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arxiv: 2605.18215 · v1 · pith:LH27GJKYnew · submitted 2026-05-18 · 💻 cs.GR

Tangent Blow-Ups for Processing Non-Manifold Geometry

Alice Petrov , Mohammad Sina Nabizadeh , Ana Dodik , Justin Solomon This is my paper

Pith reviewed 2026-05-19 23:46 UTC · model grok-4.3

classification 💻 cs.GR
keywords tangent blow-upnon-manifold geometrysingularitiesgeometry processingdifferential operatorsGrassmanniangeodesicsparameterization
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The pith

Tangent blow-ups lift points at singularities to separate them by tangent directions for stable geometry processing.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Geometry processing methods often fail at singularities such as edges, corners, and self-intersections because they assume a unique tangent plane at each point. The paper proposes the tangent blow-up to address this by lifting each point to the product of its position in space and its tangent plane from the Grassmannian. Iterating the lift separates points that share a position but differ in direction or higher contact order. Differential operators are then discretized on this new space using a product metric. This allows standard tasks to run on non-manifold inputs without custom handling at singular points.

Core claim

By lifting to the product of the ambient space and the Grassmannian of tangent planes, the tangent blow-up restores structure at singularities. Points that coincide in position but differ in tangent direction, curvature, or higher-order contact become well-separated after iteration. The construction is equipped with a product metric, and discretized versions of the gradient, divergence, and Laplacian are defined directly in the lifted domain for use in geometry processing pipelines.

What carries the argument

The tangent blow-up, which maps each point to a position-tangent plane pair and uses iteration to separate higher-order differences.

Load-bearing premise

Discretizing the product metric and operators on the lifted space keeps the separation intact and avoids creating new numerical artifacts near the original singularities.

What would settle it

A computation of the Laplacian on a lifted simple L-shaped polyline that shows large errors or instability localized at the corner would falsify the claim that the operators are stable.

Figures

Figures reproduced from arXiv: 2605.18215 by Alice Petrov, Ana Dodik, Justin Solomon, Mohammad Sina Nabizadeh.

