Nested Cylinder Representation

• Nested Cylinder Representation is a framework that uses cylindrical organization to encode geometry, symmetry, and structure across various fields such as graphics, CAD, topology, and physics.
• It is employed in 3D-aware image generation via concentric 'Swiss roll' feature cylinders to achieve multi-scale angular disentanglement and reduce rendering artifacts.
• Different disciplines interpret nesting variably—from literal concentric cylinders to structural and categorical hierarchies—illustrating its broad applicability and flexible use.

Searching arXiv for the provided works to ground the article in current records. “Nested cylinder representation” does not designate a single standardized construct across the arXiv literature. In the works considered here, it ranges from an explicit multi-scale family of concentric cylindrical feature surfaces for 3D-aware image generation, to a conformal cylinder that mediates a nested holographic correspondence, to cylindrical surface and cobordism models in combinatorics and topology, to inside-one-another cylinder geometries in scattering theory. By contrast, several CAD-oriented methods use the word “cylinder” for generalized sketch-extrude solids rather than classical circular cylinders, and these methods generally output flat unordered sets of primitives rather than a true hierarchical nesting structure (Jia et al., 21 Jul 2025, Filippas, 2024, Uy et al., 2021).

1. Terminological scope and domain-specific meanings

Across these literatures, the shared idea is not a single parameterization but the use of cylindrical organization to encode geometry, symmetry, composition, or boundary structure. In some cases the nesting is literal, as with concentric cylinders of increasing radius. In others it is structural, as when one holographic map lands on a cylinder that simultaneously serves as the boundary of another theory, or when a cylinder is the ambient surface for nested diagrammatics.

Domain	Cylinder object	Sense of “nested”
3D-aware generation	Concentric feature cylinders $F_{\theta y}^{r_i}$	Multiple radii, multi-scale composition
CAD reverse engineering	Extrusion cylinders	Usually flat, not hierarchical
Holography	$[\mathbb S^2\times\mathbb R]$	One holography inside another
Graph representation	Cylindrical rectangular dual	Periodic strip with layered order
Cobordism/category theory	Striped cylinder category $\mathrm{Cyl}$	Nested strata in a cylindrical ambient space
Casimir/scattering	One cylinder inside another	Interior multi-scattering geometry

A recurrent distinction is between explicit hierarchy and mere multi-part cylindrical decomposition. Point2Cyl and MV2Cyl are especially important on this point: both are CAD-semantic and more expressive than primitive fitting with planes, spheres, and classical cylinders, but neither infers a containment tree, parent-child relations, or an ordered constructive history (Uy et al., 2021, Hong et al., 2024). This suggests that the phrase is best interpreted locally, by field, rather than as a universal technical term.

2. Cylindrical feature fields in neural graphics

The most literal recent use of the phrase appears in “CylinderPlane: Nested Cylinder Representation for 3D-aware Image Generation” (Jia et al., 21 Jul 2025). CylinderPlane replaces Cartesian tri-planes with cylindrical-coordinate feature planes based on $(\theta,r,y)$, namely $F_{\theta y}$, $F_{r\theta}$, and $F_{yr}$. The motivation is representational rather than purely architectural: in the authors’ framing, tri-planes entangle front and back content because symmetric regions can share the same projected Cartesian feature coordinates, especially on the $xy$-plane. Cylindrical coordinates make azimuth explicit and thereby separate features across angle.

The nested component is defined as a family of concentric angular-height feature surfaces

$$F_{\theta y}=\{F_{\theta y}^{r_0},F_{\theta y}^{r_1},\ldots,F_{\theta y}^{r_N}\},\qquad r_0<r_1<\cdots<r_N.$$

Geometrically, these are concentric cylinders of increasing radius, described in the paper as resembling a “Swiss Roll.” The stated role of varying radius is multi-scale capture: different radii induce different effective angular resolutions and enable sampling from “all directions and more critical positions.” Within one CylinderPlane level, features from the three cylindrical-coordinate planes are bilinearly interpolated and summed before decoding. The exact compositing operator across nested cylinder levels is not explicitly specified in the provided text, and the paper correspondingly describes the mechanism through “combination,” “composites,” and “sampling from nested cylinder planes” rather than a closed-form aggregation rule.

