# Cone Fields and Topological Sampling in Manifolds with Bounded Curvature

## Abstract

A standard reconstruction problem is how to discover a compact set from a noisy point cloud that approximates it. A finite point cloud is a compact set. This paper proves a reconstruction theorem which gives a sufficient condition, as a bound on the Hausdorff distance between two compact sets, for when certain offsets of these two sets are homotopic in terms of the absence of μ-critical points in an annular region. We reduce the problem of reconstructing a subset from a point cloud to the existence of a deformation retraction from the offset of the subset to the subset itself. The ambient space can be any Riemannian manifold but we focus on ambient manifolds which have nowhere negative curvature (this includes Euclidean space). We get an improvement on previous bounds for the case where the ambient space is Euclidean whenever μ≤0.945 (μ∈(0,1) by definition). In the process, we prove stability theorems for μ-critical points when the ambient space is a manifold.

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1. 1.

θ∈[0,π/2].

2. 2.

In particular, for any compact set K and any bound on Hausdorff distance δ>0 there is a compact set L with zero μ-reach such that d H (K,L)<δ.

3. 3.

∥∇ x f∥ is the non-negative number $$\max\{0,\limsup_{y\to x} \frac {f(y)-f(x)}{d(y,x)}\}$$. That this is ∥∇ K (x)∥ follows from our geometric construction of ∇ K (x) and from the cosine rule.

## Abbreviations

$$\mathcal {M}$$ :

is a smooth Riemannian manifold which forms the ambient space.

A :

is a compact subset of $$\mathcal {M}$$ which we desire to reconstruct.

S :

is a noisy point cloud sample of A.

δ :

is a bound on the Hausdorff distance between two compacts sets.

$$\mathit{UT}\mathcal {M}$$ :

is the unit tangent bundle of $$\mathcal {M}$$.

$$T_{x}\mathcal {M}$$ :

is the tangent plane to $$\mathcal {M}$$ at the point x.

γ :

is a geodesic on $$\mathcal {M}$$ (usually unit speed and always constant speed).

x,y,z :

are points in $$\mathcal {M}$$.

exp x :

is the exponential map from the tangent plane at x to $$\mathcal {M}$$.

Γ(γ):

is the isometry between tangent planes induced by parallel transport along γ.

w,v :

are unit tangent vectors.

β,θ :

are angles. We mainly care about acute angles.

C(w,β):

is a cone. It is a ball in the unit tangent sphere at a point in $$\mathcal {M}$$ with center w and radius β.

W :

is a cone field. Also denoted by {(x,C(w x ,β x ))}.

W′:

denotes the complementary cone field to W when W is an acute cone field. For W above it is {(x,C(w x ,π/2−β x ))}.

X :

is a vector field.

K,L :

are compact subsets of $$\mathcal {M}$$.

d K :

is the distance function from K.

K a :

is the a-offset of K. That is, $$\{ x\in \mathcal {M}: d_{K}(x)\leq a\}$$.

K [a,b] :

is the [a,b] annulus of K. That is, {xM:ad K (x)≤b}.

K :

is the gradient vector field for d K .

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## Author information

Authors

### Corresponding author

Correspondence to Katharine Turner.

Communicated by Herbert Edelsbrunner.

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Turner, K. Cone Fields and Topological Sampling in Manifolds with Bounded Curvature. Found Comput Math 13, 913–933 (2013). https://doi.org/10.1007/s10208-013-9176-6

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### Keywords

• Distance function
• Surface and manifold reconstruction
• Deformation retraction
• Fibre bundle

• 54-C-15
• 55-S-40
• 53-B-20