Coarse Tangent Field in Multiscale Geometry

• Coarse tangent field is a generalized tangent structure that aggregates directional data over regions, capturing multiscale and anisotropic geometry beyond pointwise approximations.
• In Hilbert spaces, it uses multiresolution families and Jones-type flatness estimates to control rectifiable curve behavior by assigning linear subspaces to balls.
• In Carnot geometry and computational processing, the field is realized as graded Lie group bundles or lifted tangent projectors that robustly model singularities and non-manifold features.

“Coarse tangent field” is used for several mathematical and computational objects that retain tangent information after passing to a larger scale, an anisotropic blow-up, or a singularity-resolving lift. In geometric measure theory it denotes a multiscale assignment of subspaces to balls in a multiresolution family, constrained by Jones-type flatness estimates for rectifiable curves (David et al., 6 Aug 2025). In Carnot geometry it denotes the tangent group bundle $GM$ attached to a filtered manifold, which captures first-order anisotropic blow-up geometry (Choi et al., 2015). In geometry processing it denotes a smoothed or lifted field of tangent projectors designed to separate sheets and preserve feature structure on non-manifold data (Petrov et al., 18 May 2026). The expression therefore names a family of coarse or generalized tangent constructions rather than a single universally fixed object.

1. Terminological scope and recurring structure

Across the literature, the phrase appears in several distinct formal settings.

Setting	Object	Function
Doubling subsets of Hilbert space	$$\tau:\mathcal{F}_A\to \bigcup_{j\le k}\mathcal{L}_j$$	Controls restricted $\beta$- and $\theta$-numbers of curves
Carnot manifolds	$GM=\bigsqcup_{a\in M} GM(a)$	Encodes anisotropic first-order blow-up geometry
Non-manifold geometry processing	$\{P_i^{\mathrm{coarse}}\}$ or $\{U_i^{\mathrm{coarse}}\}$ on a lifted domain	Separates sheets and smooths tangent data

In the Hilbert-space theory, a coarse tangent field is defined on balls rather than points, and its purpose is quantitative: it bounds the collection of scales at which a rectifiable curve fails to align with prescribed directions. In Carnot geometry, the term refers to a bundle of graded nilpotent Lie groups and is tied to the blow-up of filtered tangent structures. In computational geometry, the same expression is used for a robust tangent representation on lifted spaces or discrete meshes, where singularities and noise make pointwise tangent planes ill-defined (David et al., 6 Aug 2025).

This suggests a family resemblance rather than a single definition. In every case, tangent data is not treated as a purely pointwise Euclidean object; it is aggregated, lifted, or anisotropically rescaled so that it remains meaningful in the presence of singularities, multiscale structure, or non-Euclidean local models.

2. Multiscale coarse tangent fields in Hilbert spaces

The most explicit general definition appears for subsets of Hilbert space. Let $X$ be a Hilbert space, $F\subset X$, and let $B\subset X$ be a ball. Jones’ unilateral and bilateral flatness quantities are

$$\tau:\mathcal{F}_A\to \bigcup_{j\le k}\mathcal{L}_j$$0

and

$$\tau:\mathcal{F}_A\to \bigcup_{j\le k}\mathcal{L}_j$$1

where the infimum runs over affine lines $$\tau:\mathcal{F}_A\to \bigcup_{j\le k}\mathcal{L}_j$$2. A multiresolution family is built from nested nets $$\tau:\mathcal{F}_A\to \bigcup_{j\le k}\mathcal{L}_j$$3 at scales $$\tau:\mathcal{F}_A\to \bigcup_{j\le k}\mathcal{L}_j$$4, with balls

$$\tau:\mathcal{F}_A\to \bigcup_{j\le k}\mathcal{L}_j$$5

A coarse plane field is then a map

$$\tau:\mathcal{F}_A\to \bigcup_{j\le k}\mathcal{L}_j$$6

assigning to each ball a linear subspace of dimension at most $$\tau:\mathcal{F}_A\to \bigcup_{j\le k}\mathcal{L}_j$$7. Relative to such a field, the restricted flatness numbers for a rectifiable curve $$\tau:\mathcal{F}_A\to \bigcup_{j\le k}\mathcal{L}_j$$8 are

$$\tau:\mathcal{F}_A\to \bigcup_{j\le k}\mathcal{L}_j$$9

and

$\beta$0

With inflation $\beta$1, a $\beta$2-dimensional coarse $\beta$3-tangent field for $\beta$4 is a coarse plane field on $\beta$5 such that there exists $\beta$6 with

$\beta$7

for every rectifiable curve $\beta$8 (David et al., 6 Aug 2025).

