Polar Coordinate-Based Distance

• Polar coordinate-based distance is a method that decouples radial magnitude and angular orientation to form specialized metrics, similarity scores, or loss functions.
• It is applied in diverse fields such as sequence modeling, 2D pose estimation, and 3D detection, enhancing performance by separating magnitude effects from directional discrepancies.
• Many constructions yield quasi-metrics or non-traditional distance measures that require careful balancing of radial and angular components to capture task-specific geometries.

Polar coordinate-based distance denotes a family of radial–angular constructions rather than a single standardized mathematical object. In recent arXiv literature, the phrase may refer to an explicit metric or quasi-metric on polarized features, a kernel or similarity score whose variables are magnitude and angle, a task-specific loss built from radial and angular discrepancies, or a positional parameterization in which “distance” is represented directly by a radius from a chosen origin. In several influential cases, the resulting quantity is explicitly not a metric in the Euclidean polar sense, but a similarity, overlap surrogate, or assignment cost whose geometry is nonetheless organized by radial and angular coordinates (Andrikopoulos et al., 18 Jul 2025, Gopalakrishnan et al., 5 Sep 2025, Chen et al., 2022).

1. Radial–angular decomposition as a general principle

A recurring theme is the separation of a quantity into a magnitude-like component and an orientation-like component. In sequence modeling, "Decoupling the 'What' and 'Where' With Polar Coordinate Positional Embeddings" formalizes this by assigning content to nonnegative magnitudes and position to phases. Its Polar Coordinate Position Embeddings (PoPE) define the attention logit as

$a^{\text{POPE}_{ts} = \sum_{c=1}^{d} \hat{q}_{tc}\hat{k}_{sc} \cos\!\big((s-t)\theta_c + \delta_c\big),$

so the positional comparison is a multi-frequency cosine similarity over angular differences, while content only gates the amplitude of that comparison (Gopalakrishnan et al., 5 Sep 2025).

A related decomposition appears in probing work on syntax in LLMs. "A polar coordinate system represents syntax in LLMs" treats the displacement vector between contextual token embeddings as carrying both a radial component, via

$$\hat d(\mathbf h_i,\mathbf h_j)=\|\mathbf B_P \mathbf s_{ij}\|^2,$$

and an angular component, via cosine similarity to dependency-type prototypes. In that formulation, distance captures structural relatedness, whereas orientation captures dependency type and direction (Diego-Simón et al., 2024).

The same separation motivates quantization. "PCDVQ: Enhancing Vector Quantization for LLMs via Polar Coordinate Decoupling" rewrites vector error into radial and angular terms,

$$\|\mathbf{v}-\hat{\mathbf{v}}\|^2 = (\|\mathbf{v}\|-\|\hat{\mathbf{v}}\|)^2 + 2\|\mathbf{v}\|\|\hat{\mathbf{v}}\|(1-\cos\theta),$$

and argues that direction is much more sensitive than magnitude. On separately clustering directions and magnitudes of weight vectors in LLaMA-2-7B, the reported zero-shot accuracy drops are $46.5\%$ and $2.3\%$, respectively, which motivates allocating more quantization capacity to the directional component (Yue et al., 5 Jun 2025).

These works do not use a single common metric, but they share a precise geometric idea: a phenomenon is easier to model when radial and angular effects are decoupled rather than entangled inside one Cartesian or Euclidean criterion.

2. Explicit radial–angular metrics and quasi-metrics

Some papers do define distance functions directly in polar variables. The clearest example is "Topological Social Choice: Designing a Noise-Robust Polar Distance for Persistence Diagrams," which maps a persistence point $p=(b,d)$ to

$r=\sqrt{b^2+d^2}, \qquad \theta=\arctan2(d,b),$

and defines the pointwise Polar Persistence Distance

$d_{\text{polar}(p_1,p_2)= \sqrt{(r_1-r_2)^2+\alpha\sin^2\!\left(\frac{\theta_1-\theta_2}{2}\right)}.$

The radial term compares magnitude in the birth–death plane, and the angular term is periodic, smooth, and bounded. The paper states that this object is smooth away from the origin, continuously differentiable, and locally Lipschitz on compact subsets of $\mathbb{R}^2\setminus\{(0,0)\}$, but also states that it is a quasi-metric rather than a true metric because the triangle inequality may fail (Andrikopoulos et al., 18 Jul 2025).

