Orbit Computation for Atomically Generated Subgroups of Isometries of Z^n

Isometries and their induced symmetries are ubiquitous in the world. Taking a computational perspective, this paper considers isometries of Z^n (since values are discrete in digital computers), and tackles the problem of orbit computation under various isometry subgroup actions on Z^n. Rather than just conceptually, we aim for a practical algorithm that can partition any finite subset of Z^n based on the orbit relation. In this paper, instead of all subgroups of isometries, we focus on a special class of subgroups, namely atomically generated subgroups. This newly introduced notion is key to inheriting the semidirect-product structure from the whole group of isometries, and in turn, the semidirect-product structure is key to our proposed algorithm for efficient orbit computation.

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1 Introduction

Given a metric space, an isometry, also known as a congruent transformation or a rigid(-body) transformation, is an important concept in geometry Coxeter1969 . Isometries of various kinds are ubiquitous in the world, and are also embedded as an innate preference in biological perception Goldman1986 ; Ullman1979 . Therefore, isometries are widely studied in the computational modeling of real-world data observed in different human perception modalities. Examples include vision (computer graphics and animations Carlson2017 ), audition (music Tymoczko2010 ), motion and kinematics (robotics StramigioliB2001 ), and experimental science (crystallography HahnSA1983 ; Bieberbach1911 , physics Noether1915 ). These studies are concerned with not only isometry classifications, but also the symmetries induced by various isometry subgroups. Isometry-induced symmetries, among many other types of symmetries, are strongly connected to invariance theory DerksenK2015 ; Olver1995 , and are key to computational abstraction wherein the abstracted concepts are high-level in the sense of being invariant with respect to the considered isometry subgroup YuMV2019 . Mathematically, symmetries induced by various isometry subgroups are represented by orbits under the corresponding isometry subgroup actions on the metric space. Hence, it is computationally important to have an algorithm that efficiently computes the orbits, or more precisely, the orbit-partition of the metric space.

However, to computationally identify orbits is hard in general, especially when the subgroup and/or the space are infinite. In particular, we showed in our earlier work YuMV2019 that the famous word problem for groups can be cast as a special case of the orbit computation problem, which is therefore, computationally unsolvable in the worst case Novikov1955 ; Boone1958 ; Britton1958 . Notably, when the subgroup is finite, the orbit computation problem is solvable, since obviously any pair of points in the space can be determined to be either in the same orbit or not using a finite number of checks. On the other hand, when the space is finite and the subgroup is infinite but finitely generated, the orbit computation problem is also solvable by an induction algorithm YuMV2019 , which computes the orbit-partition of the space inductively as the meet of the base orbit-partitions induced from the cyclic subgroups generated by each individual generator (base case). Nevertheless, when both the space and the subgroup are infinite (but with the subgroup being finitely generated), the induction algorithm is not, in general, accurate. Here, computing the orbit-partition of an infinite space means being able to compute the partition on any finite subset of the space. Unfortunately, in general, the induction algorithm is only accurate for some finite subset.

This paper takes a step further for isometries, solving the orbit computation problem accurately under a special class of isometry subgroups (possibly infinite) acting on an infinite metric space. In particular, since values are discrete in digital computers, we start from the (ambient) group action of —the group of isometries of —on the metric space , and consider the class of subgroups of wherein every subgroup has a finite generating set that is also atomic. Our newly introduced notion of an atomic generating set and accordingly, an atomically generated subgroup, is key to our proposed algorithm for solving the orbit computation problem exactly. The validity of the algorithm strongly relies on the semidirect product structure of , and we will show that atomically generated subgroups inherit such structure, thus making their corresponding orbit computation problem solvable. The goal of this paper is first to formalize the problem of orbit computation for atomically generated subgroups of , and then propose an algorithm to solve the problem.

