---
title: Cylindrical Gaussian Expansion Method
url: https://www.emergentmind.com/topics/cylindrical-gaussian-expansion-method
type: topic
---

# Cylindrical Gaussian Expansion Method

“Cylindrical Gaussian Expansion Method” is not a standardized name for a single method in the arXiv literature. The phrase spans several technically distinct constructions: a transform-based derivation of cylindrical Gaussian-type optical modes; a cylindrical Bessel expansion of paraxial fields such as Gaussian beams; the established Gaussian Expansion Method (GEM) for few-body Schrödinger equations, which is not formulated in cylindrical coordinates; cylindrical latent parameterizations for generating 3D Gaussian primitives in panoramic novel view synthesis; and, by contrast, some superficially similar titles in geomechanics that concern cylindrical cavity expansion rather than any Gaussian expansion. The most precise explicit mathematical framework closest to the phrase is the Hankel-transform construction of cylindrical Gaussian-type beams [2103.06988], whereas other uses require careful terminological separation [1705.08837], [1809.02619], [2603.05882], [2502.14233].

## 1. Terminological status and scope

The phrase is best treated as a family resemblance term rather than a canonical method name. In the supplied literature, four distinct meanings appear.

First, there is a genuinely cylindrical, Gaussian-centered optical construction in which Gaussian-like solutions are generated from a common Hankel-transform representation by choosing a spectral prefactor. This is the closest match to a “cylindrical Gaussian expansion” in a mathematical sense [2103.06988].

Second, there is a cylindrical-wave expansion for paraxial fields, including Gaussian beams, in which the basis functions are Bessel solutions of the Helmholtz equation. Here the field being expanded may be Gaussian, but the basis is not; accordingly, this is a cylindrical Bessel expansion of Gaussian beams rather than a Gaussian-basis expansion [1705.08837].

Third, there is the Gaussian Expansion Method in few-body quantum mechanics, where the total wave function is expanded in Gaussian basis functions over rearrangement sets of Jacobi coordinates. That method is explicitly formulated in spherical/Jacobi-coordinate language, not cylindrical coordinates [1809.02619].

Fourth, recent panoramic 3D Gaussian Splatting introduces a cylindrical triplane representation that predicts Gaussian primitives from a cylindrical coordinate lattice. This is a cylindrical formulation for Gaussian scene representation, but the paper names it a cylindrical triplane method rather than a Gaussian expansion method [2603.05882].

A frequent misconception is to treat these as interchangeable because they share the words “cylindrical,” “Gaussian,” and “expansion.” The literature does not support that conflation. Another misconception is to identify cylindrical cavity expansion in porous media with a Gaussian expansion technique; the geomechanics paper explicitly does not present anything called a “Cylindrical Gaussian Expansion Method” [2502.14233].

## 2. Hankel-transform generation of cylindrical Gaussian-type modes

The most direct mathematical realization of the phrase appears in the optical framework based on cylindrical symmetry and the Hankel transform. The paper formulates both the paraxial equation and the scalar d’Alembert equation in cylindrical coordinates, separates the azimuthal dependence as $e^{in\varphi}$, and diagonalizes the radial operator with the $n$th-order Hankel transform [2103.06988].

For the paraxial problem, the radial equation is
$$
\left(\partial_r^2+\frac1r\partial_r-\frac{n^2}{r^2}+2ik\partial_z\right)\Phi(r,z)=0,
$$
and the transform-domain evolution becomes
$$
(2ik\partial_z-s^2)\hat\Phi_n(s,z)=0.
$$
With the Gaussian spectral ansatz
$$
g_n(s)=\beta_n(s)e^{-w_0^2s^2/4},
$$
the master paraxial field is
$$
\Psi_{\mathrm p,k}(r,\varphi,z,t) = e^{ik(z-ct)}e^{in\varphi} \int_0^\infty \mathrm ds\, s\,J_n(rs)\,\beta_n(s)e^{-\alpha(z)s^2/4},
$$
where
$$
\alpha(z)=w_0^2+\frac{2iz}{k}.
$$
The exact d’Alembertian analogue is obtained by replacing $z$ with $\zeta=(z+ct)/2$ [2103.06988].