The geometric content is multiscale and Carleson-type. At most balls touching a rectifiable curve, the curve admits a line approximation whose direction is forced to lie in the assigned subspace $\beta$9, and the collection of balls where this fails has total diameter controlled by $\theta$0. The theory is built on strengthened traveling-salesman estimates, Christ cubes, and a coronization into bad cubes and stopping-time regions. Within each stopping-time region, the curve is flat, close to the underlying set, and its tangent direction varies by at most a prescribed angle. The tangent field is then selected recursively by maximizing a “goodness depth” across ancestors of dyadic cubes (David et al., 6 Aug 2025).

3. Existence theory, dimension bounds, and the pointwise counterpart

The dimension-theoretic input is the notion of badly fitting planes. A set $\theta$1 badly fits $\theta$2-planes if there exists $\theta$3 such that for every $\theta$4-dimensional plane $\theta$5 and ball $\theta$6,

$\theta$7

Equivalently, no $\theta$8-plane can pass too close to $\theta$9 at every location and scale. If $GM=\bigsqcup_{a\in M} GM(a)$0 has Nagata dimension $GM=\bigsqcup_{a\in M} GM(a)$1, then $GM=\bigsqcup_{a\in M} GM(a)$2 badly fits $GM=\bigsqcup_{a\in M} GM(a)$3-planes. This yields the main existence statements: if $GM=\bigsqcup_{a\in M} GM(a)$4 is doubling and badly fits $GM=\bigsqcup_{a\in M} GM(a)$5-planes, then every multiresolution family for $GM=\bigsqcup_{a\in M} GM(a)$6 admits a $GM=\bigsqcup_{a\in M} GM(a)$7-dimensional coarse $GM=\bigsqcup_{a\in M} GM(a)$8-tangent field for every $GM=\bigsqcup_{a\in M} GM(a)$9; if $\{P_i^{\mathrm{coarse}}\}$0 is doubling and badly fits $\{P_i^{\mathrm{coarse}}\}$1-planes, then one obtains a $\{P_i^{\mathrm{coarse}}\}$2-dimensional coarse tangent field independent of $\{P_i^{\mathrm{coarse}}\}$3; and if $\{P_i^{\mathrm{coarse}}\}$4 has Nagata dimension $\{P_i^{\mathrm{coarse}}\}$5 or Assouad dimension $\{P_i^{\mathrm{coarse}}\}$6, then it admits an $\{P_i^{\mathrm{coarse}}\}$7-dimensional coarse $\{P_i^{\mathrm{coarse}}\}$8-tangent field (David et al., 6 Aug 2025).

The converse is also quantitative. If a multiresolution family admits a $\{P_i^{\mathrm{coarse}}\}$9-dimensional coarse $\{U_i^{\mathrm{coarse}}\}$0-tangent field for sufficiently small $\{U_i^{\mathrm{coarse}}\}$1, then the underlying set badly fits $\{U_i^{\mathrm{coarse}}\}$2-planes. The theory is therefore not merely constructive; it characterizes when such fields can exist.

The same paper develops a pointwise weak tangent field. For a Borel set $\{U_i^{\mathrm{coarse}}\}$3 and $\{U_i^{\mathrm{coarse}}\}$4, a Borel assignment

$\{U_i^{\mathrm{coarse}}\}$5

is a $\{U_i^{\mathrm{coarse}}\}$6-dimensional pointwise weak tangent field if for every rectifiable curve $\{U_i^{\mathrm{coarse}}\}$7,

$\{U_i^{\mathrm{coarse}}\}$8

This extends the Alberti–Csörnyei–Preiss pointwise weak tangent line field from planar null sets to Hilbert-space settings under badly-fitting hypotheses. Every subset of a Hilbert space admits a pointwise weak tangent field of dimension at most its Nagata dimension. For porous sets in the plane, the coarse line-field theorem gives a quantitative multiscale version of the planar ACP phenomenon (David et al., 6 Aug 2025).