The method also introduces seam regularization for the periodic boundary at $\theta=-\pi$ and $[\mathbb S^2\times\mathbb R]$0, because cylindrical coordinates introduce a discontinuity absent from Cartesian planes. The representation is described as renderer-agnostic and is intended to plug into differentiable volume rendering or DMTet-based rasterization. Empirically, against PanoHead, the reported values improve from FID-front $[\mathbb S^2\times\mathbb R]$1, FID-back $[\mathbb S^2\times\mathbb R]$2, and FID-all $[\mathbb S^2\times\mathbb R]$3, with the largest gain on rear views, which is consistent with the paper’s claim that angular disentanglement reduces Janus artifacts (Jia et al., 21 Jul 2025).

A related but distinct neural shape model is “Controllable Shape Modeling with Neural Generalized Cylinder” (Zhu et al., 2024). That work does not directly propose a nested cylinder representation. Instead, it defines a generalized cylinder around a $[\mathbb S^2\times\mathbb R]$4 center curve $[\mathbb S^2\times\mathbb R]$5, equips each cross-section with an oval-shaped local frame, and evaluates a neural signed distance field in relative coordinates $[\mathbb S^2\times\mathbb R]$6. Multiple generalized cylinders can be combined by taking the minimum of their predicted signed distances, yielding a multi-part union-of-fields model. This supports explicit scaffold-driven deformation, local scaling, twisting, and part mixing, and it provides quantitative reconstruction values such as CD $[\mathbb S^2\times\mathbb R]$7 for the 1-GC variant and CD $[\mathbb S^2\times\mathbb R]$8 for the multi-GC variant, against DeepSDF at CD $[\mathbb S^2\times\mathbb R]$9 (Zhu et al., 2024). A plausible implication is that NGC supplies much of the machinery needed for a future hierarchical cylinder model, but the paper itself does not formalize concentric or parent-child nesting.

3. CAD extrusion cylinders: expressive but usually flat

In CAD-oriented reverse engineering, “cylinder” often means an extruded volume rather than a circular analytic primitive. Point2Cyl formalizes an extrusion cylinder as

$\mathrm{Cyl}$0

where $\mathrm{Cyl}$1 is the extrusion axis, $\mathrm{Cyl}$2 the sketch-plane center, $\mathrm{Cyl}$3 a normalized closed 2D sketch, $\mathrm{Cyl}$4 its scale, and $\mathrm{Cyl}$5 the extrusion extents (Uy et al., 2021). The method predicts joint per-point barrel/base-instance probabilities $\mathrm{Cyl}$6, normals $\mathrm{Cyl}$7, and then extracts the axis by a closed-form eigenvector problem, the center from barrel points, the scale from projected barrel radii, and the sketch through a learned 2D implicit field. The representation supports boolean combinations conceptually, and the appendix notes that each predicted extrusion cylinder may be positive or negative via post hoc sign inference. However, the paper is explicit that it does not infer a hierarchical containment tree, parent-child structure, or ordered constructive history. In that sense it is a flat unordered set of extrusion primitives with later boolean interpretation, not a nested symbolic system.