The rest of the paper is organized as follows. Section 2 first reviews and generalizes the notion of (inner) semidirect product—the core structure that our main results are based on. Section 3 then introduces the main mathematical objects in this paper, namely the group of isometries of and our new notion of atomically generated subgroups. Section 4 describes a distinguishing property of atomically generated subgroups—the inheritance of the semidirect product structure from the whole group—which is the key property that makes the later-introduced algorithm work. Section 5 formalizes the main computational problem in this paper—the problem of orbit computation for atomically generated subgroups of —by specifying the inputs and desired output of the problem. Continuing the problem formulation, Section 6 further specifies the computational format of the desired output of the orbit computation problem, which is through the so-called orbit-labeling maps. Section 7 presents an algorithmic roadmap that provides the big picture of the global algorithmic procedure for solving the orbit computation problem formalized earlier. The roadmap consists of five major steps, which further boil down to two stages: considering translation equivalence first and rotation equivalence in succession—the two components in the semidirect product structure inherited by all atomically generated subgroups. Accordingly, the following two sections, Sections 8 and 9 detail the two stages respectively, and further in the same pass, collectively prove the correctness of the entire algorithm. In the end, Section 10 analyzes the computational complexity of our algorithm, and discusses the possibility of parallel computing so as to speed up the computation.

Notably, we include three sections in the appendix. A provides a glossary of mathematical notation (including both the new notations and the shorthands) used in this paper. B provides a generic algorithm for generating a finite subgroup from a finite generating set. C provides proofs that are relegated from the main body of the paper.

2 Semidirect Product: Review and Generalization

In a general setting, we first review the (inner) semidirect product of two subgroups, and then generalizes it to the semidirect product of subgroups. The resulting -ary semidirect-product decomposition of a group is the core structure upon which the main results of this paper are built.

Definition 2.1.

Let be a group, and be subsets. We define the product of these subsets (which itself is a subset of ) by

(1)
Definition 2.2.

Let be a group, and be two subgroups. We write the bracket notation to mean:

  1. as a set;

  2. where denotes the normalizer of in ;

  3. where denotes the identity element in .

We call the (inner) semidirect product of and . (Note: one can check that ). Further, if , then we say is the semidirect product of and , or is a semidirect-product decomposition of .

This definition is readily generalized to subgroups as follows.

Definition 2.3.

Let be a group, and be subgroups. We define recursively (on ) by

(2)

Further, if , then we say is the -ary semidirect product of , or is a -ary semidirect-product decomposition of .

Remark 2.1.

Based on the binary bracket notation in Definition 2.2, the following information is automatically encoded in the -ary notation :

  1. recursively for all ;

  2. part of this notation is the requirement that for all , which further implies that ;

  3. for all , which further implies that for any distinct .

3 The Mathematical Objects

To set up the problem of orbit computation later (in Section 5), we first characterize the main mathematical object in this paper, namely the isometry group of denoted by ; then introduce a special class of subgroups of , namely the class of atomically generated subgroups.

3.1 The Isometry Group of :

Our ambient space is the metric space , where is the Euclidean distance. An isometry of is a function satisfying the distance-preserving property:

(3)

We use to denote the set of all isometries of , and one can check that is a group, called the isometry group of . Next, we present a characterization of via semidirect products from our earlier work YuMV2019 , and refer interested readers to that manuscript (cf. page 23–24 and page 35–37, with some terms and notations being simplified in this paper) for more details.

Inheriting properties from its counterpart (i.e. the isometry group of ), can be characterized by a semidirect product as follows:

(4)

In the above characterization:

  1. denotes the group of translations of , where a translation of is a function defined by with the parameter being called the

    translation vector

    ;

  2. denotes the group of (generalized) rotations of , where a rotation of is a function defined by with the parameter being called the rotation matrix. Important note: and the word rotation throughout this paper is a shorthand term for, more precisely, generalized rotation about the origin, which is linear and includes both proper rotation (whose rotation matrix has determinant ) and improper rotation (whose rotation matrix has determinant ).

In addition, has a property that does not have, via a finer dissection of . Repeating a semidirect product at a smaller scale, has a similar characterization that parallels Expression (4) for :

(5)

In the above characterization:

  1. denotes the group of (coordinate-wise) negations of , where a negation of is a rotation with the rotation matrix being a negation matrix—a diagonal matrix whose diagonal entries are either or . Important note: the word negation throughout this paper is a shorthand term for, more precisely, coordinate-wise negation, which negates some (possibly all) coordinates of a vector. The following are two examples of a negation matrix:

    They induce two negations of , and , such that for any , and .

  2. denotes the group of (coordinate-wise) permutations of , where a permutation of is a rotation with the rotation matrix being a permutation matrix

    —a matrix obtained by permuting the rows of an identity matrix.

