Planar Gaussians: Rendering, Mapping, and Analysis

• Planar Gaussians are Gaussian primitives with built-in planar constraints that approximate local surface patches for effective 3D reconstruction.
• They are applied in computer vision to improve depth and normal estimation, part segmentation, and articulated motion by enforcing local surface geometry.
• Variants include flattened 3D Gaussians, literal 2D Gaussians, and mathematical formulations, each serving unique roles in rendering, mapping, or theoretical analysis.

Planar Gaussians denote several closely related constructions rather than a single universal primitive. In contemporary computer vision, the term most often refers to anisotropic 3D Gaussian splats whose smallest principal scale is explicitly compressed so that each primitive approximates a local tangent plane, making rendered normals and depth geometrically useful for reconstruction, meshing, articulation, and consistency losses (Wu et al., 21 Nov 2025). In other settings, a planar Gaussian is literally a 2D Gaussian defined on a plane, as in bird’s-eye-view mapping, or a standard Gaussian whose center coordinates are constrained to a shared plane representation (Zeng et al., 10 May 2026). In mathematics, the phrase also appears in work on Gaussian densities on the plane, planar Gaussian fields, and large-$N$ “planar” asymptotics of Gaussian partition functions (Berry et al., 2016).

1. Terminological scope and common geometric idea

Across the literature surveyed here, “planar Gaussian” has at least four technically distinct meanings. This suggests that the term is best understood by the geometric role it plays, not by a single canonical formula.

Usage	Core object	Operational meaning
Planarized 3D Gaussian splat	Standard anisotropic 3D Gaussian	One principal scale is pushed toward zero, so the primitive behaves like a local surface patch
Literal planar Gaussian	2D Gaussian on a plane	Density kernel on $(x,y)$, usually with anisotropic covariance and in-plane rotation
Plane-structured Gaussian	Standard Gaussian with constrained center	The Gaussian center is parameterized to remain on a learned or detected plane
Planar Gaussian in mathematical analysis	Gaussian density or field on the plane	The adjective “planar” refers to the ambient domain, not necessarily to a flattened primitive

In the splatting literature, the common idea is to replace appearance-dominated volumetric ellipsoids by primitives that carry more explicit local surface structure. SPAGS states this directly: ordinary 3D Gaussian Splatting is poorly suited to sparse-view articulated object reconstruction because unconstrained 3D Gaussian primitives are “disorderly and irregular,” which makes reliable depth and normal estimation difficult; the remedy is to compress 3D Gaussians into planar Gaussians and optimize them with geometry-oriented regularization (Wu et al., 21 Nov 2025). REArtGS++ and GigaGS adopt essentially the same mechanism: a Gaussian remains a standard oriented anisotropic 3D primitive, but its minimum scale is explicitly penalized so that the primitive becomes a thin sheet whose shortest axis serves as a normal direction (Wu et al., 21 Nov 2025).

Other works use the adjective more literally. GSMap defines a planar Gaussian as a 2D primitive on the ground plane / BEV plane with center $\mu\in\mathbb{R}^2$, anisotropic scale $\sigma\in\mathbb{R}^2$, and rotation $\theta\in\mathbb{R}$, rendered as an oriented elliptical support region (Zeng et al., 10 May 2026). GSPlane, by contrast, does not alter Gaussian covariance at all; its “planar Gaussian” is a Gaussian whose center is re-parameterized as an affine combination of three basis points on a plane, so off-plane drift becomes impossible while the primitive remains in the planar class (Gan et al., 20 Oct 2025).

2. Surface-oriented planar Gaussians in 3D Gaussian Splatting

The most influential vision formulation treats a planar Gaussian as a constrained anisotropic 3D Gaussian. In SPAGS, the primitive $\mathcal G_i$ retains the usual Gaussian center $\mathbf p\in\mathbb R^3$, opacity $\alpha_i$, and principal scales $S_1,S_2,S_3$, but planarity is enforced by the loss

$$\mathcal{L}_{\text{scale}} = \frac{1}{N_{\mathcal G}}\sum_i^{N_{\mathcal G}} \left\| \min(S_1,S_2,S_3)\right\|.$$

Operationally, this drives the smallest principal scale toward zero, so the covariance becomes effectively rank-deficient or nearly rank-2 in practice. SPAGS further states that “Based on the planar Gaussian primitives, we can easily acquire the normal via the shortest axis and viewing direction,” and uses alpha-blended normal rendering $(x,y)$0 together with an “unbiased depth” $(x,y)$1 following PGSR (Wu et al., 21 Nov 2025).