This construction is “universal” in the precise sense that different beam families arise from small changes in the spectral prefactor $\beta_n(s)$. The paper lists the following identifications:
$$
\beta_n(s)=s^n \quad\Rightarrow\quad \text{Gaussian beam},
$$
$$
\beta_n(s)=I_n(\chi s) \quad\Rightarrow\quad \text{Bessel-Gaussian beam},
$$
$$
\beta_n(s)=J_n(\chi s) \quad\Rightarrow\quad \text{modified Bessel-Gaussian beam},
$$
$$
\beta_n(s)=s^{n+2p} \quad\Rightarrow\quad \text{elegant Laguerre-Gaussian beam},
$$
$$
\beta_n(s)=s^{n-2},\quad n\ge 2 \quad\Rightarrow\quad \gamma\text{ beam},
$$
and more generally
$$
\beta_n(s)=s^{l-1},\quad l+n>-1 \quad\Rightarrow\quad \text{Kummer-Gaussian / HyGG family}.
$$

The significance of this framework is not merely that it produces several known beams. It shows that a large class of cylindrical Gaussian-type modes can be understood as a single Bessel/Hankel spectral synthesis with Gaussian regularization and a tunable prefactor. This suggests that, when the phrase “Cylindrical Gaussian Expansion Method” is used in a mathematically strict optical sense, the most faithful interpretation is a transform-based cylindrical spectral synthesis method with Gaussian regularization rather than a finite-dimensional Gaussian basis scheme [2103.06988].

## 3. Cylindrical Bessel expansion of Gaussian and paraxial beams

A related but distinct optical construction expands paraxial focal fields into exact cylindrical Helmholtz solutions. The field is modeled as
$$
E(x,z)=e^{ikz}U(x,z),
$$
with paraxial envelope satisfying
$$
\partial_x^2 U + 2ik\,\partial_z U = 0.
$$
The key step is a Fourier expansion of the far-field angular spectrum,
$$
a_F(\theta)=\sum_{n=-\infty}^{\infty} a_n e^{in\theta},
$$
with coefficients
$$
a_n=\frac{1}{2\pi}\int_{-\pi}^{\pi} a_F(\theta)e^{-in\theta}\,d\theta.
$$
Under the paper’s paraxial assumptions, this reduces to the focal-plane sampling rule
$$
a_n=\frac{1}{2\pi}E\!\left(-\frac{n}{k},0\right).
$$
The resulting exact cylindrical-wave expansion is
$$
E(r,\theta)= 2\pi \sum_{n=-\infty}^{\infty} i^n a_n J_n(kr)e^{in\theta}.
$$
For a Gaussian beam with focal-plane profile
$$
E_{GB}(x,0)=\exp\!\left(-\frac{x^2}{w_0^2}\right),
$$
the coefficients are
$$
a_n=\frac{1}{2\pi}\exp\!\left(-\frac{n^2\theta_0^2}{4}\right),
\qquad
\theta_0=\frac{\lambda}{\pi w_0},
$$
so the Gaussian beam is represented in a Bessel basis [1705.08837].

The distinction from the previous section is essential. Here the expansion basis is cylindrical Helmholtz modes $J_n(kr)e^{in\theta}$, not Gaussian basis functions. The Gaussian character resides in the incident field and in the coefficient law, not in the basis. The paper therefore provides an accurate cylindrical expansion of Gaussian/paraxial beams for scattering, but not a Gaussian-basis expansion method [1705.08837].

This distinction matters especially in terminology. Calling this construction a “cylindrical Gaussian expansion method” is only defensible if “Gaussian” refers to the field being expanded. If “Gaussian” is intended to describe the basis family, the terminology becomes misleading.

## 4. Relation to the few-body Gaussian Expansion Method

The established Gaussian Expansion Method is a high-precision computational framework for few-body Schrödinger equations. It uses the Rayleigh–Ritz variational method for bound states, the complex-scaling method for resonant states, and the Kohn-type variational principle to the $S$-matrix for scattering states. Its central idea is to expand the total wave function in few-body Gaussian basis functions spanned over all the sets of rearrangement Jacobi coordinates [1809.02619].

For a three-body bound state, the expansion is written over all three Jacobi-coordinate sets,
$$
\Psi_{JM}= \sum_{\alpha} A_\alpha \,\Phi_\alpha^{(1)} ({\bf r}_1, {\bf R}_1) +
 \sum_{\beta} B_\beta  \, \Phi_\beta^{(2)} ({\bf r}_2, {\bf R}_2) +
 \sum_{\gamma} C_\gamma\, \Phi_\gamma^{(3)} ({\bf r}_3, {\bf R}_3),
$$
with radial Gaussians of the form
$$
\phi_{nl}(r)=N_{nl}\,r^l e^{-\nu_n r^2},
\qquad
\nu_n=1/r_n^2,
\qquad
r_n=r_1 a^{n-1},
$$
and similarly for the second Jacobi radius. A characteristic practical device is the use of Gaussian ranges in geometric progression, which the paper states works well for both short-range and long-range behavior. It also introduces complex-range Gaussians,
$$
r^l e^{-\eta_n r^2},
\qquad
\eta_n=(1+i\omega)\nu_n,
$$
to improve the representation of oscillatory states [1809.02619].