    Important note: the word permutation throughout this paper is a shorthand term for, more precisely, coordinate-wise permutation, which permutes the coordinates of a vector. The following are two examples of a permutation matrix:

    They induce two permutations of , and , such that for any , and .

From Expression (5), it is clear that the rotation group is finite, or more precisely, .

Expressions (4) and (5) reveal the semidirect-product structure at two different scales. Putting them together, we have a ternary semidirect product:

(6)

In particular, as a property of semidirect product, this means for any isometry of , say , it can be uniquely represented in the following form:

(7)

where is the translation vector, is the negation matrix, and is the permutation matrix.

3.2 Atomically Generated Subgroups

In this paper, we consider a special class of subgroups of —the class of atomically generated subgroups of —wherein every such subgroup has a so-called atomic generating set. To introduce the notion of atomic, we start with definitions in a more general setting.

Definition 3.1.

Let be a semidirect-product decomposition of . A subset is atomic (with respect to the semidirect-product decomposition), if

(8)
Definition 3.2.

Let be a semidirect-product decomposition of . A subgroup is atomically generated (with respect to the semidirect-product decomposition), if it has an atomic generating set, i.e. there exists an atomic subset such that .

Returning to our main mathematical object , we have so far introduced two semidirect-product decompositions of it, namely the binary one in Expression (4) and the ternary one in Expression (6). In the sequel, if the decomposition of is not explicitly specified, we assume it is by default the ternary decomposition in Expression (6). Therefore, a set of isometries is atomic if .

Before closing the section, we introduce a shorthand notation for referencing any component of an atomic subset, as well as any component of an atomically generated subgroup. It is designed to make such references simple, systematic, and consistent with the underlying semidirect-product decomposition.

Let be a semidirect-product decomposition of , and let denote the power set of . For every , define the function and its subscript shorthand notation by

(9)

Note that the above notation and definition apply to all subsets of . However, their main use will be for atomic subsets and atomically generated subgroups. First, for any atomic subset , it is immediate from Definition 3.1 that can be always decomposed as follows:

(10)

Indeed, one can check that Equation (10) holds if and only if is atomic. Second, for any atomically generated subgroup , we will soon see (in Section 4: Theorem 4.7) that can be always decomposed as follows:

(11)

inheriting the semidirect-product structure from . As a sanity check, notice when , Expression (11) is precisely .

In our special case when , we have the following four particular notations: for any ,

Further, for any atomic subset , we have

where again, the last two expressions will be clear after the following section.

4 Special Property of Atomically Generated Subgroups

We describe a distinguishing property of atomically generated subgroups. Since every -ary semidirect-product decomposition is recursively built from binary semidirect products, it suffices to focus on binary semidirect-product decomposition. We start from general groups with a binary semidirect-product decomposition, then generalize it to -ary decompositions, and finally apply the results to isometries.

Let be a binary semidirect-product decomposition of a group . By the definition of a semidirect product, for any , there exists a unique and a unique such that . This uniqueness allows us to define a function and a function , such that for any , . It is a known fact that is a homomorphism.

Let be any atomic subset, then by following the earlier shorthand notation in Expression (9), we can write

The main task in this section is to characterize the three subgroups below:

More specifically, we propose a three-part plan below, and present ahead of time the conclusion (stated as Theorem 4.1) that we will reach.

  1. characterize first via a generating set then via the function ;

  2. characterize first via a generating set then via the function ;

  3. characterize via and .

Theorem 4.1 (The Distinguishing Property of Atomically Generated Subgroup: Binary Case).
Let , and be atomic, then
In the above, is called the augmented generating set: augmented from through conjugation by .

To accomplish the above plan, we start with expressing elements in . We do this through three incremental steps.

  1. First, by the definition of a generating set of a subgroup,

    (12)

    for some .

  2. Second, from Equation (12), by grouping consecutive generators from and grouping consecutive generators from , we can rewrite the expression as the following alternating product:

    (13)

    where , .222From a given Equation (12), its corresponding rewritten form, Equation (13), can be made unique, by requiring that at most one —the identity element of —is in Equation (13), and if does occur, it must be either at the beginning or at the end, i.e. either or .