This flattened-splat interpretation is now used as a geometry backbone rather than merely a rendering convenience. In SPAGS, planar Gaussians sit between sparse-view selection and part segmentation: optimal sparse views are selected by a Gaussian information field, a structured initialization is produced and registered, Gaussians are converted to planar Gaussians, and then optimized in a coarse-to-fine schedule with color, depth, depth smoothness, few-shot diffusion, and view consistency; only afterward are part probabilities attached to each Gaussian via mask back-projection (Wu et al., 21 Nov 2025). The geometric importance is borne out by ablation: replacing planar Gaussians with the depth rendering pipeline from GaussianObject worsens whole-object CD from $(x,y)$2 to $(x,y)$3, dynamic-part CD-d from $(x,y)$4 to $(x,y)$5, and static-part CD-s from $(x,y)$6 to $(x,y)$7 on PartNet-Mobility (Wu et al., 21 Nov 2025).

REArtGS++ extends the same planarization into articulated motion. Its scale loss

$(x,y)$8

again collapses one axis, but the resulting normals and unbiased depth are then used in a temporal geometry constraint defined over the entire articulated motion interval via first-order Taylor expansion (Wu et al., 21 Nov 2025). The practical effect is large: in the ablation “w/o planar GS” versus “w/ all,” Axis Ang improves from $(x,y)$9 to $\mu\in\mathbb{R}^2$0, Axis Pos from $\mu\in\mathbb{R}^2$1 to $\mu\in\mathbb{R}^2$2, Part Motion from $\mu\in\mathbb{R}^2$3 to $\mu\in\mathbb{R}^2$4, CD-s from $\mu\in\mathbb{R}^2$5 to $\mu\in\mathbb{R}^2$6, CD-m from $\mu\in\mathbb{R}^2$7 to $\mu\in\mathbb{R}^2$8, and CD-w from $\mu\in\mathbb{R}^2$9 to $\sigma\in\mathbb{R}^2$0 (Wu et al., 21 Nov 2025).

GigaGS generalizes the same planar bias to large aerial scenes. It defines a flatten loss

$\sigma\in\mathbb{R}^2$1

and uses the shortest axis as the Gaussian normal $\sigma\in\mathbb{R}^2$2. From these planarized splats it renders normals and plane distances, then reconstructs depth by ray-plane intersection. The paper’s own ablation is explicit that planarity alone is insufficient: flatten regularization without stronger local and multiview constraints can reduce expressiveness, while the combination of flattening, local depth-normal consistency, and multi-view planar consistency improves surface quality (Chen et al., 2024).

3. Structured plane-aware variants and representational alternatives

A second line of work does not necessarily flatten covariance, but instead injects plane structure into coordinates, semantics, or optimization. GSPlane is the clearest example. For each planar cluster, it estimates a plane, selects three non-collinear basis points $\sigma\in\mathbb{R}^2$3, and re-parameterizes every planar Gaussian center as

$\sigma\in\mathbb{R}^2$4

Planarity is therefore a constraint on center coordinates rather than on covariance eigenvalues or rotations. The method further introduces a Dynamic Gaussian Re-classifier: it selects the top 5% of planar Gaussians ranked by average gradient magnitude and reclassifies those whose gradient exceeds the average magnitude of the top 20% non-planar Gaussians back to unconstrained $\sigma\in\mathbb{R}^2$5 form (Gan et al., 20 Oct 2025). The paper reports that the structured planar representation improves planar metrics while reducing planar vertex counts drastically; for example, on ScanNetV2, 2DGS baseline F-score $\sigma\in\mathbb{R}^2$6 becomes $\sigma\in\mathbb{R}^2$7 with “Ours-full,” and planar vertex count drops from $\sigma\in\mathbb{R}^2$8M to $\sigma\in\mathbb{R}^2$9K (Gan et al., 20 Oct 2025).

PlanarGS keeps standard 3DGS primitives but combines language-prompted planar masks, depth priors, normal priors, Gaussian flattening, and a co-planarity loss. Its flattening term is again the minimum-scale penalty

$\theta\in\mathbb{R}$0

but the more distinctive term is the plane-fit supervision: for each planar region $\theta\in\mathbb{R}$1, a plane $\theta\in\mathbb{R}$2 is fit from rendered 3D points, and rendered depth $\theta\in\mathbb{R}$3 is penalized against the induced plane depth $\theta\in\mathbb{R}$4 via

$\theta\in\mathbb{R}$5

The total loss is

$\theta\in\mathbb{R}$6

with $\theta\in\mathbb{R}$7, $\theta\in\mathbb{R}$8, $\theta\in\mathbb{R}$9, and $\mathcal G_i$0 (Jin et al., 27 Oct 2025). On MuSHRoom, PlanarGS reports Acc $\mathcal G_i$1, Comp $\mathcal G_i$2, CD $\mathcal G_i$3, F1 $\mathcal G_i$4, and NC $\mathcal G_i$5, and the ablation “w/o both co-planarity and geometric priors” degrades to CD $\mathcal G_i$6 and F1 $\mathcal G_i$7 (Jin et al., 27 Oct 2025).

