Abstract Algebra/Shear and Slope

The first terms needed are triangle, base, vertex, and area. For instance, the proposition that for a triangle of given base and area, the locus of the vertex is a line parallel to the base. Imagine that the vertex is dragged along this line, deforming the triangle. Imagine also that the whole plane is similarly deformed by a transformation taking lines to lines. This transformation is a shear mapping.

The shear mapping is expressed as a linear transformation:

$(t,x){\begin{pmatrix}1&v\\0&1\end{pmatrix}}=(t,tv+x).$

Here it is written in the kinetic interpretation with a vertical (x) space axis as time (t) evolves horizontally, such as used in time series studies.

At t=1 the shear has transformed (1,0) to (1,v), the point where a slope v line intersects t=1. Thus the parameter v in the shear transformation can be called slope.

The rectangles given by constant t and x are transformed by the shear to parallelograms, but the area of one of these parallelograms equals the area of the rectangle before transformation. Thus shear transformations preserve area.

Let $e={\begin{pmatrix}0&1\\0&0\end{pmatrix}}$ and note that e² = 0, the zero matrix, and that the shear matrix is ve plus the identity matrix. Dual numbers are used in abstract algebra to provide a short-hand for the matrix subalgebra ${\begin{pmatrix}a&b\\0&a\end{pmatrix}}:$

Definition: $N=\{a+be:a,b\in R,\ e^{2}=0\}$ is the set of dual numbers. The basis {1,e} characterizes it as a 2-algebra over R. If z = a+be, let z* = a−be, a conjugate. Then

$(a+be)(a-be)=a^{2}$ since e² = 0.

Note that zz* = 1 implies z = ± 1 + be for some b in R. Furthermore, exp(be) = 1 + be since the exponential series is truncated after two terms when applied to the e-axis. Consequently the logarithm of 1 + ve is v. Thus v can be considered the angle of 1+ve in the same way that the logarithm of a point on the unit circle is the radian angle of the point, as in Euler’s formula (exp and log are inverses).

The shear mappings acting on the plane form a multiplicative group that is isomorphic to the additive group of real numbers.

The three angles

In Euclidean plane geometry there is the trichotomy right angle, acute angle, obtuse angle. Here a trichotomy of linear motions distinguishes three species of angle.

Each of the angles pivots on its peculiar motion: rotation for circular angle, squeeze mapping for hyperbolic angle, and shear for the slope. Furthermore, each motion evolves into its peculiar algebra of complexity: the dual numbers for shear, the split-binarions for hyperbolic angle, and the rotation and circular angle correspond to the plane of division binarions, called "complex numbers" by some. In fact, in the sense of a real 2-algebra, "complex" is ambiguous: each of the division binarions, split-binarions, and dual numbers forms a plane of "complex numbers".

A property of arc length on a circle is that it stays the same under rotation. It is said that "arc length is an invariant of rotation." A segment on t=1 that is transformed by a shear has the same length after the shear as before. Similarly, a hyperbolic angle is invariant under a squeeze. These three invariances can be seen together as consequences of area-invariance of the three motions: The hyperbolic angle is the area of the corresponding hyperbolic sector to xy=1, which has minimal radius √2 to (1,1). A circular angle corresponds to the area of its sector in a circle of radius √2. Finally, the slope is equal to the area of the triangle with base on t= √2 and hypotenuse corresponding to the slope. Since squeeze, shear, and rotation are all area-preserving, their motions in their corresponding planes preserve the central angles there. The traditional term for study of angle-preservation is conformal mapping, often presuming circular angles.

These three species of angle provide a parameter for polar coordinates in each of the three 2-algebras found as subspaces of 2 × 2 real matrices.