Exact Cayley Polar Complex Numbers

Purpose

RiX has two exact representations of a complex number:

  • Cartesian form, x + y~{i};
  • Cayley polar form, Cayley(r, t).

The Cayley form stores magnitude and direction without storing a transcendental angle:

[ r = |z|, t = (/2). ]

The direction is the stereographic, tangent-half-angle, or Cayley coordinate of the unit circle. It is an exact algebraic alternative to conventional polar form. For exact rational Cartesian coordinates, conversion needs at most the positive square root of x^2 + y^2; it does not introduce π, trigonometric functions, or a floating approximation.

Construction and conversion

Convert an exact Cartesian value with one argument:

z := 1 + 1~{i};
c := .Complex.Cayley(z)       ## Cayley(sqrt2, sqrt2 - 1)
c.Cartesian()                 ## 1 + i

Construct directly from magnitude and direction with two arguments:

c := .Complex.Cayley(5, 1/2);
c.Cartesian()                 ## 3 + 4i

.Complex.Cartesian(c) is the namespace form of c.Cartesian().

For z = x + iy != 0, RiX computes the positive exact root

[ r= ]

and then

[ t==. ]

RiX uses the first expression when y != 0. This keeps the newly adjoined root in the numerator and gives an exact element of the same quadratic extension as r. The inverse conversion is

[ x=r, y=r. ]

These are identities, not numerical approximations. For example, 1 + i produces

[ r=, t=-1, ]

and substituting these exact values into the inverse formulas reduces back to x = 1, y = 1.

RiX interns a positive algebraic square-root generator for every nonsquare rational norm encountered. A perfect rational square is simplified, so 3 + 4i has r = 5 rather than a redundant root generator.

The one point at infinity

The negative real direction has theta = pi, hence finite affine t cannot represent it. Cayley direction therefore lives on the real projective line:

[ tR{}. ]

RiX exposes the single projective point as .Complex[:infinity] and formats it as Infinity:

.Complex.Cayley(-2)                 ## Cayley(2, Infinity)
.Complex.Cayley(-2).Cartesian()     ## -2

This is not a numeric infinity and arithmetic never performs ordinary numeric operations on it. There is only one such direction, so -Infinity is the same projective point.

Zero has no mathematical direction. RiX gives it the canonical storage form Cayley(0, 0) so equality, formatting, and arithmetic have one representation.

Arithmetic

Let direction composition be

[ t_1t_2=. ]

This is the tangent half-angle addition formula and is the Cayley transform of unit-complex multiplication. Multiplication is therefore

[ (r_1,t_1)(r_2,t_2)=(r_1r_2,t_1t_2). ]

The projective cases are handled explicitly:

[ , =, t/t (t). ]

Conjugation negates the finite direction and fixes infinity:

[ =(r,-t). ]

Consequently,

[ (r,t)^{-1}=(1/r,-t) ]

for nonzero r. Division composes the numerator direction with the negated denominator direction. Integer powers use exponentiation by squaring with the same Cayley multiplication, including negative powers through reciprocal.

Normal RiX operators use these rules directly:

a := .Complex.Cayley(3 + 4~{i});
b := .Complex.Cayley(1 + 1~{i});

(a * b).Cartesian()       ## -1 + 7i
(a / b).Cartesian()       ## 7/2 + 1/2i
(b^3).Cartesian()         ## -2 + 2i
a.Conjugate()             ## Cayley(5, -1/2)

When t1 and t2 belong to independent quadratic extensions, the displayed Möbius quotient can temporarily require inversion in their compositum. If the general exact-expression inverter cannot yet perform that inversion, RiX takes an exact Cartesian multiplication bridge and converts the product back to a canonical Cayley pair. No approximation is introduced. For example:

a := (5 + 3~{i}).Cayley();
b := 1/2 - 1/2~{i};
a * b                    ## Cayley(sqrt17, 4 - sqrt17)

The exact Cartesian product is 4 - i. This fallback also canonicalizes the otherwise separate magnitude factors sqrt34 * sqrt(1/2) as sqrt17.

Negation keeps the magnitude and adds the negative-real direction. For finite nonzero t, this changes t to -1/t; zero and infinity exchange places.

Addition is intrinsically less natural in polar coordinates. RiX implements + and - by converting both operands exactly to Cartesian form, adding or subtracting there, and converting the result back to Cayley form. The returned value is still Cayley:

sum := .Complex.Cayley(3 + 4~{i}) + .Complex.Cayley(1 + 2~{i});
sum.Cartesian()           ## 4 + 6i

Mixed Cayley/Cartesian arithmetic converts the Cartesian operand using the Cayley value’s configured i generator. Equality compares exact Cartesian values, so equivalent values compare equal even when their stored coordinates differ.

Methods and namespace operations

Operation Receiver form Namespace form Result form
Convert from Cartesian z.Cayley() for exact expressions .Complex.Cayley(z) Cayley
Construct coordinates — .Complex.Cayley(r, t) Cayley
Convert to Cartesian c.Cartesian() .Complex.Cartesian(c) Cartesian
Conjugate c.Conjugate() .Complex.Conjugate(c) Cayley
Reciprocal c.Inverse() .Complex.Inverse(c) Cayley
Components c.Re(), c.Im() .Complex.Re(c), .Complex.Im(c) exact scalar
Magnitude c.Magnitude() .Complex.Magnitude(c) exact real
Direction c.Direction() .Complex.Direction(c) exact real or Infinity
Squared norm c.NormSquared() .Complex.NormSquared(c) exact real

Arg is intentionally not an alias for Direction: Arg conventionally means a radian angle, while Cayley direction is tan(Arg/2). A future real and trigonometric subsystem may provide Arg without changing this representation.

Algebraic background: the Pell-type relation

For rational x and y, let r^2=q and suppose r is genuinely quadratic. Then finite direction can be written

[ t=ur+v, u,vQ, ]

with

[ u=1/y, v=-x/y. ]

It follows that

[ q u2-v2=1. ]

Equivalently, t has field norm -1 under the conjugation r -> -r. This is a useful characterization of rational Cartesian points in Cayley coordinates, but RiX does not impose it on general Cayley values: exact algebraic Cartesian coordinates are valid too.

Current exactness boundary

The first implementation computes a new magnitude generator when x^2 + y^2 is rational. This covers all Gaussian rational inputs and the primary exact use case. If the norm squared itself contains algebraic or transcendental generators, conversion reports that a rational norm squared is currently required. Supporting general real-algebraic root isolation belongs to the future algebraic-number layer; silently approximating such a magnitude would violate the contract of this type.

Back to top