AtlasGS is plane-aware in yet another sense. It builds on 2DGS-style surfel Gaussians with attributes $\mathcal G_i$8, $\mathcal G_i$9, $\mathbf p\in\mathbb R^3$0, $\mathbf p\in\mathbb R^3$1, and semantic logits $\mathbf p\in\mathbb R^3$2, then regularizes them using Atlanta-world plane indicators for floor, ceiling, and gravity-aligned wall structure (Zhang et al., 29 Oct 2025). The explicit 3D regularizer

$\mathbf p\in\mathbb R^3$3

is complemented by a 2D local surface term from rendered depth normals. On ScanNet, the ablation “w/o $\mathbf p\in\mathbb R^3$4” yields CD $\mathbf p\in\mathbb R^3$5 and F-score $\mathbf p\in\mathbb R^3$6, whereas the full model reports CD $\mathbf p\in\mathbb R^3$7 and F-score $\mathbf p\in\mathbb R^3$8 (Zhang et al., 29 Oct 2025).

4. Dynamic, articulated, and optical extensions

Planar Gaussians are especially useful when geometry must remain coherent under articulation or under layered image formation. In articulated reconstruction, SPAGS associates each Gaussian with a part probability

$\mathbf p\in\mathbb R^3$9

so part labels are back-projected onto surface-like primitives rather than amorphous volumetric blobs (Wu et al., 21 Nov 2025). This makes planar Gaussians not just a meshing device but also the substrate for part-level reasoning.

REArtGS++ adds a time-continuous constraint that depends directly on the planar interpretation. It renders planar-Gaussian normals $\alpha_i$0 and unbiased depth $\alpha_i$1, estimates a depth-derived normal $\alpha_i$2, and aligns both the normals and their temporal derivatives: $\alpha_i$3 The canonical state $\alpha_i$4 is chosen because no motion is applied there and $\alpha_i$5 does not participate in back-propagation, reducing memory and preventing motion-accumulation error (Wu et al., 21 Nov 2025).

TR-Gaussians represent a different planar extension. Here the Gaussians themselves are standard 3DGS primitives with covariance $\alpha_i$6, but the planar structure is introduced through learnable reflection planes $\alpha_i$7 that mirror Gaussians to synthesize planar reflection and transmission (Liu et al., 17 Nov 2025). The final image is

$\alpha_i$8

with $\alpha_i$9 given by a Fresnel-based view-dependent term multiplied by a rendered reflection mask (Liu et al., 17 Nov 2025). The paper is explicit that the planar aspect is not that the Gaussians themselves become 2D planar splats; instead, planes act as global operators over a 3D Gaussian set. On RTR, TR-Gaussians reports $S_1,S_2,S_3$0 PSNR, $S_1,S_2,S_3$1 SSIM, and $S_1,S_2,S_3$2 LPIPS, versus 3DGS at $S_1,S_2,S_3$3, while retaining real-time rendering at $S_1,S_2,S_3$4 FPS and training in $S_1,S_2,S_3$5 minutes (Liu et al., 17 Nov 2025).

5. Literal 2D planar Gaussians in mapping and vectorization

GSMap uses the term in the most literal geometric sense: a planar Gaussian is a 2D anisotropic kernel on the BEV plane. One Gaussian is

$S_1,S_2,S_3$6

with density

$S_1,S_2,S_3$7

and covariance parameterized as

$S_1,S_2,S_3$8

Each HD map element is an ordered sequence $S_1,S_2,S_3$9, and the ordered centers $$\mathcal{L}_{\text{scale}} = \frac{1}{N_{\mathcal G}}\sum_i^{N_{\mathcal G}} \left\| \min(S_1,S_2,S_3)\right\|.$$0 are connected sequentially to form continuous vectorized map elements (Zeng et al., 10 May 2026).

The central technical point is that the same primitive supports both raster and vector supervision. Rasterization is defined by

$$\mathcal{L}_{\text{scale}} = \frac{1}{N_{\mathcal G}}\sum_i^{N_{\mathcal G}} \left\| \min(S_1,S_2,S_3)\right\|.$$1

