Abstract Algebra/Shear and Slope

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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:

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 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

Definition: is the set of dual numbers. The basis {1,e} characterizes it as a 2-algebra over R. If z = a+be, let z* = abe, a conjugate. Then

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

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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.

Planar rotations with complex, hyperbolic and dual numbers.

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.