The moving frame method for iterated-integrals: orthogonal invariants

Geometric features, robust to noise, of curves in Euclidean space are of great interest for various applications such as machine learning and image analysis. We apply the Fels-Olver's moving frame method (for geometric features) paired with the log-signature transform (for robust features) to construct a set of integral invariants under rigid motions for curves in $\mathbb{R}^d$ from the iterated-integral signature. In particular we show that one can algorithmically construct a set of invariants that characterize the equivalence class of the truncated iterated-integrals signature under orthogonal transformations which yields a characterization of a curve in $\mathbb{R}^d$ under rigid motions (and tree-like extensions) and an explicit method to compare curves up to these transformations.

curvature have been used [ HO13 ] for this purpose, and more recently integral invariants of curves have been of interest [ FKK08 ,DR19 ]. In this work we construct a rigid motion-invariant representation of a curve through its iterated-integrals signature by applying the Fels-Olver moving frame method. We show that this yields sets of integral invariants that characterize the truncated iterated integral signature up to rigid motions.
In [ Che54a ,Che58 ], the author used the collection of all iterated integrals to characterize smooth curves, and in [ HL10 ] the author extended this result to more irregular curves. The modern term for this collection of iterated integrals of a curve is the iterated-integrals signature. It has since been used in various applications such as constructing features for machine learning tasks (see [ CK16 ] and references therein) and shape analysis [ CLT19 ,LG20 ].
The Fels-Olver moving frame method, introduced in [ FO99 ], is a modern generalization of the classical moving frame method formulated by Cartan [ Car35 ]. In the general setting of a Lie group G acting on a manifold M a moving frame is defined as a G-equivariant map from M to G. A moving frame can be re-interpreted as a choice of cross-sections to the orbits of G, and hence a unique canonical form for elements of M under G. Thus the moving frame method provides a framework for algorithmically constructing G-invariants on M that characterize orbits and for determining equivalence of submanifolds of M under G.
The moving frame method has been used to construct differential invariants of smooth planar and spatial curves under Euclidean, affine, and projective transformations, and, in certain cases, these differential invariants lead to a differential signature which can be used to classify curves under these transformation groups [ COS`98 ]. The differential signature has been applied in a variety of image science applications from automatic jigsaw puzzle assembly [ HO14 ] to medical imaging [ GS17 ]. Also in the realm of image science, the moving frame method has been used to construct invariants of grayscale images [ Bou13 ,TOT19 ].
We consider the induced action of the orthogonal group of rotations on the log-signature of a curve, which provides a compressed representation of a curve obtained by applying the log transform to the iterated-integrals signature, and provide an explicit cross-section for this action. We show that for most curves and any truncation of the curve's log-signature, the orbit is characterized by the value on this cross-section. As a consequence a curve is completely determined up to rigid motions and tree-like extensions by the invariantization of its iterated-integrals signature induced by this cross-section.
This yields a constructive method to compare curves up to rigid motions and to evaluate rotation invariants that characterize the iterated-integrals signature under rotations. These invariants are constructed from integrals on the curve, and hence are likely to be more noise-resistant than their differential counterparts such as curvature. One can easily set up an artificial example where this is visible. Consider for instance the circle of radius n´3 {2 given by the parameterization γ : r0, 1s Ñ R 2 where γptq " pxptq, yptqq "ˆc osp2πntq n 3{2 , sinp2πntq n 3{2˙, which as n Ñ 8, converges to the constant curve (at the origin). Now the curvature of this curve does not converge (in fact, it blows up). In contrast, the iterated integrals do all converge (to zero) since γ converges in variation norm. Then, the invariants built out of the iterated integrals (Section 4 ) also converge to their value on the zero-curve. On this toy example these integral invariants are hence more "stable". Additionally, in contrast to the methods in [ DR19 ], the resulting set of integral invariants is shown to uniquely characterize the curve under rotations, and moreover, does so in a minimal fashion. Since the iterated-integrals signature of a curve is automatically invariant to translations, this provides rigid motion-invariant features of a curve which can be used for applications such as machine learning or shape analysis.
