Geometry/Unified Angles

Slope, hyperbolic and circular angle placed on the same footing:


The hyperbolic angle is somewhat obscure but has been described in the wikibook Calculus. This chapter of Geometry will clarify the connection to circular angle, the one measured from zero to 360 degrees since Alexandria. The unification requires accepting differences of slope as a third angle type. One can use the phrase "angle of arc" in the unification, as hyperbolic and circular arcs arise and line segments serve as "arcs" for the third angle type. The arcs also connote motion along the arc, such as rotation of a circular arc about its center, permuting the points of the extended arc. These motions can be described by 2x2 real matrices, the kind used in linear algebra. Some reference to group theory is made: an additive group of angles corresponds under exponential isomorphism to a multiplicative group. Division in the group corresponds to subtraction of angles. Using standard positioning, the differences indicate directed angles. For instance, two oppositely directed angles map with exp to multiplicative inverses.


Traditionally circular arc has been measured as the ratio of its length to the radius, but here we use the area of sector of the arc when the radius squared is two, or r = √ 2. Then the circumference becomes an arc measured to be π r2 = 2π. A fractional sector has proportional area and gives the corresponding angle of arc.

For the hyperbolic angle, the area bounded by xy=1 measures an angle between (a,1/a) and (b,1/b), 0<a<b, to be log b − log a = log (b/a). The identification of sector area with   follows from an exchange of triangles with area one-half. To the area under y=1/x between a and b, add triangle {(0,0), (1,0), (1,1)} and take away triangle {(0,0), (x,0), (x,1/x)}.

The third angle type is easily described using slopes. A point (x,y), x>0, determines a slope m=y/x, and indicates an angle between the x-axis and the ray from the origin to (x,y). A triangle of area m is formed by the x-axis, the slope m line, and x= √2: A=(1/2)(√2)(m √2). For any two points on x=√2 an angle at the origin has magnitude equal to the area formed by rays to the points.


Each type of angle has a planar motion that moves it but keeps its magnitude constant. Since rotation leaves length invariant, area is also invariant, so circular angle is not modified in magnitude by rotation. The motions of the other angles are NOT length-preservers, but they are area-preservers.

In the case of hyperbolic angle the motion squeezes a square into a rectangle of the same area. A seeming paradox arises with area and the hyperbola y = 1/x: take the harmonic series   The terms in the sum become very small yet there is no upper bound on the sequence of partial sums. This divergence indicates that there is an infinite area between a hyperbola and its asymptotes.

Start with one wing, the unit area   In fact, there is a wing of hyperbolic angle between en and en+1 for any n, so the number of wings is infinite. A step from one wing to the next is made by the linear transformation that squeezes the unit square to a rectangle of length e and height 1/e. This feature of y=1/x was presented in 1647 by G. de Saint-Vincent as a characteristic of the quadrature of the hyperbola, and provided a geometric expression of natural logarithm, a more common designation of area also associated with hyperbolic angle, and here reinforced by wing measure.

A shear mapping takes a rectangle to a parallelogram of the same area as the rectangle. This motion of the plane increases or decreases the slopes of lines through the origin by a constant amount. The arc of the third type of angle is a segment on x= √2 that moves up or down with shearing, but the triangle with this segment as base, and apex at the origin, has constant area.

In the ring M(2,R) of 2x2 real matrices, the ones with determinant equal to 1 preserve area, and are elements of the special linear group SL(2,R), which forms a type of unit sphere in M(2,R). Three types of subgroups of SL(2,R) arise as exponential images of angles, which also express the planar motions characteristic of each angle type.

Though Leonard Euler is associated with the correspondence for circular angle, the other angles have been absorbed into a more general study of general linear groups GL(n,F) over a field F, and also of tangent vectors at the group identity, initiated by Sophus Lie. Indeed, the algebra of tangents at 1 is called Lie algebra.