MV2Cyl transfers this extrusion-cylinder idea to multi-view images. Its primitive is

$\mathrm{Cyl}$8

with axis $\mathrm{Cyl}$9, center $(\theta,r,y)$0, height $(\theta,r,y)$1, sketch $(\theta,r,y)$2, and scale $(\theta,r,y)$3 (Hong et al., 2024). The method trains separate 2D U-Net-based surface and curve segmenters, lifts them into 3D existence and attribute fields, then reconstructs per-instance extrusion parameters from surface support and base-curve support. The central claim is that surface-only extraction is suboptimal because of occlusion and segmentation difficulty, while curves sharpen sketch recovery. The reported results include, on Fusion360, E.A. $(\theta,r,y)$4, E.C. $(\theta,r,y)$5, E.H. $(\theta,r,y)$6, Fit Cyl. $(\theta,r,y)$7, and Fit Glob. $(\theta,r,y)$8; on DeepCAD, E.A. $(\theta,r,y)$9, E.C. $F_{\theta y}$0, E.H. $F_{\theta y}$1, Fit Cyl. $F_{\theta y}$2, and Fit Glob. $F_{\theta y}$3 (Hong et al., 2024).

A common misconception is that such extrusion-cylinder decompositions are already nested-cylinder representations. The literature here argues otherwise. Point2Cyl explicitly outputs an unordered set of extrusion volumes, and MV2Cyl likewise outputs a set $F_{\theta y}$4 rather than a feature tree. Because sketches are sets of non-self-intersecting closed loops, annular or hollow profiles are plausibly compatible with the parameterization, but MV2Cyl also explicitly states a limitation in predicting binary operations across primitives (Uy et al., 2021, Hong et al., 2024).

4. Cylinders as combinatorial and categorical ambient spaces

In graph representation theory, “Rectangular Duals on the Cylinder and the Torus” develops an exact characterization of when a graph embedded on a cylinder admits a cylindrical rectangular dual (Biedl et al., 8 Jun 2025). The representation lives on a flat cylinder modeled as a rectangular strip with identified left and right sides. A cylindrical rectangular dual is a contact representation by interior-disjoint rectangles such that no four rectangles share a point and the union of the rectangles forms a strip from the left to the right side of the cylinder. The paper proves that a cylindrical graph admits such a representation iff it is a properly triangulated cylindrical graph (PTC), and that recognition and construction can be done in linear time. The cylinder here is not volumetric but periodic, and the relevant “nesting” is layered ordering around the cylinder, encoded combinatorially by cylindrical regular edge labelings.

The low-dimensional categorical analogue appears in “Nested cobordisms, Cyl-objects and Temperley-Lieb algebras” (Calle et al., 2024). There, a nested $F_{\theta y}$5-manifold is a cylinder $F_{\theta y}$6 containing a 1-dimensional embedded submanifold, yielding the striped cylinder category $F_{\theta y}$7. Objects are circles with marked points $F_{\theta y}$8, and morphisms are generated by identities, twists $F_{\theta y}$9, births $F_{r\theta}$0, and deaths $F_{r\theta}$1, subject to a complete set of relations including the contractible-circle relation, snake relation, twist–birth compatibility, twist–death compatibility, and Dehn twist relation $F_{r\theta}$2. A $F_{r\theta}$3-object in a category $F_{r\theta}$4 is then a functor $F_{r\theta}$5. This links striped-cylinder geometry to affine Temperley–Lieb modules, cyclic objects, a doubling construction analogous to edgewise subdivision, and a cylindrical bar construction. The nesting is literally stratified: a 1-manifold sits inside a cylindrical 2-manifold, and the representation theory records that ambient containment.

5. Physics: nested cylinders as shared boundaries and interior geometries

In “Nested Holography,” the cylinder is $F_{r\theta}$6, more precisely its conformal class $F_{r\theta}$7, obtained from a spin-orbit duality that maps a massive theory in $F_{r\theta}$8 to data on a three-dimensional cylinder (Filippas, 2024). The basic kinematic statement is

$F_{r\theta}$9

with cylinder radius set by the Pauli–Lubanski invariant, classically $F_{yr}$0 and quantum mechanically $F_{yr}$1. The paper interprets the cylinder as null infinity and argues that the same conformal cylinder is also the boundary of the universal cover of compactified $F_{yr}$2. The nesting is therefore holographic rather than concentric: flat-space holography lands on the same boundary structure that serves ordinary AdS/CFT. The summary formula is a double bridge,

$F_{yr}$3

The paper further matches partition functions for free fermions across $F_{yr}$4, $F_{yr}$5, and $F_{yr}$6, stating $F_{yr}$7 up to dimensionless $F_{yr}$8-volume factors.