The theory is not an $\{U_i^{\mathrm{coarse}}\}$9 replacement for Jones’ traveling salesman theorem. There exists a compact, porous set $X$0 such that for any coarse line field $X$1, one can find a rectifiable curve $X$2 with

$X$3

Thus restricted $X$4-numbers do not satisfy uniform $X$5 summability even in a porous planar setting (David et al., 6 Aug 2025).

4. Carnot-manifold coarse tangent fields

In Carnot geometry, the coarse tangent field is the tangent group bundle. A Carnot manifold is a smooth manifold $X$6 equipped with a filtration

$X$7

compatible with Lie brackets in the sense that

$X$8

The associated graded bundle is

$X$9

and at each point $F\subset X$0 one obtains a graded nilpotent Lie algebra

$F\subset X$1

Its simply connected integration is the tangent Carnot group $F\subset X$2. The bundle

$F\subset X$3

is the tangent group bundle, with anisotropic dilations

$F\subset X$4

The group law is given by the Baker–Campbell–Hausdorff series truncated at step $F\subset X$5. When $F\subset X$6, the fibers are Abelian and $F\subset X$7 (Choi et al., 2015).

This bundle is explicitly identified as the coarse tangent field of a Carnot manifold. It models the first-order geometry after anisotropic blow-up, and privileged or E-Carnot coordinates make that blow-up precise. In such coordinates, zooming by $F\subset X$8 produces an asymptotic regime in which the manifold looks like the tangent Carnot group. For equiregular sub-Riemannian structures, the metric tangent cone at $F\subset X$9 is $B\subset X$0 up to natural identifications (Choi et al., 2015).

The differential calculus adapted to this field is the Carnot differential. If $B\subset X$1 is a Carnot manifold map, then at each $B\subset X$2 its differential induces graded linear maps and hence a Lie algebra morphism, which exponentiates to a group morphism

$B\subset X$3

respecting the dilations. In Carnot coordinates, $B\subset X$4 is osculated by $B\subset X$5:

$B\subset X$6

and on graded nilpotent Lie groups this agrees with the Pansu derivative (Choi et al., 2015).

The global deformation device is the Carnot tangent groupoid

$B\subset X$7

which smoothly deforms $B\subset X$8 to $B\subset X$9. This makes precise the statement that $$\tau:\mathcal{F}_A\to \bigcup_{j\le k}\mathcal{L}_j$$00 is the “true” tangent space in the Carnot setting, while also accounting for the group structure of tangent objects.

5. Computational and discrete realizations

In non-manifold geometry processing, the tangent blow-up framework defines a practical coarse tangent field on singular data. The lifted domain is

$$\tau:\mathcal{F}_A\to \bigcup_{j\le k}\mathcal{L}_j$$01

where each tangent subspace is encoded by its symmetric orthogonal projector $$\tau:\mathcal{F}_A\to \bigcup_{j\le k}\mathcal{L}_j$$02. The product metric is

$$\tau:\mathcal{F}_A\to \bigcup_{j\le k}\mathcal{L}_j$$03

and admits the isometric Euclidean embedding

$$\tau:\mathcal{F}_A\to \bigcup_{j\le k}\mathcal{L}_j$$04

A coarse tangent field is defined there as a simplified, aggregated tangent field that assigns well-defined tangents at singularities by separating and/or duplicating incident sheets, suppresses fine-scale noisy variation through smoothing in the lifted domain, and preserves sharp feature-level differences. The basic construction computes $$\tau:\mathcal{F}_A\to \bigcup_{j\le k}\mathcal{L}_j$$05-nearest neighbors in the lifted space, diffuses $$\tau:\mathcal{F}_A\to \bigcup_{j\le k}\mathcal{L}_j$$06 under a graph Laplacian for time $$\tau:\mathcal{F}_A\to \bigcup_{j\le k}\mathcal{L}_j$$07, and retracts the averaged matrices back to rank-$$\tau:\mathcal{F}_A\to \bigcup_{j\le k}\mathcal{L}_j$$08 projectors by top-$$\tau:\mathcal{F}_A\to \bigcup_{j\le k}\mathcal{L}_j$$09 eigenspaces, producing $$\tau:\mathcal{F}_A\to \bigcup_{j\le k}\mathcal{L}_j$$10 and optionally $$\tau:\mathcal{F}_A\to \bigcup_{j\le k}\mathcal{L}_j$$11 (Petrov et al., 18 May 2026).

