Geometry/Unified Angles

Taking area of a region as a primitive notion, three species of angles are given a common basis. The magnitude of a slope, a hyperbolic or circular angle is determined by the area of an appropriate sector.

Preview: From locus to group

Given the base of a triangle, what is the locus of the vertex if the triangular area is to be kept constant? The answer, a line parallel to the base, is a feature of Euclidean geometry. This classically stated evocation of an image has been translated into group theory with its own terms. Furthermore, with analytic geometry, the image has expression with linear algebra as shear mapping. The collection of shear mappings forms a one-parameter group, in this case a transformation group that expresses a group action.

Another image is the rectangle {(0,0), (x,0), (0,y), (x,y)} in the Cartesian plane. What is the locus of (x,y) if the rectangle is to have constant area? The curve given by xy = constant is called a rectangular hyperbola. In this case, for c > 0, the squeeze mapping (x,y) → ( c x, y/c ) permutes the hyperbola – the area of the rectangle after transformation is the same as the area before. The collection of squeeze mappings forms a group with elements corresponding to c. Applied to the whole plane, a squeeze is a group action. Here is another one-parameter group.

Introduction

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 analogy between circular and hyperbolic functions as determined by corresponding sectors was noted by w:Robert Baldwin Hayward in 1892.

The unification here requires differences of slope as a third angle species. 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 species. 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 with determinant 1. 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 are mapped by the exponential function to multiplicative inverses.

Signed areas

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 circular angle of arc. The notion that angle of arc should be measured with an area ratio was expressed by w:Alexander Macfarlane in his 1894 essay on the definitions of trigonometric functions (page 9): "true analytical argument for the circular ratios is not the ratio of the arc to the radius, but the ratio of twice the area of the sector to the square of the radius."

Hyperbolic sectors corresponding to natural logarithm are constructed according to whether x is greater or less than one. A variable right triangle with area 1/2 is $V=\{(x,1/x),\ (x,0),\ (0,0)\}.$ The isosceles case is $T=\{(1,1),\ (1,0),\ (0,0)\}.$ The natural logarithm is known as the area under y = 1/x between one and x. A positive hyperbolic angle is given by the area of $\int _{1}^{x}{\frac {dt}{t}}+T-V.$ A negative hyperbolic angle is given by the negative of the area $\int _{x}^{1}{\frac {dt}{t}}+V-T.$ This convention is in accord with a negative natural logarithm for x in (0,1). Since T and V each have area 1/2, their difference is zero, so the hyperbolic angle is given by the natural logarithm.

The third angle species 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.

For this angle, the arc is a line segment that can be taken as the base of a triangle. A classical theorem of Euclidean geometry is "If the base and the area of a triangle be given, the locus of its vertex is a straight line parallel to the base." (see Robert Potts (1865) Euclid's Element of Geometry, page 285.)

Motions

Each species 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 species of angle 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 $\{\sum _{1}^{m}{\tfrac {1}{n}}\}_{m=1}^{\infty }.$ 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 $\int _{1}^{e}{\tfrac {dx}{x}}=\log e.$ 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 this third species 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 species.

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.