None of this is cylindrical. The method is organized around Jacobi coordinates, spherical harmonics, angular-momentum coupling, and multiple rearrangement channels. The paper explicitly states that it does not formulate GEM in cylindrical coordinates and does not discuss a “cylindrical Gaussian Expansion Method” explicitly [1809.02619].

A plausible implication is that the phrase can nevertheless acquire a derivative meaning by analogy. The paper notes that the Gaussian basis expansion itself is coordinate-agnostic in principle, and it cautiously infers that one could imagine a cylindrical adaptation using cylindrically adapted Gaussian basis functions. That step, however, is not carried out in the paper. Accordingly, the few-body GEM is foundational background for Gaussian basis expansions, but not an instance of a cylindrical Gaussian expansion method in the strict literature-based sense [1809.02619].

## 5. Cylindrical Gaussian parameterization in panoramic 3D Gaussian Splatting

A modern computer-vision usage arises in panoramic novel view synthesis, where Gaussian primitives are generated from a cylindrical latent scene representation. The method introduces a camera-centered cylindrical triplane representation with three feature planes
$$
F_{r\theta},\quad F_{\theta z},\quad F_{zr},
$$
defined over local cylindrical coordinates $(r,\theta,z)$. For each camera, the volume branch samples a dense cylindrical grid of size
$$
N_r \times N_\theta \times N_z,
$$
with default settings
$$
(N_r, N_z, N_\theta) = (16, 64, 128),
$$
and a cylindrical support covering 10 m radius, 10 m height, and full $360^\circ$ azimuth. The triplane stores three two-dimensional feature planes instead of a dense three-dimensional field, reducing complexity from
$$
O(\Theta \cdot Z \cdot R)
$$
to
$$
O(\Theta \cdot Z + Z \cdot R + R \cdot \Theta)
$$
[2603.05882].

At a cylindrical grid point, the fused feature is obtained by summing the three plane features,
$$
\mathbf{f}_{r\theta z} = \mathbf{f}''_{r\theta}(k, i) + \mathbf{f}''_{\theta z}(i, j) + \mathbf{f}''_{zr}(j, k).
$$
An MLP then predicts local Gaussian attributes,
$$
\{\boldsymbol{\delta}_\text{local}, \mathbf{S}_\text{local}, \mathbf{R}, \alpha\}_\text{volume} = \text{MLP}(\mathbf{f}_{r\theta z}),
$$
where $\boldsymbol{\delta}_\text{local} = (\delta_r,\delta_\theta,\delta_z)$ and $\mathbf{S}_\text{local} = (S_r,S_\theta,S_z)$. The Gaussian center is transformed to Cartesian coordinates by
$$
(x', y', z') = (-(r+\delta_r)\sin(\theta+\delta_\theta), \quad z+\delta_z, \quad -(r+\delta_r)\cos(\theta+\delta_\theta)),
$$
and the scale is transformed through the Jacobian of the cylindrical-to-Cartesian map,
$$
\mathbf{S}' = |\mathbf{J}| \cdot \mathbf{S}_{\text{local}},
$$
with
$$
\mathbf{J} = \begin{pmatrix}
-\sin(\theta + \delta_\theta) & -(r + \delta_r)\cos(\theta + \delta_\theta) & 0 \\
0 & 0 & 1 \\
-\cos(\theta + \delta_\theta) & (r + \delta_r)\sin(\theta + \delta_\theta) & 0
\end{pmatrix}.
$$
The final Gaussian parameter set is
$$
\{\mathbf{x}', \mathbf{S}', \mathbf{R}, C, \alpha\}_{\text{volume}}
$$
[2603.05882].

In this context, “cylindrical Gaussian expansion” is not the paper’s terminology, but it is technically apt in a functional sense: a cylindrical latent grid is decoded into a dense set of 3D Gaussian primitives. The paper explicitly emphasizes that the cylindrical representation changes the scene feature representation and the Gaussian generation parameterization, while rendering remains standard 3DGS-style splatting after conversion to Cartesian coordinates [2603.05882].

The empirical evidence in the paper reinforces that the cylindrical parameterization is not a cosmetic coordinate change. In the volume-branch-only ablation on Matterport3D 2.0m, the cylindrical version achieves PCC 0.782, WS-PSNR 22.17, SSIM 0.782, and LPIPS 0.210, compared with 0.581, 19.22, 0.633, 0.398 for spherical and 0.564, 16.77, 0.495, 0.545 for Cartesian [2603.05882]. This suggests that, in machine-vision practice, the phrase may increasingly refer to cylindrical coordinate systems used to generate or organize Gaussian scene primitives rather than to classical basis expansions.