A more literal inside-one-another cylinder geometry appears in Casimir theory. For massless fermions with MIT bag boundary conditions, one paper studies two parallel cylinders with radii $F_{yr}$9, center offset $xy$0, and minimum gap $xy$1 (Teo, 2015). The exact energy has the TGTG form

$xy$2

with interior translation coefficients $xy$3, which distinguish the nested case from the exterior geometry. The small-gap asymptotic is

$xy$4

The scalar-field generalization in $xy$5-dimensional Minkowski space uses the same nested-cylinder geometry, now with TGTG matrices built from interior $xy$6-Bessel translations and an outer-cylinder interior scattering matrix $xy$7; the leading term scales as $xy$8 and coincides with proximity force approximation (Teo, 2015).

6. Cylindrical organization in geometric analysis and symplectic geometry

In symplectic embedding theory, the relevant nesting problem is whether the standard cylinder

$xy$9

fits into

$$F_{\theta y}=\{F_{\theta y}^{r_0},F_{\theta y}^{r_1},\ldots,F_{\theta y}^{r_N}\},\qquad r_0<r_1<\cdots<r_N.$$0

“Sharp symplectic embeddings of cylinders” proves the endpoint $$F_{\theta y}=\{F_{\theta y}^{r_0},F_{\theta y}^{r_1},\ldots,F_{\theta y}^{r_N}\},\qquad r_0<r_1<\cdots<r_N.$$1, and together with earlier results yields the sharp criterion

$$F_{\theta y}=\{F_{\theta y}^{r_0},F_{\theta y}^{r_1},\ldots,F_{\theta y}^{r_N}\},\qquad r_0<r_1<\cdots<r_N.$$2

(Pelayo et al., 2013). The proof combines explicit spiraling maps, a punctured-torus reduction, Hamiltonian separation of overlaps, and a smooth-family limiting theorem. Here “nested cylinder representation” refers neither to hierarchy nor to surface coding, but to a precise intermediate-dimensional embedding threshold.

Two rigidity results use cylinders to isolate and characterize surfaces. “New characterizations of the helicoid in a cylinder” studies a compact helicoid piece $$F_{\theta y}=\{F_{\theta y}^{r_0},F_{\theta y}^{r_1},\ldots,F_{\theta y}^{r_N}\},\qquad r_0<r_1<\cdots<r_N.$$3 inside a solid cylinder

$$F_{\theta y}=\{F_{\theta y}^{r_0},F_{\theta y}^{r_1},\ldots,F_{\theta y}^{r_N}\},\qquad r_0<r_1<\cdots<r_N.$$4

and analyzes it through concentric subcylinders

$$F_{\theta y}=\{F_{\theta y}^{r_0},F_{\theta y}^{r_1},\ldots,F_{\theta y}^{r_N}\},\qquad r_0<r_1<\cdots<r_N.$$5

(Lee, 2021). Via the coarea formula, shell-by-shell comparison on $$F_{\theta y}=\{F_{\theta y}^{r_0},F_{\theta y}^{r_1},\ldots,F_{\theta y}^{r_N}\},\qquad r_0<r_1<\cdots<r_N.$$6, and the fact that helices are shortest connecting curves on the developed cylinder, the paper proves area-minimizing and uniqueness characterizations. This is a genuine nested-shell analysis: the helicoid is reconstructed from helical traces on all concentric cylindrical shells.