This work is structured as follows. In Section 2 we provide background on the iterated-integrals signature and the moving frame method, as well as some facts about algebraic group and invariants. In Section 3 and consider the orthogonal action on the second order truncation of the log-signature over the complex numbers. Using tools from algebraic invariant theory, we construct the linear space which will form the basis for the cross-section in the following section. We also provide an explicit set of polynomial invariants that characterize the second order truncation of the log-signature under the orthogonal group. In Section 4 we construct the moving frame map for paths in R d . In particular, in Section 4.1 , we outline our procedure and results in simple language for paths in R 2 and in Section 4.3 we introduce sufficient conditions for the resulting moving frame invariants to be polynomial, showing these conditions are satisfied for some values of d. In Section 5 we detail the moving frame map for planar and space curves, compute some of the resulting integral invariant functions, and illustrate this procedure on a particular curve. Finally in Section 6 we discuss some of the interesting questions that arise as a result of our work.

Preliminaries
2.1. The tensor algebra. Let d ě 1 be an integer. A word, or multi-index, over the alphabet t1, . . . , du is a tuple w " pw 1 , . . . , w n q P t1, . . . , du n for some integer n ě 0, called its length which is denoted by |w|. As is usual in the literature, we use the short-hand notation w " w 1¨¨¨wn , where the w i , words of length one, are called letters. The concatenation of two words v, w is the word vwv 1¨¨¨vn w 1¨¨¨wm of length |vw| " n`m. Observe that this product is associative and non-commutative. There is a unique element of length zero, called the empty word and denoted by e. It satisfies we " ew " w for all words w. If we denote by T pR d q the real vector space spanned by words, the bilinear extension of the concatenation product endows it with the structure of an associative (and non-commutative) algebra. We also note that T pR d q admits the direct sum decomposition There is a commutative product on T pR d q, known as the shuffle product, recursively defined by e ¡ www ¡ e and vi ¡ wj -pv ¡ wjqi`pvi ¡ wqj.
The commutator bracket ru, vs -uv´vu endows T pR d q with the structure of a Lie algebra. The free Lie algebra over R d , denoted by gpR d q, can be realized as the following subspace of T pR d q, where W 1 -span R t1, . . . , du -R d and W n`1 -rW 1 , W n s. There are multiple choices of bases for gpR d q, but we choose to work with the Lyndon basis. A Lyndon word is a word h such that whenever h " uv, with u, v ‰ e, then u ă v for the lexicographical order. We denote the set of Lyndon words over the alphabet t1, . . . , du by L d . In particular, h with |h| ě 2 is Lyndon if and only if there exist non-empty Lyndon words u and v such that u ă v and h " uv. Although there might be multiple choices for this factorization, the one with v as long as possible is called the standard factorization of h. The Lyndon basis b h is recursively defined by setting b i " i and b h " rb u , b v s for all Lyndon words h with |h| ě 2, where h " uv is the standard factorization.
Example 2.1. Suppose d " 2. The Lyndon words up to length 4, their standard factorizations and the associated basis elements are In particular, we have no growth requirement for the coefficients xF, wy P R. The above expression is meant only as a notation for treating the values of F on words as a single object. This space can be endowed with a multiplication given, for F, G P T ppR d qq, by Observe that since there is a finite number of pairs of words u, v such that uv " w, the coefficients of F G are well defined for all w, so the above formula is an honest element of T ppR d qq. It turns out that this product is dual to the deconcatenation coproduct ∆ : in the sense that xF G, wy " xF b G, ∆wy for all words. This formula is nothing but eq. ( 2.1 ) componentwise.
There are two distinct subsets of T ppR d qq that will be important in what follows. The first one is the subspace gppR d qq of infinitesimal characters, formed by linear maps F such that xF, u ¡ vy " 0 whenever u and v are non-empty words, and such that xF, ey " 0. It can be identified with the dual space It is a Lie algebra under the commutator bracket rF, Gs " F G´GF . The second one is the set G ppR d qq of characters, i.e., linear maps F such that xF, u ¡ vy " xF, uyxF, vy for all u, v P T pR d q.
We may define an exponential map exp : gppR d qq Ñ G ppR d qq by its power series exppF q - On a single word, the map is given by and since F vanishes on the empty word, all terms with n ą |w| also vanish, so that the sum is always finite. Therefore, exppF q is a well defined element of T ppR d qq. It can be shown that the image of exp is equal to G ppR d qq and that it is a bijection onto its image, with inverse log : G ppR d qq Ñ gppR d qq defined by logpGq - where ε is the unique linear map such that xε, ey " 1 and zero otherwise. Finally, we remark some freeness properties of the tensor algebra and its subspaces. Below, denotes the reduced tensor algebra over R d . The following result can be found in [ FPT16 , Corollary 2.1].