"Polar Coordinate-Based 2D Pose Prior with Neural Distance Field" introduces a different explicit distance. Each 2D pose is represented by 15 joint connection vectors of the form $(\cos\varphi_i,\sin\varphi_i,r_i)$, and the proposed whole-pose distance is

$$\hat d(\mathbf h_i,\mathbf h_j)=\|\mathbf B_P \mathbf s_{ij}\|^2,$$0

The paper explicitly characterizes this as a non-geodesic arc-radius distance that separates angular and radial discrepancies. It is used to retrieve nearest real poses, construct a local prior pose, and provide scalar supervision for a neural distance field (Gan et al., 6 May 2025).

A third construction is more indirect. "Circular Clustering with Polar Coordinate Reconstruction" does not define a single named distance on $$\hat d(\mathbf h_i,\mathbf h_j)=\|\mathbf B_P \mathbf s_{ij}\|^2,$$1, but maps a point to cylindrical coordinates

$$\hat d(\mathbf h_i,\mathbf h_j)=\|\mathbf B_P \mathbf s_{ij}\|^2,$$2

unwraps the cylinder to

$$\hat d(\mathbf h_i,\mathbf h_j)=\|\mathbf B_P \mathbf s_{ij}\|^2,$$3

and then performs clustering with Euclidean distance in the reconstructed space. The effective distance in one unwrapped copy is therefore

$$\hat d(\mathbf h_i,\mathbf h_j)=\|\mathbf B_P \mathbf s_{ij}\|^2,$$4

while periodicity is handled by repeating periods so that points near $$\hat d(\mathbf h_i,\mathbf h_j)=\|\mathbf B_P \mathbf s_{ij}\|^2,$$5 and $$\hat d(\mathbf h_i,\mathbf h_j)=\|\mathbf B_P \mathbf s_{ij}\|^2,$$6 can still become neighbors (Sun et al., 2023).

Work	Mathematical object	Metric status
PPD	$$\hat d(\mathbf h_i,\mathbf h_j)=\|\mathbf B_P \mathbf s_{ij}\|^2,$$7	Quasi-metric (Andrikopoulos et al., 18 Jul 2025)
Pose prior	$$\hat d(\mathbf h_i,\mathbf h_j)=\|\mathbf B_P \mathbf s_{ij}\|^2,$$8	Distance used operationally (Gan et al., 6 May 2025)
Circular clustering	Euclidean distance after cylindrical reconstruction	Implicit effective distance, not a named metric (Sun et al., 2023)

3. Similarity kernels, overlap surrogates, and non-metric comparisons

A substantial part of the literature uses polar geometry without defining a distance function in the strict metric sense. PoPE is explicit on this point: its attention score is the real part of a complex inner product, so the positional term is a sum of cosine kernels over angular differences, not a Euclidean or polar distance. The paper states that the induced query–key comparison is “a radial-amplitude-weighted angular similarity over relative offsets, not a conventional distance” (Gopalakrishnan et al., 5 Sep 2025).

In radio interferometric reconstruction, "PolarRec: Radio Interferometric Data Reconstruction with Polar Coordinate Representation" likewise does not define a full polar metric between sample locations. Instead, it uses standard complex visibility error

$$\hat d(\mathbf h_i,\mathbf h_j)=\|\mathbf B_P \mathbf s_{ij}\|^2,$$9

and multiplies it by a radial weight

$$\|\mathbf{v}-\hat{\mathbf{v}}\|^2 = (\|\mathbf{v}\|-\|\hat{\mathbf{v}}\|)^2 + 2\|\mathbf{v}\|\|\hat{\mathbf{v}}\|(1-\cos\theta),$$0

The resulting Radial Visibility Loss emphasizes higher-radius, higher-frequency components while leaving the base error in complex-value space (Wang et al., 2023).

Remote-sensing detection offers two further non-metric constructions. "Objects detection for remote sensing images based on polar coordinates" defines a center-relative representation $$\|\mathbf{v}-\hat{\mathbf{v}}\|^2 = (\|\mathbf{v}\|-\|\hat{\mathbf{v}}\|)^2 + 2\|\mathbf{v}\|\|\hat{\mathbf{v}}\|(1-\cos\theta),$$1 and introduces the Polar Ring Area Loss

$$\|\mathbf{v}-\hat{\mathbf{v}}\|^2 = (\|\mathbf{v}\|-\|\hat{\mathbf{v}}\|)^2 + 2\|\mathbf{v}\|\|\hat{\mathbf{v}}\|(1-\cos\theta),$$2

This is an area-based discrepancy in polar space, meant to couple radius error and angle error rather than treat them independently. The paper also notes a limitation: if either the radius error or the angle error is exactly zero, the ring area becomes zero, so the loss must remain auxiliary (Zhou et al., 2020).

