Singular Arrange and Traverse Algorithm for Computing Reeb Spaces of Bivariate PL Maps

Abstract

We present an exact and efficient algorithm for computing the Reeb space of a bivariate PL map. The Reeb space is a topological structure that generalizes the Reeb graph to the setting of multiple scalar-valued functions defined over a shared domain, a situation that frequently arises in practical applications. While the Reeb graph has become a standard tool in computer graphics, shape analysis, and scientific visualization, the Reeb space is still in the early stages of adoption. Although several algorithms for computing the Reeb space have been proposed, none offer an implementation that is both exact and efficient, which has substantially limited its practical use. To address this gap, we introduce singular arrange and traverse, a new algorithm built upon the arrange and traverse framework. Our method exploits the fact that, in the bivariate case, only singular edges contribute to the structure of Reeb space, allowing us to ignore many regular edges. This observation results in substantial efficiency gains on datasets where most edges are regular, which is common in many numerical simulations of physical systems. We provide an implementation of our method and benchmark it against the original arrange and traverse algorithm, showing performance gains of up to four orders of magnitude on real-world datasets.

Figures (10)

• Figure 1: Example tetrahedral mesh (a) and its image in the range under a bivariate PL map (b).
• Figure 2: Examples of the change of connectivity of four fibers at the indefinite edge $ab$ and the definite edges $av_1$ and $bv_5$. The Reeb space of $f$ has three sheets (blue, yellow and green) -- one for each fiber component. To visualize the Reeb space we map its sheets to the range and overlay them.
• Figure 3: The key stages of the arrange and traverse algorithm hristovArrangeTraverseAlgorithm2025 for \ref{['fig:domain-regular-arrangement']}.
• Figure 4: The singular set (a) of \ref{['fig:domain-regular-arrangement']} and its image in the range (b). The arrangement $\bar{A}$ of the singular segments has four bounded faces, each subdivided by the faces of the full arrangement.
• Figure 5: The fiber graph class of the face $\bar{F_2}$ is the collection of the fiber graphs of faces of the full arrangement that are in $\bar{F_2}$. All yellow (and respectively blue) fiber components correspond to one another; they change continuously from one to another as a fiber point moves around $\bar{F_2}$. All corresponding fiber graph components form a component of the fiber graph class of $\bar{F_2}$.

Theorems & Definitions (9)

• Definition 1: Fiber Graph Class
• Definition 2: Fiber Graph Class Component
• Definition 3: Fiber Graph Class Component Correspondence
• Definition 4: Singular Correspondence Graph
• Lemma 5: Essential Faces
• Lemma 6: Essential Fiber graphs
• Lemma 7: Singular Correspondence Graph
• Corollary 8: Reeb space
• Lemma 9