Because the first lift only separates coincident points with distinct tangent planes, the paper also introduces an iterated, curvature-aware lift. The projector derivative $$\tau:\mathcal{F}_A\to \bigcup_{j\le k}\mathcal{L}_j$$12 and its compact representation

$$\tau:\mathcal{F}_A\to \bigcup_{j\le k}\mathcal{L}_j$$13

encode second-order geometry, and a level-2 embedding can separate points with identical position and tangent but different curvature. For surfaces in $$\tau:\mathcal{F}_A\to \bigcup_{j\le k}\mathcal{L}_j$$14, the ambient lifted dimension is $$\tau:\mathcal{F}_A\to \bigcup_{j\le k}\mathcal{L}_j$$15 at level 1 and $$\tau:\mathcal{F}_A\to \bigcup_{j\le k}\mathcal{L}_j$$16 at level 2 (Petrov et al., 18 May 2026).

Discrete surface analogues pursue the same coarse objective through combinatorics and subdivision. A discrete gradient line field on a compact oriented surface is a pair $$\tau:\mathcal{F}_A\to \bigcup_{j\le k}\mathcal{L}_j$$17, where $$\tau:\mathcal{F}_A\to \bigcup_{j\le k}\mathcal{L}_j$$18 is a cell decomposition and $$\tau:\mathcal{F}_A\to \bigcup_{j\le k}\mathcal{L}_j$$19 is a Morse matching on the vertex-edge part of the Hasse diagram. Critical faces are determined by the number $$\tau:\mathcal{F}_A\to \bigcup_{j\le k}\mathcal{L}_j$$20 of unmatched boundary edges, with index

$$\tau:\mathcal{F}_A\to \bigcup_{j\le k}\mathcal{L}_j$$21

and the Euler formula becomes

$$\tau:\mathcal{F}_A\to \bigcup_{j\le k}\mathcal{L}_j$$22

The reduced radial graph then serves as a coarse tangent or line-field skeleton on the surface (Lewiner et al., 2017).

For triangle meshes, subdivision directional fields provide a linear, structure-preserving refinement of face-based tangent directional fields. The key representation uses halfedge-based scalar quantities in a space $$\tau:\mathcal{F}_A\to \bigcup_{j\le k}\mathcal{L}_j$$23, and the subdivision operator satisfies commuting identities such as

$$\tau:\mathcal{F}_A\to \bigcup_{j\le k}\mathcal{L}_j$$24

As a consequence, curl-free fields are reproduced precisely, while divergence-free fields are reproduced in the weak sense (Custers et al., 2018).

6. Analytic and physical extensions

In coarse-grained quantum field theory, the term acquires an information-geometric meaning. The coarse tangent field is the scale-dependent notion of a direction in the manifold of equilibrium states generated by perturbations of the Hamiltonian, pushed forward by a coarse-graining channel, and measured by a contractive quantum information metric. The central bilinear form is

$$\tau:\mathcal{F}_A\to \bigcup_{j\le k}\mathcal{L}_j$$25

where $$\tau:\mathcal{F}_A\to \bigcup_{j\le k}\mathcal{L}_j$$26 is the Kubo–Mori source metric and $$\tau:\mathcal{F}_A\to \bigcup_{j\le k}\mathcal{L}_j$$27 is the adjoint recovery map. In the regime $$\tau:\mathcal{F}_A\to \bigcup_{j\le k}\mathcal{L}_j$$28, all contractive quantum metrics coincide with the classical Fisher metric, and the norm of the quartic interaction exhibits scale dependence consistent with Wilsonian relevance criteria (Bény, 2015).

A different analytic generalization appears on manifolds with a tangent Lie structure. There one has a bundle of Lie groups

$$\tau:\mathcal{F}_A\to \bigcup_{j\le k}\mathcal{L}_j$$29

with Lie algebroid $$\tau:\mathcal{F}_A\to \bigcup_{j\le k}\mathcal{L}_j$$30, a maximal compact-subgroup bundle $$\tau:\mathcal{F}_A\to \bigcup_{j\le k}\mathcal{L}_j$$31, and a fixed fiberwise linear isomorphism $$\tau:\mathcal{F}_A\to \bigcup_{j\le k}\mathcal{L}_j$$32. The “coarse tangent framework” is then a field of cosymbols on the group $$\tau:\mathcal{F}_A\to \bigcup_{j\le k}\mathcal{L}_j$$33-algebras of the fibers, assembled by operator integration into a coarse order-zero pseudodifferential calculus. Hypoellipticity is expressed by invertibility of the cosymbol modulo compacts, and the index theorem takes the form

$$\tau:\mathcal{F}_A\to \bigcup_{j\le k}\mathcal{L}_j$$34

in $$\tau:\mathcal{F}_A\to \bigcup_{j\le k}\mathcal{L}_j$$35 (Kasparov, 2024).