“A characterization of the grim reaper cylinder” proves that a connected properly embedded translator in $$F_{\theta y}=\{F_{\theta y}^{r_0},F_{\theta y}^{r_1},\ldots,F_{\theta y}^{r_N}\},\qquad r_0<r_1<\cdots<r_N.$$7 with uniformly bounded genus on compact sets, which is $$F_{\theta y}=\{F_{\theta y}^{r_0},F_{\theta y}^{r_1},\ldots,F_{\theta y}^{r_N}\},\qquad r_0<r_1<\cdots<r_N.$$8-asymptotic outside a solid cylinder to two half-planes, is either flat or the grim reaper cylinder (Martin et al., 2015). The cylinder

$$F_{\theta y}=\{F_{\theta y}^{r_0},F_{\theta y}^{r_1},\ldots,F_{\theta y}^{r_N}\},\qquad r_0<r_1<\cdots<r_N.$$9

isolates a compact nonlinear core, while the exterior half-plane ends force a slab-width conclusion $\theta=-\pi$0, $\theta=-\pi$1. A related planar-section rigidity theorem shows that if a complete connected smooth surface has the central cross-cut property and one plane cuts it in a clean figure-8, then the surface is a geometric cylinder over a central figure-8 (Solomon, 2015). In both cases, cylindrical organization is not merely descriptive; it is rigidity-generating.

7. Analytical, algebraic, and packing-theoretic extensions

The phrase “representation” also appears in non-geometric senses. “On the representation of cylinder functions” derives a mixed integral–finite-sum formula for $\theta=-\pi$2,

$\theta=-\pi$3

where $\theta=-\pi$4 is the regularized incomplete gamma function and $\theta=-\pi$5 is a finite correction term (Micheli, 2019). The nesting here is analytical: an outer integral contains a special-function kernel that itself depends on the integration variable, and the full representation combines that integral with a finite corrective sum. The paper develops analogous formulas for $\theta=-\pi$6, $\theta=-\pi$7, $\theta=-\pi$8, and $\theta=-\pi$9.

In algebraic geometry, “Isomorphisms between cylinders over Danielewski surfaces” studies cylinders $[\mathbb S^2\times\mathbb R]$00 over special Danielewski surfaces and shows that two such cylinders are isomorphic whenever the bases have the same number of multiple origins (Moser-Jauslin et al., 2020). The mechanism is Danielewski’s fiber-product trick: both cylinder structures are realized through a common fiber product over the affine line with $[\mathbb S^2\times\mathbb R]$01 origins, so the same threefold is simultaneously a cylinder over two different surfaces. The explicit construction uses a coordinate $[\mathbb S^2\times\mathbb R]$02, an extended locally nilpotent derivation, and a slice $[\mathbb S^2\times\mathbb R]$03, yielding $[\mathbb S^2\times\mathbb R]$04.

A final contrast comes from hard-sphere packing inside a cylinder. For diameter ratio $[\mathbb S^2\times\mathbb R]$05, the densest structures are entirely cylinder-touching and can be constructed by sequential deposition on a single coaxial shell of radius $[\mathbb S^2\times\mathbb R]$06, with contact relation

$[\mathbb S^2\times\mathbb R]$07

(Chan, 2011). This is a strong shell-based cylindrical representation, but not a nested radial multilayer. The paper is explicit that above $[\mathbb S^2\times\mathbb R]$08 the surface-touching shell becomes hollow and no longer captures the true densest 3D packings. This sharpens a general lesson across the literature: cylindrical organization, concentric shells, and genuine hierarchical nesting are related but distinct notions.

A coherent synthesis is therefore possible only at a high level. In neural graphics, nested cylinder representation usually means concentric multi-scale feature cylinders. In CAD reverse engineering, “cylinder” often means arbitrary-profile extrusion volume, and nesting is typically absent. In topology and graph theory, the cylinder is an ambient periodic surface or cobordism arena. In holography, it is a shared conformal boundary. In scattering theory, it is an interior geometry of one cylinder inside another. Across these uses, the common thread is cylindrical structure as an organizing medium; the meaning of “nested” depends entirely on whether the relevant layer is geometric, symbolic, categorical, or physical.