while vectorization simply reads out the ordered centers. Training combines classification, point regression, and raster loss: $$\mathcal{L}_{\text{scale}} = \frac{1}{N_{\mathcal G}}\sum_i^{N_{\mathcal G}} \left\| \min(S_1,S_2,S_3)\right\|.$$2 The best ablation settings reported are $$\mathcal{L}_{\text{scale}} = \frac{1}{N_{\mathcal G}}\sum_i^{N_{\mathcal G}} \left\| \min(S_1,S_2,S_3)\right\|.$$3, $$\mathcal{L}_{\text{scale}} = \frac{1}{N_{\mathcal G}}\sum_i^{N_{\mathcal G}} \left\| \min(S_1,S_2,S_3)\right\|.$$4, and rasterization resolution $$\mathcal{L}_{\text{scale}} = \frac{1}{N_{\mathcal G}}\sum_i^{N_{\mathcal G}} \left\| \min(S_1,S_2,S_3)\right\|.$$5 (Zeng et al., 10 May 2026). Integrated into MapTR, GSMap improves nuScenes val performance at 24 epochs from $$\mathcal{L}_{\text{scale}} = \frac{1}{N_{\mathcal G}}\sum_i^{N_{\mathcal G}} \left\| \min(S_1,S_2,S_3)\right\|.$$6, $$\mathcal{L}_{\text{scale}} = \frac{1}{N_{\mathcal G}}\sum_i^{N_{\mathcal G}} \left\| \min(S_1,S_2,S_3)\right\|.$$7 for MapTR to $$\mathcal{L}_{\text{scale}} = \frac{1}{N_{\mathcal G}}\sum_i^{N_{\mathcal G}} \left\| \min(S_1,S_2,S_3)\right\|.$$8 and $$\mathcal{L}_{\text{scale}} = \frac{1}{N_{\mathcal G}}\sum_i^{N_{\mathcal G}} \left\| \min(S_1,S_2,S_3)\right\|.$$9 for MapTR + GSMap, while retaining near-identical efficiency: $(x,y)$00 FPS and $(x,y)$01M parameters versus $(x,y)$02 FPS and $(x,y)$03M for MapTR (Zeng et al., 10 May 2026).

This literal 2D usage is conceptually important because it separates “planar Gaussian” from “flattened 3D Gaussian.” In GSMap, the Gaussian is not a 3D splat that has been compressed; it is a native 2D primitive whose center doubles as a vector vertex and whose support doubles as a raster occupancy kernel.

6. Mathematical meanings: densities, fields, convexity, and planar asymptotics

In mathematics, planar Gaussian terminology refers to the ambient plane or to planar asymptotics, not to surface-oriented splats. The double-Gaussian isoperimetric problem studies the density

$(x,y)$04

on $(x,y)$05. The main planar theorem is conditional: among linear boundaries, only horizontal and vertical lines are stationary, and vertical lines enclose given area with less perimeter than horizontal lines; therefore, if planar isoperimetric regions are half-spaces, they must be bounded by vertical lines (Berry et al., 2016).

A different convex-geometric usage appears in the planar Gaussian-Minkowski problem, where one seeks a convex body whose Gaussian surface area measure matches prescribed data on $(x,y)$06. For density $(x,y)$07, the support function $(x,y)$08 satisfies

$(x,y)$09

For constant data, the paper proves that if the Gaussian surface area measure is proportional to spherical Lebesgue measure, the body must be a centered disk; it then uses this uniqueness result to prove existence of smooth small solutions by a degree-theoretic argument (Chen et al., 2023).

Planar Gaussian fields form yet another branch. For smooth stationary planar Gaussian fields, four discretisation schemes recover different levels of topological information about level sets. The mesh conditions are sharply stratified: $(x,y)$10 for full topology in each face, $(x,y)$11 for full topology in $(x,y)$12, $(x,y)$13 for the random plane wave up to visible ambiguities, and $(x,y)$14 for approximating the mean number of excursion domains and hence the Nazarov–Sodin constant (Beliaev et al., 2017). For stationary planar Gaussian fields with trends, exact formulas are available for the expected number of local maxima and the height distribution of local maxima: $(x,y)$15 with the Hessian contribution $(x,y)$16 reduced to generalized chi-squared densities because GOE-based isotropic machinery no longer applies directly in the non-isotropic, non-centered case (Cheng, 2023).

Finally, “planar” can refer to large-$(x,y)$17 asymptotics rather than planar geometry. For Gaussian distributions on Riemannian symmetric spaces, the partition function admits a genus expansion

$(x,y)$18

so the large-$(x,y)$19 limit is governed by the planar sector $(x,y)$20. In that sense the paper develops a theory of Gaussian models whose normalization constants are controlled by planar random-matrix and ribbon-graph asymptotics, rather than by Euclidean planar support (Heuveline et al., 2021).

Taken together, these lines of work show that “planar Gaussian” is a technically dense but context-dependent phrase. In reconstruction and rendering it usually means a surface-biased Gaussian primitive, obtained either by collapsing one covariance axis or by constraining Gaussian coordinates to a plane. In mapping it can mean a literal 2D Gaussian on a plane. In mathematical analysis it often means a Gaussian density or field defined on the plane, or a Gaussian partition function whose asymptotics are organized by the planar large-$(x,y)$21 limit.