Computing handle and tunnel loops with knot linking

doi:10.1016/j.cad.2009.01.001

Computer-Aided Design

Volume 41, Issue 10, October 2009, Pages 730-738

https://doi.org/10.1016/j.cad.2009.01.001 Get rights and content

Abstract

Many applications seek to identify features like ‘handles’ and ‘tunnels’ in a shape bordered by a surface, embedded in three dimensions. To this end, we define handle and tunnel loops on surfaces which can help identify these features. We show that a closed surface of genus $g$ always has $g$ handle and $g$ tunnel loops induced by the embedding. For a class of shapes that retract to graphs, we characterize these loops by a linking condition with these graphs. These characterizations lead to algorithms for detection and generation of these loops. We provide an implementation with applications to feature detection and topology simplification to show the effectiveness of the method.

Introduction

Many applications need to identify features such as ‘handles’ and ‘tunnels’ induced by the embedding of a connected closed surface in three dimensions. Quantitatively, their numbers can be computed from Betti numbers, for which efficient algorithms are known [1]. However, these numbers cannot provide any qualitative information about these features. To this end, we define a class of loops on $M$ , called handle and tunnel loops, that help identifying these features in the shape bounded by $M$ . Intuitively, a loop is a handle if it spans a disk (surface) in the bounded space bordered by $M$ . If one cuts $M$ along such a loop and fills the boundary with that disk, one eliminates a handle. Similarly, a tunnel loop spans a disk (surface) in the unbounded space bordered by $M$ , and its removal eliminates a tunnel. Fig. 1 shows three handle and three tunnel loops on a CAD surface. In this paper, we provide a formal definition of handle and tunnel loops in terms of homology groups, and provide topological analyses that lead to the algorithms for their detection and generation. Our algorithm can be a basis for applications that require one to recognize features such as handles in a shape and tunnels in its complement, or to simplify a shape topologically by eliminating insignificant handles and tunnels [2], [3], [4], [5], [6].

Researchers have looked into the problem of computing nontrivial loops on surfaces with various conditions. Vegter and Yap [7] and Dey and Schipper [8] gave linear time algorithms to compute polygonal schema whose removal cuts the surface into a disk. Erickson and Har-Peled [9] showed that computing graphs of shortest length whose removal cut the surface into a disk is NP-hard. Verdière and Lazarus [10] gave an algorithm for computing a system of loops on a surface which is shortest among the homotopy class of a given system. Yin, Jin, and Gu used universal covering spaces [11] to compute shortest cycles in a homotopy class. Erickson and Whittlesey [12] gave a greedy algorithm to compute the shortest system of loops, among all systems of loops, relaxing the homotopy condition.

The above works were mainly concerned with computing a set of non trivial loops while optimizing some metric on the surface. Our goal is different. We seek to compute only specific loops that are handles and/or tunnels. One fundamental difference is that the aforementioned works do not take into account the embedding $M \to R^{3}$ whereas handle and tunnel loops become meaningful only for embedded surfaces $M \subset R^{3}$ . Moreover, a loop may change its classification if the embedding changes.

We formalize the ideas of handle and tunnel loops and provide an existence proof for them. We argue that the notion of handle and tunnel loops loses its intuitive meaning if the surface has a knotted embedding. We define graph retractable surfaces that avoid these knotted embeddings. These are surfaces whose interior and exterior deformation retract to graphs called core graphs. We present algorithms to detect and generate handle and tunnel loops on such surfaces. Specifically, the main contributions of this paper are:

Definition and existence. We provide a formal definition of handle and tunnel loops and prove their existence.

Detection. We characterize handle and tunnel loops on graph retractable surfaces in terms of their linking with the core graphs. This leads to an algorithm for detecting handle and tunnel loops.

Generation. We show that there exists a special class of handle and tunnel loops that link minimally with the core graphs and present an algorithm to compute them.

Implementation. We present an implementation of our algorithm which incorporates the geometry of the surface more intimately. The results of our implementation show that the method is effective in practice.

Application. We apply our algorithm to the problem of computing ‘handle’ and ‘tunnel’ features in shapes which can further be used for topology simplification. Again, the results show the effectiveness of the method.

Section snippets

Preliminaries

We state some standard concepts from topology. For a more detailed introduction, the interested reader may consult Munkres [13] or Hatcher [14].

Let $X$ be any topological space. A singular $k$ -simplex is defined as a continuous map $σ : Δ^{k} \to X$ where $Δ^{k}$ is the standard $k$ -simplex which is the convex hull of ${e_{i}}_{i = 1}^{k + 1}$ . Each $e_{i}$ is a vector in $R^{k + 1}$ and its $j$ th component is $δ_{i j}$ , where $δ_{i j} = 1$ if $i = j$ and 0 otherwise. A $k$ -chain is a finite linear combination of singular $k$ -simplices. In this paper we assume the

Definition and existence

Let $M$ be a connected, closed (compact and without boundary), and orientable surface. The genus $g$ of $M$ is the maximum number of disjoint simple loops whose removal does not disconnect $M$ . Two closed, connected and orientable surfaces are homeomorphic if and only if they have the same genus. To make our argument simple, let $M$ sit inside a three sphere $S^{3}$ , which is the compactification of $R^{3}$ . Being embedded in $S^{3}$ , the surface $M$ has to be orientable. It separates $S^{3}$ into two parts. Given an

