Directional curvature and medial axis

Abstract: The medial axis $M_X$ of a closed set $X\subset \mathbb{R}<sup>n$ is the set of points from the ambient space that admit more than one closest point in $X$. We study the problem of reaching the singularities, i.e. of characterising the points of the set $\overline{M_X}\cap X$. In order to tame the geometry, we assume that $X$ is definable in a polynomially bounded structure and obtain a general criterion based on a generalisation of the notion of superquadraticity previously introduced by Birbrair and Denkowski for $C<sup>1$-smooth hypersurfaces and extended to any codimension by Białożyt. We do not require any smoothness as we achieve our goal by introducing a notion of directional curvature in some naturally chosen camber directions. This allows us in particular to complete the study of the plane case.

• The paper introduces directional curvature and extends superquadraticity to define precise criteria for medial axis reachability of singularities.
• It establishes necessary and sufficient conditions for when singular points of definable sets lie in the closure of the medial axis.
• The work employs o-minimal techniques and shape functions, offering insights applicable in computational geometry and singularity theory.

Directional Curvature and Medial Axis: A Technical Summary

Introduction and Problem Statement

This work addresses the geometric and analytic properties of the medial axis $M_X$ of a closed, definable subset $X \subset \mathbb{R}^n$, with particular focus on singularity reaching phenomena. The medial axis, which consists of points in $\mathbb{R}^n$ admitting more than one closest point in $X$, is a fundamental object in geometric modeling, pattern recognition, and singularity theory. The central problem is the precise characterization of points in the intersection $\overline{M_X}\cap X$, i.e., singularities "reached" by the closure of the medial axis.

The authors extend the toolkit for studying $M_X$ by introducing the notion of directional curvature in naturally chosen "camber directions" and by generalizing the concept of superquadraticity beyond the smooth category, providing necessary and sufficient conditions for the reaching of singularities by the medial axis.

Foundational Definitions and Geometric Framework

Let $X\subset \mathbb{R}^n$ be closed and definable in a polynomially bounded o-minimal structure. The set of points with more than one closest point, the medial axis, is denoted as $M_X := \{ a \in \mathbb{R}^n \mid \# m(a) > 1 \}$ where $m(a):= \{ x \in X \mid \|a-x\| = d(a,X) \}$ is the (possibly multivalued) nearest-point map.

A key challenge is understanding under what geometric or analytic conditions a singularity $x\in X$ satisfies $X \subset \mathbb{R}^n$0; that is, when is the singularity "detectable" or "reachable" by $X \subset \mathbb{R}^n$1\mathscr{C}^2X⊂RnX \subset \mathbb{R}^n2M_X,</sup>compellinganalysistofocusonpointsoflowerregularity.</p><p>Acrucialgeometricinvariantisthe<strong>Peano<ahref="https://www.emergentmind.com/topics/tangent−cone"title=""rel="nofollow"data−turbo="false"class="assistant−link"x−datax−tooltip.raw="">tangentcone</a></strong>,</sup> compelling analysis to focus on points of lower regularity.</p> <p>A crucial geometric invariant is the <strong>Peano <a href="https://www.emergentmind.com/topics/tangent-cone" title="" rel="nofollow" data-turbo="false" class="assistant-link" x-data x-tooltip.raw="">tangent cone</a></strong> X \subset \mathbb{R}^n$3 and the associated normal cone $X \subset \mathbb{R}^n$4, providing the set of directions relevant for probing the tangency and "flatness" of $X \subset \mathbb{R}^n$5 at $X \subset \mathbb{R}^n$6. The authors introduce the notions $X \subset \mathbb{R}^n$7 (the $X \subset \mathbb{R}^n$8-singular locus) and $X \subset \mathbb{R}^n$9 (the set of $\mathbb{R}^n$0-regular but $\mathbb{R}^n$1-singular points), partitioning the locus of potential reaches by $\mathbb{R}^n$2.

Superquadraticity, Directional Curvature, and General Curvature

The paper extends the notion of superquadraticity—a condition previously understood for hypersurfaces—to arbitrary codimensions and singular points, thereby covering broader geometric settings. For a germ $\mathbb{R}^n$3, superquadraticity is measured via a growth function $\mathbb{R}^n$4 comparing the normal component's distance to the tangent plane as a function of the tangential distance, with the critical slope governed by exponents $\mathbb{R}^n$5 such that $\mathbb{R}^n$6 as $\mathbb{R}^n$7.

To operationalize this in arbitrary dimension and at singular points, the authors define:

• Directional curvature $\mathbb{R}^n$8 along a "camber direction" $\mathbb{R}^n$9, quantifying the optimal exponent $X$0 such that the germ of $X$1 around 0 is contained in a $X$2-directional sector where the $X$3-component is bounded by $X$4, with $X$5 in the orthogonal complement of $X$6.
• General curvature $X$7, the minimal exponent such that the $X$8-component (in a suitable coordinate splitting) is $X$9 for $\overline{M_X}\cap X$0 near 0.

It is demonstrated that $\overline{M_X}\cap X$1, coupling the various directional curvatures to a global invariant (Theorem \ref{theta}). Superquadraticity corresponds to $\overline{M_X}\cap X$2, generalizing previous results for hypersurfaces.

Figure 1: Unlike the parabola, the curve $\overline{M_X}\cap X$3 is only $\overline{M_X}\cap X$4-smooth (not $\overline{M_X}\cap X$5-smooth) at the origin, illustrating the type of singularity where $\overline{M_X}\cap X$6 can reach the origin.

