Azimuthal Equidistant Projection

• Azimuthal Equidistant Projection is a mapping that preserves radial distances and azimuthal angles, providing a direct link between geodesic and Euclidean coordinates.
• The projection operates identically on spherical, Euclidean, and hyperbolic manifolds, enabling efficient rendering via modular tessellation and geometric updates.
• Its applications in atom probe tomography and non-Euclidean visualization offer robust, low-error reconstructions compared to conventional projection methods.

The azimuthal equidistant projection is a mathematical mapping that sends points from a manifold of constant curvature to the Euclidean plane such that geodesic (radial) distances from a chosen origin are preserved and azimuthal (angular) relationships remain invariant. This projection provides a direct and linear relationship between intrinsic geodesic coordinates and projected Euclidean coordinates, enabling applications in both geometric rendering—particularly of non-Euclidean spaces—and in physical reconstruction pipelines, including atom probe tomography. Its suitability arises from exact distance-preserving properties along all radii and azimuthal fidelity, features retained regardless of whether the source geometry is spherical, hyperbolic, or Euclidean.

1. Mathematical Definition and Properties

Let $(M,g)$ denote a simply-connected two-dimensional Riemannian manifold of constant curvature $K$. Select a base point $O$. Every point $P\in M$ is associated with geodesic polar coordinates $(r,\theta)$:

• $r$ is the geodesic distance from $O$ to $P$;
• $\theta$ is the azimuthal direction of the initial geodesic, relative to a fixed reference ray.

The azimuthal equidistant projection $\pi: M\to\mathbb{R}^2$ maps

$K$0

The projected radial coordinate $K$1 satisfies $K$2, making the projection exactly radial distance preserving for any $K$3 (Osudin et al., 2019).

2. Invariance with Respect to Curvature

The projection is remarkable in that it operates identically for spherical ($K$4), Euclidean ($K$5), and hyperbolic ($K$6) geometries. For both spherical and hyperbolic cases, the law of cosines (spherical or hyperbolic) in a degenerate triangle (one leg vanishes) reduces to $K$7, so the geodesic distance is directly mapped to planar radius. For spherical manifolds with radius $K$8, the central angle is $K$9, yielding chord-length $O$0. In hyperbolic manifolds of curvature $O$1, the hyperbolic angle is $O$2 and again the projected radius $O$3 (Osudin et al., 2019).

This result is nontrivial since most map projections treat positive and negative curvature in incompatible ways. The azimuthal equidistant mapping does not require a curvature- or geometry-specific formula at the point of planar projection—only the correct geodesic distance is required.

3. Computational Workflow and Algorithms

The azimuthal equidistant projection enables an efficient and modular computational architecture for both graphics and data reconstruction. The canonical implementation in a graphics/physics engine consists of four steps (Osudin et al., 2019):

1. Storage: Every object is encoded in intrinsic geodesic polar coordinates $O$4 about its object center.
2. Geometric Update: Object motion and placement are computed using spherical/hyperbolic trigonometry (law of cosines; see Corollary 3.1, 3.2 in (Osudin et al., 2019)).
3. Tessellation: Edges are subdivided by computing geodesic midpoints, using repeated law-of-cosines applications (Corollaries 3.3, 3.4).
4. Projection: Each world-space $O$5 coordinate is mapped to $O$6 via $O$7, $O$8.

The critical feature is that the projection step is independent of $O$9, allowing the underlying curvature to be modified in real time without the need for re-projecting Euclidean coordinates; only the (intrinsic) geodesic distances are recalculated. The computational complexity per frame, for $P\in M$0 shapes, $P\in M$1 vertices, and $P\in M$2 subdivisions per edge, is $P\in M$3; trigonometric calculations constitute the dominant constant factor.

4. Applications in Tomographic Reconstruction and Imaging

The azimuthal equidistant projection finds critical application in fields such as atom probe tomography (APT), where it provides an analytic mapping between ion launch angles on a spherical-cap specimen and impact positions on a planar detector (Geuser et al., 2016). In the APT framework:

• Each ion is characterized by launch angles $P\in M$4 (polar and azimuth);
• The detector hit is recorded at radius $P\in M$5, preserving the launch azimuth $P\in M$6;
• The mapping from impact location back to surface coordinates is explicit: $P\in M$7.

This projection addresses deficiencies of the pseudo-stereographic (PS) projection, yielding reduced distortion in both angular and distance measurements across wide angular fields of view, a more stable image-compression factor (ICF), and decoupling parameter tuning from nonlinear effects. Quantitative comparisons show the equidistant model's ICF and local magnification remain nearly constant within a few percent, while the PS model exhibits variations up to $P\in M$8 over the detector area (Geuser et al., 2016).

5. Robustness, Error Propagation, and Limitations

Analysis of robustness demonstrates that the azimuthal equidistant projection is less sensitive to errors in calibration parameters (e.g., projection constant $P\in M$9, specimen radius $(r,\theta)$0, and projection center), with experimental fits showing $(r,\theta)$1 constant within $(r,\theta)$2 (vs. $(r,\theta)$3 for the PS model), and lower residuals in fit quality. Electrostatic simulations confirm the linearity of the $(r,\theta)$4–$(r,\theta)$5 relation to within $(r,\theta)$6 across a range of realistic specimen parameters (Geuser et al., 2016).

However, several practical limitations apply:

• The cost of geodesic tessellation increases cubically in the worst case for highly curved objects ($(r,\theta)$7).
• The hyperbolic plane is formally infinite; in graphics, a practical cut-off radius (e.g., the screen circle) is necessary (Osudin et al., 2019).
• On the CPU, trigonometric function evaluation is a performance bottleneck; suggested mitigations include parallelization, GPU offloading, lookup tables for sources of trig values, and adaptive tessellation for regions of low curvature.
• Extension to non-constant curvature or full 3D mappings would necessitate local integration of curvature, rather than the fixed-$(r,\theta)$8 law-of-cosines analytic approach.

6. Comparative Advantages and Scientific Impact

The azimuthal equidistant projection is linear, conserves azimuths exactly, and preserves inter-feature angular relationships across the mapped field. Its use in APT reconstruction improves distance and angular fidelity, especially for wide-angle data or materials requiring high planarity accuracy. The simplicity of requiring only a single projection constant $(r,\theta)$9 and its applicability for both spherical and hyperbolic spaces without modification enhance robustness and applicability. In rendering non-Euclidean geometries in real time, its $r$0-independence at projection decouples graphical updates from curvature changes, enabling efficient, dynamic visualization of variable-curvature spaces (Osudin et al., 2019, Geuser et al., 2016).

Comparison Model	Radial Mapping Law	Local ICF Stability
Azimuthal Equidistant	$r$1	Stable ($r$2)
Pseudo-Stereographic	$r$3	Varies ($r$4)

A plausible implication is that for computational visualization, scientific tomography, and other applications requiring high angular and metric fidelity under parameter uncertainty, the azimuthal equidistant projection should be considered the default mapping over traditional point-projection models.