson (1803–1882)