"Anchor Free remote sensing detector based on solving discrete polar coordinate equation" goes further toward overlap geometry. It approximates each rotated bounding box by an inner ellipse, samples the radial function over $$\|\mathbf{v}-\hat{\mathbf{v}}\|^2 = (\|\mathbf{v}\|-\|\hat{\mathbf{v}}\|)^2 + 2\|\mathbf{v}\|\|\hat{\mathbf{v}}\|(1-\cos\theta),$$3 angular bins, and defines

$$\|\mathbf{v}-\hat{\mathbf{v}}\|^2 = (\|\mathbf{v}\|-\|\hat{\mathbf{v}}\|)^2 + 2\|\mathbf{v}\|\|\hat{\mathbf{v}}\|(1-\cos\theta),$$4

The resulting JIoU is a discretized polar overlap similarity, not a classical distance metric; the paper uses $$\|\mathbf{v}-\hat{\mathbf{v}}\|^2 = (\|\mathbf{v}\|-\|\hat{\mathbf{v}}\|)^2 + 2\|\mathbf{v}\|\|\hat{\mathbf{v}}\|(1-\cos\theta),$$5 as a regression loss (Shi et al., 2023).

These formulations share a common pattern: polar coordinates organize the comparison, but the resulting mathematical object is a kernel, weighted reconstruction error, area discrepancy, or overlap ratio rather than a metric with the usual axioms.

4. Radial distance as a positional parameter in detection and perception

In multi-camera 3D detection, polar coordinate-based distance often means that distance from the ego vehicle is represented directly as a radial variable rather than implicitly inside Cartesian coordinates. "Polar Parametrization for Vision-based Surround-View 3D Detection" parameterizes each object center as

$$\|\mathbf{v}-\hat{\mathbf{v}}\|^2 = (\|\mathbf{v}\|-\|\hat{\mathbf{v}}\|)^2 + 2\|\mathbf{v}\|\|\hat{\mathbf{v}}\|(1-\cos\theta),$$6

with the decoded radial distance

$$\|\mathbf{v}-\hat{\mathbf{v}}\|^2 = (\|\mathbf{v}\|-\|\hat{\mathbf{v}}\|)^2 + 2\|\mathbf{v}\|\|\hat{\mathbf{v}}\|(1-\cos\theta),$$7

The paper argues that radial distance $$\|\mathbf{v}-\hat{\mathbf{v}}\|^2 = (\|\mathbf{v}\|-\|\hat{\mathbf{v}}\|)^2 + 2\|\mathbf{v}\|\|\hat{\mathbf{v}}\|(1-\cos\theta),$$8 is associated with object size in the image, azimuth $$\|\mathbf{v}-\hat{\mathbf{v}}\|^2 = (\|\mathbf{v}\|-\|\hat{\mathbf{v}}\|)^2 + 2\|\mathbf{v}\|\|\hat{\mathbf{v}}\|(1-\cos\theta),$$9 is associated with camera index and pixel position, and the circular perception region avoids the asymmetries of a rectangular Cartesian range. In label assignment, the matching cost explicitly includes

$46.5\%$0

which makes radial distance a first-class optimization variable (Chen et al., 2022).

"PolarFormer: Multi-camera 3D Object Detection with Polar Transformer" uses the same idea at the representation level. In its bird’s-eye-view formulation,

$46.5\%$1

so distance from the ego or camera origin becomes the radial axis of the BEV map. The paper argues that this radial geometry better matches wedge-shaped camera observations, but also shows that direct regression loss in polar coordinates performs poorly because of angular discontinuity: Cartesian prediction plus Cartesian loss gives $46.5\%$2 mAP / $46.5\%$3 NDS, polar prediction plus polar loss gives $46.5\%$4 mAP / $46.5\%$5 NDS, and polar prediction plus Cartesian loss gives $46.5\%$6 mAP / $46.5\%$7 NDS (Jiang et al., 2022).