  3. Third, from Equation (13), by leveraging the semidirect-product decomposition of , we can further rewrite as follows:

    That is, given any expressed as in Equation (13), we can write:

    (14)

In Equation (14), is indeed an element in , since implies . Yet, the expression for is a bit messy, making it less obvious that is indeed an element in . To make this expression easier to parse, we introduce the following conjugation function.

Definition 4.1.

Given any , define the conjugation function by for any .

Remark 4.1.

Two properties are immediate from the above definition:

By using conjugation functions, the expression for in Equation (14) can be systematically rewritten as follows:

where the last equality repeatedly uses the two properties in Remark 4.1. Thus, it is now clear that , since the fact that is a normal subgroup of implies that all terms in the product, i.e. , , , , , are elements in , and so is the product itself.

Moving towards characterizing and , we first introduce an important definition.

Definition 4.2.

Define the following augmented generating set:

We will soon prove: while generates ; instead of , generates . To do that, we first present two lemmas in the following, whose proofs are straightforward and are thus relegated to Appendices C.1 and C.2, respectively.

Lemma 4.1.

.

Lemma 4.2.

.

Theorem 4.2.

.

Proof.

For any , implies ; implies by Lemma 4.1. Therefore, , i.e. .

Conversely, for any , by Definition 4.2,

for some , . The fact that implies that for all , which further implies that . On the other hand, for all , which implies that . Therefore, , i.e. . ∎

Theorem 4.3.

.

Proof.

For any , implies ; implies . Therefore, , i.e. .

Conversely, by Lemma 4.1 and Theorem 4.2. ∎

Remark 4.2.

So far, we have accomplished the first part of the plan and have characterized via both a generating set and the function :

Theorem 4.4.

.

Proof.

For any , implies ; implies by Lemma 4.2. Therefore, , i.e. .

Conversely, implies and , thus,

This further implies that . ∎

Theorem 4.5.

.

Proof.

For any , implies ; implies . Therefore, , yielding .

Conversely, by Lemma 4.2 and Theorem 4.4. ∎

Remark 4.3.

So far, we have accomplished the second part of the plan and have characterized via both a generating set and the function :

We proceed to the third part of the plan to characterize via and . We start with a more general setting: let be any subgroup of , and consider the following two objects and . Below are a few facts about and , linked via the restriction .

  1. is a homomorphism (as the restriction of the homomorphism ).

  2. The image .

  3. The kernel .

  4. (by the first group isomorphism theorem).

The above properties hold for all and particularly, for atomically generated for some atomic subset . However, we can say more about atomically generated subgroups.

Theorem 4.6.

Let be an atomically generated subgroup, then has the following semidirect-product decomposition: , which by Remark 4.3, is equivalent to .

Proof.

It is clear that and , since by definition , , and . Further, (a general property we listed earlier). It remains to be shown that .

The fact that immediately implies . Conversely, for any , we have . Therefore, . Then, by Definition 2.2, . ∎

Remark 4.4.

It is important that the subgroup in Theorem 4.6 is atomically generated, which guarantees that is also a subgroup of (Theorem 4.5). This is, in general, not true for any arbitrary . In particular, for some , is not even a subset of . For example, let denote the group of (invertible) affine transformations of , denote the group of translations of , and

denote the group of (invertible) linear transformations of

; further, for any , let denote the affine transformation defined by . Consider

In this case, and .

So far, we have reached the conclusion in Theorem 4.1 (whose proof is immediate from Theorem 4.6 together with Remarks 4.2 and 4.3). By induction, we can further generalize it to any -ary semidirect-product decompositions. This is stated in the following theorem whose proof is relegated to Appendix C.3.

Theorem 4.7 (The Distinguishing Property of Atomically Generated Subgroup: General Case).
Let , and be atomic. Then, has a similar semidirect-product decomposition:
(15) (16)
In Equation (16), the augmented generating set is consistently defined as follows:
(17) (18)
In particular, .

The Distinguishing Property in the Case of Isometries

Consider the main mathematical object in this paper. Recall in Section 3.1, we concluded that

or collectively, we can write (cf. the bracket notation (2) in Definition 2.3)

Therefore, for any , we can represent it as for some unique , , , and . This uniqueness allows us to define , , , , such that for any , .

We apply Theorem 4.7 to the above semidirect-product decompositions regarding and its subgroups. For any atomic subset ,

(19)
(20)

or collectively,

(21)

where

(22)
(23)
(24)
(25)

In Equation (22),