Geometry
A two-dimensional real vector space with the Minkowski inner product is called 1+1 dimensional Minkowski space, often denoted R1,1. Just as much of the geometry of the Euclidean plane R2 can be described with complex numbers, the geometry of the Minkowski plane R1,1 can be described with split-complex numbers.
The set of points
is a hyperbola for every nonzero a in R. The hyperbola consists of a right and left branch passing through (a, 0) and (−a, 0). The case a = 1 is called the unit hyperbola. The conjugate hyperbola is given by
with an upper and lower branch passing through (0, a) and (0, −a). The hyperbola and conjugate hyperbola are separated by two diagonal asymptotes which form the set of null elements:
These two lines (sometimes called the null cone) are perpendicular in R2 and have slopes ±1.
Split-complex numbers z and w are said to be hyperbolic-orthogonal if <z, w> = 0. While analogous to ordinary orthogonality, particularly as it is known with ordinary complex number arithmetic, this condition is more subtle. It forms the basis for the simultaneous hyperplane concept in spacetime.
The analogue of Euler's formula for the split-complex numbers is
This can be derived from a power series expansion using the fact that cosh has only even powers while that for sinh has odd powers. For all real values of the hyperbolic angle θ the split-complex number λ = exp(jθ) has norm 1 and lies on the right branch of the unit hyperbola. Numbers such as λ have been called hyperbolic versors.
Since λ has modulus 1, multiplying any split-complex number z by λ preserves the modulus of z and represents a hyperbolic rotation (also called a Lorentz boost or a squeeze mapping). Multiplying by λ preserves the geometric structure, taking hyperbolas to themselves and the null cone to itself.
The set of all transformations of the split-complex plane which preserve the modulus(or equivalently, the inner product) forms a group called the generalized orthogonal group O(1,1). This group consists of the hyperbolic rotations — which form a subgroup denoted SO+(1,1) — combined with four discrete reflections given by
- and
The exponential map
sending θ to rotation by exp(jθ) is a group isomorphism since the usual exponential formula applies:
If a split-complex number z does not lie on one of the diagonals, then z has a polar decomposition.
Read more about this topic: Split-complex Number
Famous quotes containing the word geometry:
“I am present at the sowing of the seed of the world. With a geometry of sunbeams, the soul lays the foundations of nature.”
—Ralph Waldo Emerson (1803–1882)
“... geometry became a symbol for human relations, except that it was better, because in geometry things never go bad. If certain things occur, if certain lines meet, an angle is born. You cannot fail. It’s not going to fail; it is eternal. I found in rules of mathematics a peace and a trust that I could not place in human beings. This sublimation was total and remained total. Thus, I’m able to avoid or manipulate or process pain.”
—Louise Bourgeois (b. 1911)
“The geometry of landscape and situation seems to create its own systems of time, the sense of a dynamic element which is cinematising the events of the canvas, translating a posture or ceremony into dynamic terms. The greatest movie of the 20th century is the Mona Lisa, just as the greatest novel is Gray’s Anatomy.”
—J.G. (James Graham)