The Reaching Problem for Medial Axis

For regular points, the Nash Lemma guarantees that $\overline{M_X}\cap X$7 avoids neighborhoods of points with non-degenerate second-order tangency ($\overline{M_X}\cap X$8). Singular points and certain degenerate regular points remain potential targets for $\overline{M_X}\cap X$9.

The study provides a refined criterion for reachability: if there exists a camber direction $M_X$0 at $M_X$1 such that $M_X$2, then $M_X$3 (Theorem \ref{kierunek}). Equivalently, $M_X$4 is sufficient for reachability in the germ. The proof exploits definable selection and \L ojasiewicz inequalities, leveraging o-minimality to extract precise exponents.</p> <p>The <strong>reaching radius</strong> $M_X$5, together with its directional and limiting variants, is also invoked as an analytic characterization: $M_X$$6 holds generally, but the directional curvature framework yields more geometric insight.</p> <p><img src="https://images.emergentmind.com/paper-images/2604-26490/horizontally_filled.png" alt="Figure 2" title="" class="markdown-image" loading="lazy"></p> <p><p class="figure-caption">Figure 2: The necessity of horizontal filling for the plane case; on the left, the origin is not achievable by$$M_X$7 if $M_X$8 is horizontally filled, while on the right, the origin is achievable otherwise.

Plane Case: Sharp Criteria and Separation at the Critical Slope

In the planar case ($M_X$9), the authors present sharp results under mild "horizontal filling" conditions. If $X\subset \mathbb{R}^n$0, then $X\subset \mathbb{R}^n$1; if $X\subset \mathbb{R}^n$2, then $X\subset \mathbb{R}^n$3 (Theorems \ref{plane geq 2} and \ref{plane &lt;2}). The proofs use constructive geometric arguments involving tangent disks to the branches of $X\subset \mathbb{R}^n$4 and separations provided by the associated curvature exponents.

Figure 3: Illustrates the central set and the geometric locus where $X\subset \mathbb{R}^n$5 may approach the origin in the planar filled case.

Figure 4: Configuration of mutually disjoint medial balls in the planar setup, establishing constraints on $X\subset \mathbb{R}^n$6 and sequence convergence.

Figure 5: Shows covering geometry and exclusion of origin reachability by $X\subset \mathbb{R}^n$7 in the paraboloid-bound case ($X\subset \mathbb{R}^n$8).</p></p> <h2 class='paper-heading' id='higher-dimensional-cases-and-limitations'>Higher Dimensional Cases and Limitations</h2> <p>Extension to higher dimensions reveals subtleties not present in the planar case. The authors provide counterexamples (Proposition \ref{higher geq 2}) demonstrating that, for $X\subset \mathbb{R}^n$9, non-superquadraticity ($M_X := \{ a \in \mathbb{R}^n \mid \# m(a) > 1 \}$0) does not universally prevent $M_X := \{ a \in \mathbb{R}^n \mid \# m(a) > 1 \}$1 from accumulating at the origin. The construction leverages a union of cones and specific cuts, with $M_X := \{ a \in \mathbb{R}^n \mid \# m(a) > 1 \}$2 still "glued" at the origin in directions not present in $M_X := \{ a \in \mathbb{R}^n \mid \# m(a) > 1 \}$3, indicating that the interaction between geometry and the medial axis is more intricate in higher dimensions.

Figure 6: Depicts a three-dimensional configuration where $M_X := \{ a \in \mathbb{R}^n \mid \# m(a) > 1 \}$4 reaches the origin due to nontrivial global geometry, despite superquadraticity in local sections.

Analytic Tools: Shape Functions and Order Theory

The authors formalize shape functions $M_X := \{ a \in \mathbb{R}^n \mid \# m(a) > 1 \}$5, measuring the extremal displacement in direction $M_X := \{ a \in \mathbb{R}^n \mid \# m(a) > 1 \}$6 for points at Euclidean distance $M_X := \{ a \in \mathbb{R}^n \mid \# m(a) > 1 \}$7 from the origin along the tangent cone. The order of vanishing of these functions at the origin directly recovers the relevant curvature invariants, systematically organizing the possible geometric germ types.

Furthermore, the proofs clarify the relationship between the order of growth of the normal component functions and the geometry detected by $M_X := \{ a \in \mathbb{R}^n \mid \# m(a) > 1 \}$8, drawing on selection theorems and the structure theory of definable sets.

Implications and Future Directions

This work provides comprehensive necessary and sufficient criteria—expressed via explicit curvature exponents—for when singularities of definable sets are reached by their medial axes, resolving open problems in the analytic and subanalytic, o-minimal context. The introduction of directional curvature, with explicit computation via shape functions, equips singularity theory with discriminants accessible for automated and algorithmic checking, especially pertinent in computational geometry or singularity classification.

The paper demonstrates that, at least in tame geometric settings, the reachability of singularities by $M_X := \{ a \in \mathbb{R}^n \mid \# m(a) > 1 \}$9 is governed by fine analytic data (curvature exponents), with clear phase transitions at the critical value $m(a):= \{ x \in X \mid \|a-x\| = d(a,X) \}$0. For higher dimensions, the necessity of global geometric hypotheses is highlighted, suggesting further possible research into topological obstructions, stratification effects, and the role of filling conditions.

Conclusion

The authors' framework systematically generalizes the theory of the medial axis and its interaction with singularities, incorporating directional and general curvature as precise quantitative invariants in the tame geometry category. The theorems unify and extend prior scattered results, clarify the structure of attainable singularities, and lay the groundwork for further investigations in higher codimension and dimension, as well as in algorithmic applications to medial axis computation in semialgebraic and subanalytic contexts.