Proposition 2.2. Let φ : T`pR d q Ñ R e be a linear map. There exists a unique extensionφ : T pR d q Ñ T pR e q such that pφ bφq˝∆ " ∆˝φ and π˝φ " φ, where π : T pR e q Ñ R e denotes the projection of T pR e q onto R e , orthogonal to Re and À ną2 span R tw : |w| " nu. Moreover, it is given bỹ By transposition, we obtain a unique map Φ : T ppR e qq Ñ T ppR d qq such that ΦpF Gq " ΦpF qΦpGq for all F, G P T ppR e qq. In particular, 2.2. The iterated-integrals signature. The iterated-integrals signature of (smooth enough) paths was introduced by Chen for homological considerations on loop space, [ Che54b ]. It played a vital role in the rough path analysis of Lyons, a pathwise approach to stochastic analysis, [ Lyo98 ]. Recently it has found applications in statistics and machine learning, where it serves as a method of feature extraction for possibly non-smooth time-dependent data. Let Z " pZ 1 , . . . , Z d q : r0, 1s Ñ R d be an absolutely continuous path. 1 Given a word w " w 1¨¨¨wn , define xIISpZq, wy -ż¨¨¨ż 0ăs1ă¨¨¨ăsnă1 9 Z w1 ps 1 q¨¨¨9 Z wn ps n q ds 1¨¨¨d s n P R.
This definition has a unique linear extension to T pR d q. We obtain thus an element IISpZq P T ppR d qq, called the iterated-integrals signature (IIS) of Z. It was shown by Ree [ Ree58 ] that the coefficients of IISpZq satisfy the so-called shuffle relations: xIISpZq, vyxIISpZq, wy " xIISpZq, v ¡ wy.
In other words, IISpZq P G ppR d qq.
As a consequence of the shuffle relation one obtains that the log-signature logpIISpZqq is a Lie series, i.e., an element of gppR d qq. Moreover, the identity IISpZq " expplogpIISpZqqq holds. The log-signature therefore contains the same amount of information as the signature itself; it in fact is a minimal (linear) depiction of it. 2 The entire iterated-integrals signature IISpZq is an infinite dimensional object, and hence can never actually be numerically computed. We now provide more detail on the truncated, finite-dimensional setting.
For each integer N ě 1, the subspace I n Ă T ppR d qq generated by formal series such that xF, wy " 0 for all words with |w| ď N is a two-sided ideal, that is, the inclusion I n T ppR d qq`T ppR d qqI n Ă I n holds. Therefore, the quotient space T ďn ppR d qq -T ppR d qq{I n inherits an algebra structure from T ppR d qq. Moreover, it can be identified with the direct sum 1 One can get away with much less regularity, see [ Lyo98 ]. Since our considerations are purely algebraic, there is no loss in restricting to 'smooth' paths.
We denote by proj ďn : T ppR d qq Ñ T ďn ppR d qq the canonical projection. Denote with g ďn ppR d qq the free step-N nilpotent Lie algebra (over R d ). It can be realized as the following subspace of T ďn ppR d qq, see [ FV10 ,Section 7.3], where, as before W 1 -span R ti : i " 1, . . . , du -R d and W n`1 -rW 1 , W n s. In the case of N " 2 this reduces to where we denote with sopd, Rq the space of skew-symmetric dˆd matrices. Indeed, an isomorphism is given by We remark that the coefficients c i and c ij are the coordinates 3 with respect to the Lyndon basis (see Example 2.1 ). The linear space g ďn ppR d qq is in bijection to its image under the exponential map. This image, denoted G ďn pR d q -exp g ďn ppR d qq, is the free step-N nilpotent group (over R d ). It is exactly the set of all points in T ďn pR d q that can be reached by the truncated signature map, that is (see [ FV10 ,Theorem 7.28]) Equivalently, the log-signature completely fills out the Lie algebra gppR d qq. We have where c h pZq " xISSpZq, ζ h y for uniquely determined ζ h P T pR d q. This inspires us to also denote the coordinates of an arbitary c ďn P g ďn ppR d qq by c h , were analogously Example 2.3 (Moment curve). We consider the moment curve in dimension 3, which is the curve Z : r0, 1s Ñ R 3 given as X t -pt, t 2 , t 3 q.