In soft-matter modeling, the phrase appears in an embedded, particle-based setting rather than as an abstract tangent structure. A coarse-grained vesicle model assigns to each membrane particle a unit normal-like vector $$\tau:\mathcal{F}_A\to \bigcup_{j\le k}\mathcal{L}_j$$36 and a unit nematic director $$\tau:\mathcal{F}_A\to \bigcup_{j\le k}\mathcal{L}_j$$37 with the tangent-plane constraint $$\tau:\mathcal{F}_A\to \bigcup_{j\le k}\mathcal{L}_j$$38. The pair potential includes a term proportional to $$\tau:\mathcal{F}_A\to \bigcup_{j\le k}\mathcal{L}_j$$39, so tangent-plane nematic alignment is evaluated in three dimensions. Because neighboring tangent planes on a curved surface are differently oriented, this produces an extrinsic curvature–director coupling that drives morphology transitions from nearly spherical vesicles to prolates and then to tubes as the nematic coupling $$\tau:\mathcal{F}_A\to \bigcup_{j\le k}\mathcal{L}_j$$40 increases (Geng et al., 2011).

7. Limitations, misconceptions, and open directions

A common misconception is that a coarse tangent field is merely a noisy estimate of pointwise tangents. The Hilbert-space theory shows otherwise: the field is defined on balls, not points, and its purpose is a Carleson-controlled restriction on admissible directions for all rectifiable curves. The Carnot version is likewise not a rough approximation to $$\tau:\mathcal{F}_A\to \bigcup_{j\le k}\mathcal{L}_j$$41; it is a bundle of graded nilpotent Lie groups, and when $$\tau:\mathcal{F}_A\to \bigcup_{j\le k}\mathcal{L}_j$$42 its fibers are non-Abelian. The computational versions are not just smoothed normals; they may duplicate points across sheets or lift tangent data to Grassmannians and higher-order contact spaces (David et al., 6 Aug 2025).

Within the Hilbert-space theory, several open problems remain. Independence of $$\tau:\mathcal{F}_A\to \bigcup_{j\le k}\mathcal{L}_j$$43 is proved for line fields when the set badly fits $$\tau:\mathcal{F}_A\to \bigcup_{j\le k}\mathcal{L}_j$$44-planes, but is open in higher codimension. Uniform $$\tau:\mathcal{F}_A\to \bigcup_{j\le k}\mathcal{L}_j$$45 summability for restricted $$\tau:\mathcal{F}_A\to \bigcup_{j\le k}\mathcal{L}_j$$46 fails, and the optimal exponent $$\tau:\mathcal{F}_A\to \bigcup_{j\le k}\mathcal{L}_j$$47 remains open. The role of Hilbertian structure is substantial: the proofs use Pythagorean identities and matching square exponents, so extension to more general Banach spaces is unresolved. Other questions concern coarse tangent fields for collections of balls not tied to an underlying set, and the development of a fuller coarse differential calculus (David et al., 6 Aug 2025).

Computational variants have their own constraints. Tangent blow-ups are sensitive to tangent estimation near singularities, level-2 lifting increases dimensionality substantially, and exact $$\tau:\mathcal{F}_A\to \bigcup_{j\le k}\mathcal{L}_j$$48-nearest-neighbor search degrades in the higher lifted dimension. Discrete line fields assume compact oriented surfaces, while subdivision directional fields are developed for triangle meshes and do not specially treat sharp features (Petrov et al., 18 May 2026).

Taken together, these literatures show that coarse tangent fields are best understood as a technical family of tangent surrogates adapted to settings where ordinary pointwise tangent spaces are insufficient: multiscale metric geometry, filtered and sub-Riemannian geometry, non-manifold geometry processing, coarse analysis, and embedded physical models. The shared theme is not uniform formalism but the replacement of classical tangent data by a scale-aware, anisotropic, or lifted structure that remains meaningful under singularity, curvature, or coarse observation.