##  [Reach](/reach-0) 

 Definition

For a closed subset A of Euclidean space, the reach is the supremal r≥0 such that every point at distance less than r from A has a unique nearest point in A; equivalently the radius of the largest tubular neighborhood on which the nearest-point projection is well-defined and single-valued.

 

 

 

 

 

 





## Principle

Principle

Quantifies extrinsic regularity by ruling out nearby self-approach and high curvature: reach controls curvature, medial axis distance and guarantees a tubular neighborhood where projection to the set is a Lipschitz (indeed smooth for smooth sets) retraction.

 

 

 

 

 





## Demonstration

Demonstration

For a smooth embedded curve or hypersurface with bounded curvature, the reach is at least the reciprocal of the maximal principal curvature; for a round sphere of radius R embedded in Euclidean space the reach equals R (the distance to the medial axis being the center).

 

 

 

 

## Misapplication

Misapplication

Using Hausdorff closeness or C^1 regularity as a substitute for positive reach — a C^1 curve can have arbitrarily small reach due to near self-approach — or confusing reach with intrinsic injectivity or convexity radii that are defined differently.

 

 

 

 

 





## Consequence

Consequence

Positive reach yields a tubular neighborhood diffeomorphic to a normal bundle, well-defined orthogonal projection, stability under perturbation scales, and enables curvature measures and geometric inference; many geometric reconstruction results require a positive lower bound on reach.

 

 

 

 

## Reversal

Reversal

The medial axis (set of points with multiple nearest neighbors) is the complement within the tubular neighborhood of the domain of uniqueness; reversing the notion locates where projections fail or where curvature constraints are violated.

 

 

 

 

 





## Boundary

Boundary

An extrinsic Euclidean concept applying to subsets of Euclidean space (or Riemannian manifolds via isometric embedding); it requires global control of embedding geometry and does not directly apply to abstract Riemannian manifolds without reference to an ambient Euclidean space.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Often compared to injectivity and convexity radii for embedded manifolds: reach is extrinsic and concerns nearest-point uniqueness, while injectivity and convexity radii are intrinsic geodesic notions; conflation leads to errors in reconstruction or analysis of neighborhoods.

 

 

 

 

 





## Synthesis

Synthesis

The reach is the largest radius of a tubular neighborhood around an Euclidean subset on which the nearest-point map is single-valued: an extrinsic measure of boundary regularity that ties curvature bounds and medial-axis avoidance to the existence of smooth projection maps.