Exact Cayley polar form

Polar form without an angle

Ordinary polar form stores a magnitude and an angle. RiX's Cayley form instead stores Cayley(r, t), where r is the exact nonnegative magnitude and t = tan(theta/2) is the stereographic half-angle coordinate.

For 1 + i, both coordinates stay algebraic:

Runnable RiXRun cell

z := 1 + 1~{i}; c := .Complex.Cayley(z); {: c, c.Magnitude(), c.Direction() }

The result is Cayley(sqrt2, sqrt2 - 1). The positive sqrt2 generator is adjoined exactly; no decimal approximation or symbolic angle is introduced.

Exact conversion in both directions

For Cartesian x + iy, RiX computes r = sqrt(x^2 + y^2) and t = (r - x)/y. Conversion back uses only field arithmetic:

x = r(1 - t^2)/(1 + t^2) and y = r(2t)/(1 + t^2).

Runnable RiXRun cell

c := .Complex.Cayley(5, 1/2); z := c.Cartesian(); roundTrip := .Complex.Cayley(z); {: z, roundTrip }

This reconstructs 3 + 4i, and the round trip returns Cayley(5, 1/2). The namespace spelling .Complex.Cartesian(c) is equivalent to the receiver method.

Multiplication is direction composition

If directions are t1 and t2, the product direction is (t1 + t2)/(1 - t1*t2). This is the tangent half-angle addition law. Magnitudes simply multiply.

Runnable RiXRun cell

a := .Complex.Cayley(3 + 4~{i}); b := .Complex.Cayley(1 + 1~{i}); product := a * b; quotient := a / b; cube := b^3; {: product.Cartesian(), quotient.Cartesian(), cube.Cartesian() }

Multiplication, division, integer powers, reciprocal, negation, and conjugation all stay in Cayley coordinates.

If two directions use independent square-root generators and the exact inverter cannot simplify the Möbius denominator directly, RiX takes an exact Cartesian bridge and returns the canonical Cayley result. This remains exact:

Runnable RiXRun cell

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

The result is Cayley(sqrt17, 4 - sqrt17) and 4 - i, with no floating approximation.

Addition takes the Cartesian bridge

Complex addition is simplest in Cartesian coordinates. RiX converts exactly, adds there, and converts the result back. The operator still returns a Cayley value.

Runnable RiXRun cell

a := .Complex.Cayley(3 + 4~{i}); b := .Complex.Cayley(1 + 2~{i}); sum := a + b; difference := a - b; {: sum, sum.Cartesian(), difference.Cartesian() }

Conjugation and inspection

Conjugation keeps the magnitude and negates the finite direction. Component methods reconstruct exact values only when requested.

Runnable RiXRun cell

c := .Complex.Cayley(3 + 4~{i}); {: c.Conjugate(), c.Re(), c.Im(), c.NormSquared(), c.Inverse().Cartesian() }

Direction is deliberately not called Arg: it is tan(Arg/2), not an angle measured in radians.

The projective infinity direction

The negative real axis corresponds to t = Infinity. This is one projective direction, not floating-point infinity. RiX branches on it algebraically:

Runnable RiXRun cell

negative := .Complex.Cayley(-1); positiveAgain := negative * negative; rotated := negative * .Complex.Cayley(1 + 1~{i}); {: negative, positiveAgain, rotated.Cartesian() }

Here Infinity composed with itself gives direction zero, matching (-1)(-1) = 1.

Current boundary

The first implementation adjoins a magnitude root when the Cartesian norm squared is rational. That covers Gaussian rational values. General real-algebraic root isolation is future work; RiX reports this boundary instead of approximating silently.