## 6. Distinction from cylindrical cavity expansion in geomechanics

An important negative case is provided by the CPTU geomechanics paper. Despite the title’s focus on cylindrical cavity expansion, the paper does not present anything called a “Cylindrical Gaussian Expansion Method.” Its actual contribution is a hydro-mechanically coupled cylindrical cavity expansion solution for porous soils under partially drained conditions, developed for interpreting CPTU excess pore-water pressure and penetration-rate normalization [2502.14233].

The formulation idealizes the cone tip as time-dependent expansion of a cylindrical cavity in plane strain, with the soil modeled as a perfectly elastoplastic Tresca material and pore-water flow governed by Darcy’s law. The problem is split into elastic and plastic zones separated by a moving elastoplastic boundary, which makes it a Stefan problem with dynamic boundary conditions. The analytical device is a variable transformation introducing the similarity-like coordinate
$$
\bar \xi = \frac{r}{2\sqrt{C_{hp}t}},
$$
which yields a solvable transformed equation in the plastic zone. The resulting expressions involve exponential integrals, for example
$$
U = A_e\, E_1\!\left(\frac{r^2}{4C_{he}t}\right)
$$
in the elastic zone [2502.14233].

This matters because the phrase “Gaussian expansion” could be misread into the diffusion-like similarity structure or into the appearance of exponential-integral functions. The paper explicitly rules out that interpretation: there is no Gaussian trial function, Gaussian kernel, Gaussian quadrature-based cavity approximation, or Gaussian expansion of fields. The only remotely Gaussian-like element is the standard diffusion scaling $r/\sqrt{t}$, which is not a Gaussian expansion technique [2502.14233].

The misconception is therefore categorical, not merely terminological. Cylindrical cavity expansion in this paper belongs to hydro-mechanical PDE analysis with a moving boundary; it is unrelated to Gaussian-basis expansions, Gaussian beam constructions, or Gaussian scene representations.

## 7. Conceptual synthesis and recurrent misconceptions

Across the cited literature, the common thread is not a single algorithm but the interaction of cylindrical geometry with Gaussian structure. The cylindrical component may refer to the coordinate system, symmetry class, latent scene support, or modal basis; the Gaussian component may refer to a basis family, a spectral envelope, a physical beam profile, or a rendered primitive. The “expansion” component may denote a transform synthesis, a modal series, a variational basis expansion, or the generation of many Gaussian primitives from a structured representation.

These distinctions give rise to several recurrent misconceptions.

**Gaussian beam versus Gaussian basis**: a cylindrical expansion of Gaussian beams in Bessel modes is not the same as a Gaussian-basis expansion. The former is exemplified by the Bessel series
$$
E(r,\theta)=2\pi\sum_n i^n a_n J_n(kr)e^{in\theta},
$$
whereas the latter would require Gaussian basis functions as the expansion family [1705.08837].

**Cylindrical symmetry versus cylindrical formulation**: few-body GEM uses Gaussian bases but is not cylindrical, because its architecture is built around Jacobi coordinates and spherical harmonics rather than cylindrical coordinates [1809.02619].

**Cylindrical Gaussian generation versus classical basis expansion**: the cylindrical triplane method in panoramic 3DGS generates Gaussian primitives from cylindrical latent features, but this is not a variational Gaussian basis method in the sense of few-body quantum mechanics [2603.05882].

**Cylindrical cavity expansion versus Gaussian expansion**: diffusion-type scaling and exponential-integral solutions in porous-media mechanics do not constitute a Gaussian expansion method [2502.14233].

Taken together, the literature supports a precise editorial conclusion. In the narrowest mathematical sense, the closest explicit exemplar of a Cylindrical Gaussian Expansion Method is the Hankel-transform framework for deriving cylindrical Gaussian-type light modes [2103.06988]. In broader contemporary usage, the phrase may also serve as a descriptive shorthand for cylindrical formulations that organize, generate, or expand Gaussian entities, such as panoramic 3D Gaussian Splatting [2603.05882]. It should not, however, be used indiscriminately for cylindrical Bessel expansions of Gaussian beams [1705.08837], for the few-body Gaussian Expansion Method in its standard Jacobi-coordinate form [1809.02619], or for cylindrical cavity expansion in geomechanics [2502.14233].

Source: https://www.emergentmind.com/topics/cylindrical-gaussian-expansion-method