Graph retractable surface

Although Theorem 1 assures the existence of $g$ handle loops and $g$ tunnel loops on all connected closed surfaces in $S^{3}$ , they do not bear intuitive meaning of handles and tunnels in ‘knotted’ surfaces. Fig. 3 shows such a surface. It is obtained by thickening a trefoil knot, namely $M$ is the boundary of the product, $K \times D$ , of a trefoil $K$ and a 2-disk $D$ . The red loop in Fig. 3(a) is obtained by projecting the trefoil knot on the surface $M$ . In contrary to the natural intuition this loop is not a tunnel

Graph complement basis through linking

Since we assume both $I$ and $O$ deformation retract to core graphs, we study the first homology groups of graphs and their complements in $S^{3}$ . Consider a knot ¹ $K \subset S^{3}$ . Since $K$ is homeomorphic to a circle $S^{1}$ , one can assign an orientation to it. It is known that $H_{1} (K) = Z$ and $[K]$ is the generator for $H_{1} (K)$ . We also know that $H_{1} (S^{3} ∖ K) = Z$ . Let $J$ be another knot in $S^{3}$ that is disjoint from $K$ . The union $J \cup K$ is a link. Consider a regular projection of link $J \cup K$ to a plane. Each

Minimally linked loops

In this section, we give more constructive statements about the handle and tunnel loops for a graph retractable surface. These statements relate the linking numbers of the loops with the core graphs to their characterization and existence. Theorem 3 characterizes the handle and tunnel loops in terms of linking numbers. Theorem 4 provides the existence of a special class of handle and tunnel loops which link minimally with the core graphs. This leads to an algorithm to compute a set of $2 g$ loops

Topological algorithms

In this section, we present algorithms to detect and generate handle and tunnel loops for a graph retractable surface.

Assume $\hat{I}$ and $\hat{O}$ are given. In a preprocessing step we compute a set of knots ${K_{i}}_{i = 1}^{2 g}$ from $\hat{I}$ and $\hat{O}$ using their spanning trees as indicated in Section 5. Assuming that $\hat{I}$ and $\hat{O}$ have $t$ edges altogether, computation of $K_{i}$ s take $O (t)$ time.

Detection: Let $ℓ$ be any given loop with $s$ edges on $M$ . By Theorem 3, if $ℓ$ links $\hat{I}$ and not $\hat{O}$ , it is a handle loop. If $ℓ$ links $\hat{O}$ and not $\hat{I}$ ,

Incorporating geometry

In this section, we give an implementation of the algorithm, assuming that $M$ is presented as a piecewise linear surface. The algorithm described in Section 7 computes topologically correct handle and tunnel loops without considering any geometric information. Hence the computed loops may not be very good geometrically. The implementation presented in this section incorporates geometry into the algorithm, namely it computes two sets of loops with small size. Although it does not guarantee that

Experimental results

We implemented our algorithm in C++. Fig. 8 shows only the tunnel and the handle loops for some complicated models. As we can see, all detected loops are geometrically relevant. More examples are shown for feature detection and topology simplification in Fig. 11 and Fig. 12 which we discuss later.

In Table 1 we provide the times for our code on different models. The test machine has 2.66 GHz Intel Xeon CPU and 4 GB memory. We observe that curve skeleton computation dominates the computation

Application

In this section we apply handle and tunnel loops to compute actual handle or tunnel features for shapes which can further be used for removing insignificant topologies.

Feature detection. The basic idea is to sweep the handle or tunnel loops over the surface appropriately. To compute a tunnel feature, we start with a tunnel loop. We run Dijkstra’s shortest path algorithm for multiple sources, where the starting points are the vertices on the tunnel loop. At any generic step, Dijkstra’s algorithm

Conclusions and future work

In this work, we define and prove the existence of loops on surfaces which represent handles and tunnels for the shape bordered by the surface. We characterize these loops on graph retractable surfaces, using linking with core graphs. These characterizations lead to algorithms for detecting and generating handle and tunnel loops.

Several open questions arise as a result of this research. Our implementation does not guarantee that the computed tunnel or handle loops are the shortest in length.

Acknowledgments

We acknowledge that several models used in this paper are taken from AIM@SHAPE repository. We also acknowledge Cindy Grimm for the Knotty Cup and the Flower models.

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Computing handle and tunnel loops with knot linking

Abstract

Introduction

Section snippets

Preliminaries

Definition and existence

Graph retractable surface

Graph complement basis through linking

Minimally linked loops

Topological algorithms

Incorporating geometry

Experimental results

Application

Conclusions and future work

Acknowledgments

Comput Aided Geom Design

Structure preserving CAD model repair

Comput Graphics Forum

Removing excess topology from isosurfaces

ACM Trans Graphics

Topology repair of solid models using skeletons

IEEE Trans Visual Comput Graphics

A new technique to compute polygonal schema for 2-manifolds with application to null-homotopy detection

Discrete Comput Geom

Optimally cutting a surface into a disk

Discrete Comput Geom

Optimal system of loops on an orientable surface

Discrete Comput Geom