Polar-Cartesian Transforms

📐 Definition

For a complex number $z = x + iy$, the polar coordinates $(r,\theta)$ are

$$r = \sqrt{x^2 + y^2}, \qquad \theta = \operatorname{atan2}(y,x), \qquad z = r e^{i\theta}$$

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Conversely, given $(r,\theta)$,

$$x = r\cos\theta, \qquad y = r\sin\theta$$

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Domain & Codomain

Rectangular inputs $(x,y) \in \mathbb{R}^2$ map to $(r,\theta)$ with $r \ge 0$ and $\theta$ modulo $2\pi$; the inverse map takes $r \ge 0$ and $\theta \in \mathbb{R}$ to $\mathbb{R}^2$.

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⚙️ Key Properties

Consistency with Magnitude and Phase

The polar coordinates match the canonical magnitude and phase:

$$r = |z|, \qquad \theta = \operatorname{Arg}(z)$$

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Jacobian

For the map $(r,\theta)\mapsto(x,y)$ with $x=r\cos\theta$, $y=r\sin\theta$, the Jacobian determinant is

$$\left|\frac{\partial(x,y)}{\partial(r,\theta)}\right| = r$$

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Periodicity in Angle

Angles differing by $2\pi k$ (with $k\in\mathbb{Z}$) represent the same point: $(r,\theta)\sim(r,\theta+2\pi k)$.

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🎯 Special Cases and Limits

At $r=0$, the phase is undefined; away from the origin the transform is smooth. For purely real $z>0$, $\theta=0$; for purely imaginary $z=iy$ with $y>0$, $\theta=\tfrac{\pi}{2}$.

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🔗 Related Functions

Complex Magnitude Radial component for polar coordinates

Complex Phase Angular component for polar coordinates

Complex Multiplication & Division Composition via polar coordinates

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Usage in Oakfield

Applications in Oakfield

Oakfield uses polar/Cartesian ideas in three main places:

• Visualization: GPU views compute phase and magnitude from stored (re, im) values (atan + length) for phase- and energy-colored renders.
• Phase rotations / spectral factors: several operators build complex unit factors from angles (e.g. stimulus “rotation” parameters and spectral phase multipliers like dispersion’s cexp(I * phase)), which is effectively $(r,\theta)\mapsto(x,y)$ with $r=1$.
• Polar-mode operators: analytic_warp and remainder have a polar complex mode that converts (re, im) into “magnitude + direction” internally, applies a scalar update to the magnitude response, then writes it back along the current phase direction via $z/|z|$.

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Historical Foundations

1. 📜 Geometry of Complex Numbers

   Polar coordinates for complex numbers follow from the geometric interpretation of $\mathbb{C}\cong\mathbb{R}^2$, enabling a clean separation into radius and angle.

2. 🌍 Modern Perspective

   In numerical workflows, polar–Cartesian transforms are used to apply operations that act naturally on amplitude or phase while preserving the underlying complex structure.

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📚 References

• Ahlfors, Complex Analysis
• Needham, Visual Complex Analysis