We calculate, as an example, xISSpZq, 32y " The entire step-2 truncated signature is proj ď2 IISpZq "¨» - These are often referred to as coordinates of the first kind, see [ Kaw11 ,OM01 ] and the step-2 truncated log-signature is proj ď2 log IISpZq "¨» - 2.3. Invariants. In this work we are interested in functions on paths that factor through the signature and that are invariant to a group G acting on the path's ambient space R d . We will mainly focus on G " O d pRq, acting linearly on R d . The action of A P G on an R d -valued path Z is given by AZ : r0, 1s Ñ R d , t Þ Ñ AZ t .
Using Proposition 2.2 , we can extend the action of G on R d to a diagonal action on words. The matrix A J acts on single letters by and we set φ A J pwq " 0 whenever |w| ě 2. By Proposition 2.2 , this induces an endomorphismφ (2.5) In particular,φ A J puiq "φ A J puqφ A J piq for all words u and letters i P t1, . . . , du. In order to be consistent with the notation in [ DR19 ], we will denote its transpose map (Φ A in Proposition 2.2 ) just by A : T ppR d qq Ñ T ppR d qq.
Lemma 2.4. The mapφ A J : T pR d q Ñ T pR d q is a shuffle morphism, that is, Proof. We proceed by induction on |u|`|v| ě 0. If |u|`|v| " 0 then necessarily u " v " e, and the identity becomesφ which is true by definition. Now, suppose that the identity is true for all words u 1 , v 1 with |u 1 |`|v 1 | ă n. If |u|`|v| " n we suppose, without loss of generality, that u " u 1 i, v " v 1 j for some (possibly empty) words u 1 , v 1 with |u 1 |`|v 1 | ă n. Theñ i. Let F P G ppR d qq, and u, v be words. Then that is, A¨F P G ppR d qq. ii. Since A¨pF Gq " pA¨F qpA¨Gq, A is automatically a Lie morphism. Now we check that A¨F P gppR d qq whenever F P gppR d qq. It is clear that xA¨F, ey " xF, ey " 0. Now, if u, v are non-empty words, then In particular, we easily see that (see also [ DR19 , Lemma 3.3]) IISpA¨Zq " A¨IISpZq. (2.6) The same is true for the truncated versions, and we note that, in the special case of g ď2 ppR d qq, under the isomorphism in eq. ( 2.4 ), the action has the simple form A¨pv, M q " pAv, AM A J q, (2.7) where the operations on the right-hand side are matrix-vector resp. matrix-matrix multiplication. It follows from Corollary 2.6 and ( 2.6 ) that logpIISpAZqq " A¨logpIISpZqq. As already remarked, log is a bijection (with inverse exp). To obtain invariant expressions in terms of IISpZq it is hence enough to obtain invariant expressions in terms of logpIISpZqq. Going this route has the benefit of working on a linear object. To be more specific, IISpZq is, owing to the shuffle relation, highly redundant. As an example in d " 2, Now, both of these expressions are invariant to O 2 pRq. The left-hand-side is a nonlinear expressions in the signature, whereas the right-hand-side is a linear one. To not have to deal with this kind of redundancy we work with the log-signature. We note that in [ DR19 ] the linear invariants of the signature itself are presented. Owing to the shuffle relation, this automatically yields (all) polynomial invariants. But, as just mentioned, it also yields a lot of redundant information.
Its step-2 truncated signature is proj ď2 IISpY q "¨» - In the present work, we consider general, nonlinear expressions of the log-signature. That way, we use the economical form of the log-signature, while still providing a complete -in a precise sense -set of nonlinear invariants.
2.4. Moving frame method. We now provide a brief introduction to the Fels-Olver moving frame method introduced in [ FO99 ], a modern generalization of the classical moving frame method formulated by Cartan [ Car51 ]. For a comprehensive overview of the method and survey of many of its applications see [ Olv18 ,FO98 ]. We will assume in this subsection that G is a finite dimensional Lie group acting smoothly 4 on an m-dimensional manifold M .
Definition 2.8. A moving frame for the action of G on M is a smooth map ρ : M Ñ G such that ρpg¨zq " ρpzq¨g´1.