Figure 1
Figure 1. Figure 1: In the figure, different metrics give different patterns of heat diffusion. At singularities and intersections, standard methods (left) [SC20] treat all geometric components equally. Our tangent blow-up representation enriches each point with geometric information such as tangents (middle) and curvature (right) by iteratively lifting data into a higher-dimensional space where singularities are resolved and… view at source ↗
Figure 3
Figure 3. Figure 3: Spectral and curvature analysis of Banchoff’s Klein bot￾tle, a self-intersecting nonorientable surface [Ban76; Bus12]. Top row: the three smallest non-trivial eigenvectors of our lifted Lapla￾cian (§6.1). Bottom row: Gaussian curvature, squared mean curva￾ture, and the Frobenius norm of the second fundamental form, all computed from the unoriented projector field (§7.3) . Our contributions are: • A discret… view at source ↗
Figure 4
Figure 4. Figure 4: A schematic cartoon illustrating the tangent blow-up of a figure-eight curve. In the base space, the curve crosses itself at x0. After lifting to the product space R n × Gr(d,n), the singular point is replaced by two lifted points (x0,T1↗) and (x0,T2↘). The projection π (defined in §3.2) is an isomorphism everywhere away from the set of singular points. 3.1. Continuous Construction We begin by making our c… view at source ↗
Figure 5
Figure 5. Figure 5: Examples of stratified spaces. The Grassmannian and its projector representation. To repre￾sent and compute over (point, tangent) pairs, we need a space that parametrizes all possible tangent planes. This is the Grassmannian Gr(d,n), the compact smooth manifold of d-dimensional linear subspaces of R n [Lee03]. The Grassmannian admits multiple con￾crete computational representations [BZA24]. We opt to repre… view at source ↗
Figure 7
Figure 7. Figure 7: The projector derivative on a smooth surface. As the basepoint moves along the curve with tangent v (blue arrow), the tangent plane rotates from T1 to T2 to T3. At x1, the projector deriva￾tive Pv(w) (red arrow) measures the component of the tangent vec￾tor w (green arrow) that leaves the tangent plane, which is the vec￾tor valued second fundamental form II(v,w). 4. The Discrete Tangent Blow-Up We now pres… view at source ↗
Figure 8
Figure 8. Figure 8: The discrete tangent blow-up of a self-intersecting figure￾eight curve. Top row: k-nearest neighbors of a query point (red), computed in the ambient space R 2 (left) and in the lifted embedding R 2 ×Gr(1,2) (right). Bottom rows: entries of the tangent projector P ∈ Gr(1,2), shown component-wise over the curve. Points at the crossing share spatial position but have distinct tangents, yielding different valu… view at source ↗
Figure 9
Figure 9. Figure 9: A line and parabola sharing position (left) and tangent (center) at the origin. The Frobenius norm of the vector valued sec￾ond fundamental form ∥II∥F distinguishes the intersection (right). blow-ups, jet spaces, and higher-order differential invariants such as torsion is left to future work. 5. Theoretical Analysis Assuming all intersections are transverse and singularities are thus resolved after one lif… view at source ↗
Figure 10
Figure 10. Figure 10: Same-sheet vs. cross-sheet lifted distances on a figure-eight curve. (a) The curve with crossing point marked. (b) Lifted distance dL as a function of Euclidean distance ∥p−q∥. Same-sheet pairs (red) vanish with spatial distance; cross-sheet pairs (blue) are bounded below by the separation floor √ 1 2 √ αδ (dashed). The second guarantee is that the lifted image of each smooth sheet is regular enough to su… view at source ↗
Figure 11
Figure 11. Figure 11: Ablation over the projector weight α on the first lift of a mushroom point cloud [ZJ16]. Heat is placed at a single source point on the middle cap. At α = 0 the kernel ignores tangents entirely and heat bleeds across all sheets; as α grows, diffusion remains on each smooth sheet. At higher values of α, diffusion becomes highly sensitive to underlying tangents. divergence operators for vector field computa… view at source ↗
Figure 13
Figure 13. Figure 13: Lifted spectral segmentation on two non-manifold point clouds. Left: the self-intersecting icicles model (Thingi10k [ZJ16]). Right: the piece-wise smooth fandisk [HDD*94]. Lifted spectral segmentation recovers individual smooth components. (§7.2), and curvature estimation (§6.3). We set α = 1.0 for all ex￾amples, except in [PITH_FULL_IMAGE:figures/full_fig_p010_13.png] view at source ↗
Figure 12
Figure 12. Figure 12: Geodesic distance via the heat method, with the source point marked in cyan. The nonmanifold Laplacian [SC20] (left) dif￾fuses across self-intersections, while the lifted Laplacian (right) is restricted to geometrically distinct components. 3. Take the trace: (divYb X)i = tr(S −1 i Ji), where Ji = [J 1 i ;··· ; J d i ]. 6.3. Curvature Having estimated Bi ≈ Bxi at each sample (§4.2), we now extract classic… view at source ↗
Figure 14
Figure 14. Figure 14: Spectral parameterization of two self-intersecting nonorientable surfaces: Banchoff’s Klein bottle [Ban76; Bus12] (top) and a five-branch Boy-like surface [PS81; Bus12] (bottom two rows). Left to right: of the three smallest non-trivial eigen￾vectors of the lifted Laplacian Lsym, followed by the induced UV checker overlay. Because the lifted Laplacian is built from projec￾tors, the eigenmaps extend smooth… view at source ↗
Figure 16
Figure 16. Figure 16: Curvature estimation error on the Klein bottle with R = 2 and (u, v) ∈ [0,2π) 2 , sampled at N = 100,000 points, as a func￾tion of singularity distance (defined in §7.3). Each sample is binned by its ambient distance to the nearest cross-sheet neighbor (25 bins, median point plotted with interquartile range shaded). Bot￾tom row: per-point error in curvature magnitude |∥II∥F − ∥IIgt∥F | rendered on the sur… view at source ↗
Figure 19
Figure 19. Figure 19: Mean squared curvature H2 on a self-intersecting pro￾peller model (Thingi10k [ZJ16]). Left: jet fitting. Center: CNC. Right: blow-up (ours). Jet fitting and CNC exhibit high-magnitude estimates (yellow) along intersections and edges. The discrete blow-up recovers consistent per-component curvature: the cylin￾drical base carries uniform magnitude, and the flat blade faces read near zero. No singularity det… view at source ↗
Figure 21
Figure 21. Figure 21: Curvature magnitude ∥II∥F on a column model (Three￾DScans [Lar12]) with thin walls. Left: CNC reports high curvature along thin edges where oriented normals reverse between opposite faces. Right: the blow-up estimator reports low curvature at these edges because the tangent plane itself has little rotation. of RAM, and an NVIDIA GeForce RTX 3070). In particular, exact kD-tree queries in the lifted ambient… view at source ↗
Figure 22
Figure 22. Figure 22: Geodesic distances on two intersecting planes (N=6000, k=20, α=1.0, σx=σu=0.5). Top row: ground-truth tangent frames; bottom row: PCA-estimated tangents (kpca=30). Columns show increasing positional noise σ ∈ 0,0.01,0.02,0.04,0.08,0.16. Grey points are unreachable (infinite geodesic distance), indicating correct sheet separation. The lifted heat method is robust to positional noise, but degrades when tang… view at source ↗
read the original abstract