In these works, “distance” is not primarily a metric between two arbitrary objects. It is a coordinate attached to each object location, chosen because the underlying image evidence is more naturally radial than orthogonal.

5. Discrete, probabilistic, and task-dependent uses

Several papers move away from continuous geometry altogether and use polar structure to define discrete path lengths, probability laws, or efficient search spaces. "The Cartesian Shortcut: Re-evaluate Vision Reasoning in Polar Coordinate Space" reformulates visual reasoning tasks on discrete polar grids made of rings and angular sectors. It explicitly states that Cartesian grid distances are translated to unit edge counting in polar space, so distance becomes a graph-theoretic quantity on the cell complex rather than an analytic formula in $46.5\%$8 and $46.5\%$9. The reported performance collapse from roughly $2.3\%$0–$2.3\%$1 on Cartesian layouts to $2.3\%$2–$2.3\%$3 on polar equivalents indicates that many current models are not topology-invariant even when task logic is preserved (Hu et al., 11 May 2026).

"Distance Distribution Between Two Random Nodes in Arbitrary Polygons" uses polar decomposition in a different sense. Letting the relative displacement between two uniformly sampled points be $2.3\%$4, it derives the distance PDF

$2.3\%$5

Here polar coordinates reduce a four-dimensional geometric probability problem to an angular integral over overlap areas of translated regions. The paper applies this framework to arbitrary convex or concave polygons, including coincident, overlapping, and disjoint cases (Pure et al., 2019).

In computational geometry, "Fast Algorithm for Finding Maximum Distance with Space Subdivision in $2.3\%$6" uses polar subdivision into eight sectors around the center of the axis-aligned bounding box to eliminate points that cannot be endpoints of the farthest pair. The final distance remains ordinary Euclidean distance, but angular sectorization accelerates the search by pruning non-extreme candidates (Skala et al., 2017).

These examples show that polar coordinate-based distance can denote not only a formula over $2.3\%$7, but also a discrete edge-counting rule, an overlap-integral representation of a distance distribution, or an angular subdivision scheme that supports exact Euclidean optimization.

6. Terminological divergence, caveats, and recurrent limitations

The expression is not stable across fields, and several papers explicitly mark that divergence. In coding theory, "An Efficient Construction Method Based on Partial Distance of Polar Codes with Reed-Solomon Kernel" states that its “distance” is a coding-theoretic partial distance associated with kernel rows and synthesized channels, not geometric distance in polar coordinates (Zhao et al., 2023). In operator algebra, "Distance to regular elements and polar decompositions in a $2.3\%$8-algebra" studies norm distance to regular elements via cut-downs and internal polar decompositions, where “polar” refers to polar decomposition $2.3\%$9, not coordinate geometry (Thiel, 26 Nov 2025). A plausible implication is that the phrase should be interpreted locally within each discipline rather than as a universal term of art.

Even within genuinely coordinate-based work, several limitations recur. PoPE states that its comparison is periodic and therefore aliasing-prone at any single frequency, so positional information depends on the collection of frequencies rather than one channel alone (Gopalakrishnan et al., 5 Sep 2025). PPD is explicitly not a true metric, requires an exclusion radius $p=(b,d)$0 near the origin, and has a less developed stability theory than bottleneck or Wasserstein distances (Andrikopoulos et al., 18 Jul 2025). The pose-prior distance is operationally useful, but the paper notes that, as written, the angular term uses $p=(b,d)$1, so the formula is not obviously symmetric in its two arguments (Gan et al., 6 May 2025). Polar Parametrization for surround-view detection introduces $p=(b,d)$2 because radial values can dominate angular terms numerically, which means that radial and angular coordinates require explicit balancing in both matching and loss (Chen et al., 2022). P-RSDet’s Polar Ring Area Loss cannot replace direct regression losses because it vanishes whenever either radius error or angle error is exactly zero, even if the other is wrong (Zhou et al., 2020).

A persistent misconception is that any polar coordinate-based distance must be the classical Euclidean expression derived from $p=(b,d)$3, $p=(b,d)$4, and $p=(b,d)$5. The recent literature does not support that identification. Some methods use explicit radial–angular metrics, some use kernels or overlap ratios, some encode distance as a task variable rather than as a pairwise metric, and some use “polar” in a non-coordinate sense entirely. The common element is not a single formula, but the decision to organize comparison, representation, or optimization around a separation between radial extent and angular structure.