Many geometry processing pipelines implicitly assume their input data is a manifold, or is sampled from one, with a unique tangent plane at every point. Geometric data, however, routinely contains sharp features like edges, corners, self-intersections, branching junctions, and other singularities, rendering standard methods ill-defined at these points. To bring geometry processing to these and other singular spaces, we introduce the ``tangent blow-up,'' a representation inspired by algebraic geometry that restores structure at singularities by lifting to the product of the ambient space and the Grassmannian of tangent planes. After iterating this construction, points that coincide in position but differ in tangent direction, curvature, or higher-order contact become well-separated. We equip the tangent blow-up with a product metric and define discretized differential operators, such as the gradient, divergence, and Laplacian, directly in the lifted domain. We demonstrate our framework across geodesic computation, segmentation, surface parameterization, and curvature estimation.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript introduces the tangent blow-up, a representation that lifts geometric data to the product of the ambient space and the Grassmannian of tangent planes (with iteration for higher-order contact) in order to separate points that coincide in position but differ in tangent direction at singularities. It equips the lifted space with a product metric and defines discretized differential operators (gradient, divergence, Laplacian) directly in this domain, with demonstrations on geodesic computation, segmentation, surface parameterization, and curvature estimation.

Significance. If the discretization of the product metric and operators is shown to preserve separation without reintroducing coupling at singularities, the framework would offer a principled, unified approach to geometry processing on non-manifold and singular data, reducing reliance on ad-hoc feature handling. The algebraic-geometry inspiration and explicit lifting construction are strengths that could support reproducible implementations if accompanied by clear discretization details.

major comments (1)
  1. [Discretization of operators (abstract and demonstration sections)] The central claim that discretized gradient, divergence, and Laplacian can be defined directly in the lifted domain (as stated in the abstract) is load-bearing for all demonstrations. The manuscript must provide the explicit discretization of the product metric on discrete inputs (meshes or point clouds) and show that lifted copies with distinct tangent planes remain separated under the chosen neighborhood or interpolation scheme; without this, standard nearest-neighbor or cotangent discretizations risk re-coupling the copies and undermining the separation property.
minor comments (1)
  1. [Abstract] The abstract would benefit from a single sentence clarifying the supported input representations (e.g., triangle meshes, point clouds) for which the discretization is implemented.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful review and for identifying the need for greater explicitness in the discretization of the product metric and operators. We address the major comment below and will strengthen the manuscript accordingly.

read point-by-point responses

  1. Referee: [Discretization of operators (abstract and demonstration sections)] The central claim that discretized gradient, divergence, and Laplacian can be defined directly in the lifted domain (as stated in the abstract) is load-bearing for all demonstrations. The manuscript must provide the explicit discretization of the product metric on discrete inputs (meshes or point clouds) and show that lifted copies with distinct tangent planes remain separated under the chosen neighborhood or interpolation scheme; without this, standard nearest-neighbor or cotangent discretizations risk re-coupling the copies and undermining the separation property.

    Authors: We agree that explicit discretization details are essential to substantiate the separation property. The manuscript defines the product metric as the sum of the Euclidean distance in ambient space and the canonical metric on the Grassmannian, with operators obtained by lifting the standard cotangent or finite-element stencils to this metric. Because neighborhoods and weights are computed directly in the product space, points that coincide in position but differ in tangent plane are separated by a positive Grassmannian distance and therefore receive distinct neighbors and weights. To make this fully reproducible, we will add a dedicated subsection with pseudocode for mesh and point-cloud lifting, explicit formulas for the lifted cotangent weights, and a short numerical verification (including a simple self-intersection example) confirming that distinct tangent copies remain decoupled under the chosen scheme. revision: yes

Circularity Check

0 steps flagged

No circularity: tangent blow-up is a definitional lift with independent discretization claims

full rationale

The paper defines the tangent blow-up explicitly as a lift to the product of ambient space and Grassmannian (iterated as needed), equips it with a product metric, and states that differential operators are then defined directly on the lifted domain. This is a constructive representation rather than a fitted model or prediction that reduces to target data by construction. No equations or steps in the provided abstract or description equate the claimed operators or separation property to parameters chosen from the same data; the discretization is presented as a subsequent implementation choice whose validity is demonstrated on applications rather than assumed tautologically. No self-citation chains or uniqueness theorems imported from prior author work are invoked to force the construction. The derivation chain is therefore self-contained as a new geometric representation.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The framework rests on standard properties of the Grassmannian and product metrics from differential geometry, plus the assumption that the lifting can be discretized without destroying separation. No free parameters or invented physical entities are introduced in the abstract.

axioms (2)
  • standard math The Grassmannian of tangent planes is a well-defined manifold that can be paired with Euclidean space to form a product space with a natural metric.
    Invoked when defining the lift and the product metric.
  • domain assumption Iterated lifting separates points that share position but differ in tangent direction or higher-order contact.
    Central to the claim that singularities become well-separated.

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    Relation between the paper passage and the cited Recognition theorem.

    We introduce the tangent blow-up, a representation inspired by algebraic geometry that restores structure at singularities by lifting to the product of the ambient space and the Grassmannian of tangent planes... We equip the tangent blow-up with a product metric and define discretized differential operators... directly in the lifted domain.

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Reference graph

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