In general one can define a moving frame as a smooth G-equivariant map ρ : M Ñ G. For simplicity we assume G acts on itself by right multiplication; this is often referred to as a right moving frame. A moving frame can be constructed through the use of a cross-section to the orbits of the action of G on M .
Definition 2.9. A cross-section for the action of G on M is a submanifold K Ă M such that K intersects each orbit transversally at a unique point.
Definition 2.10. The action of G is free if the stabilizer G z of any point z P M is trivial, i.e.
where id P G denotes the identity transformation.
The following result appears in much of the previous literature on moving frames (see, for instance, [ Olv01 , Thm. 2.4]).
Theorem 2.11. Let G be an action on M and assume that (˚) The action is free, and around each point z P M there exists arbitrarily small neighborhoods whose intersection with each orbit is path-wise connected. If K is a cross-section, then the map ρ : M Ñ G defined by sending z to the unique group element g P G such that g¨z P K is a moving frame.
Remark 2.12. The equivariance of the map ρ : M Ñ G such that ρpzq¨z P K can be seen from the fact that ρpzq¨z " ρpg¨zq¨pg¨zq for any g P G. Since G is free this implies that ρpzq " ρpg¨zq¨g, and hence ρ satisfies Definition 2.8 .
Similarly, in this setting, a moving frame ρ specifies a cross-section defined by K " tρpzq¨z P M u. This construction can be interpreted as a way to assign a "canonical form" to points z P M under the action of G, thus producing invariant functions on M under G.
Definition 2.13. Let ρ : M Ñ G be a moving frame. The invariantization of a function F : M Ñ R with respect to ρ is the invariant function ιpF q defined by ιpF qpzq " F pρpzq¨zq.
Given a moving frame ρ and local coordinates z " pz 1 , . . . , z m q on M , the invariantization of the coordinate functions ιpz 1 q, . . . , ιpz m q are the fundamental invariants associated with ρ. In particular we can compute ιpF q by ιpF qpz 1 , . . . , z m q " F pιpz 1 q, . . . , ιpz m qq. Since ιpIqpzq " Ipzq for any invariant function I, the fundamental invariants provide a functionally generating set of invariants for the action of G on M . In general, we will call a set of invariants I " tJ 1 , . . . , J m u fundamental if it functionally generates all invariants, i.e. for any invariant I there is a function I 1 such that Now, suppose further that G is an r-dimensional Lie group and that ρ is the moving frame associated to a coordinate cross-section K defined by equations z 1 " c 1 , . . . , z r " c r for some constants c 1 , . . . , c r . Then the first r fundamental invariants are the phantom invariants c 1 , . . . , c r , while the remaining m´r invariants tI 1 , . . . , I m´r u form a functionally independent generating set. In this case we can see that two points z 1 , z 2 P M lie in the same orbit if and only if I 1 pz 1 q " I 1 pz 2 q, . . . , I s pz 1 q " I s pz 2 q.
Example 2.14. Consider the canonical action of SO 2 pRq on R 2 ztp0, 0qu. This action satisfies the assumptions of Theorem 2.11 and a cross-section to the orbits is given by The unique group element taking a point to the intersection of its orbit with K is the rotation (see Figure 1 ) The fundamental invariants associated with the moving frame ρ : R 2 ztp0, 0qu Ñ SO 2 pRq are given by ιpxq " 0 ιpyq " a x 2`y2 . Thus any invariant function for this action can be written as a function of ιpyq, the Euclidean norm. One can check that indeed for an invariant Ipx, yq, one has Ipx, yq " Ip0, a x 2`y2 q. This additionally implies that two points are related by a rotation if and only if they have the same Euclidean norm.
In practice it is difficult, or impossible, to find a global cross-section, and thus a global moving frame, to the orbits of G on M . For instance in the above example, the origin was removed from R 2 to ensure freeness of the action. If the action of G on M satisfies condition (˚) from Theorem 2.11 , then the existence of a local moving frame around each point z P M is guaranteed by [ FO99 ,Thm. 4.4]. In this case the moving frame is a map ρ : U Ñ V from a neighborhood z P U of M to a neighborhood of the identity in V Ă G. The fundamental set of invariants produced are also local in nature and thus only guaranteed to be invariant on U for elements g P V .
The condition (˚) in Theorem 2.11 can be relaxed in certain cases. In [ HK07b , Sec. 1] the authors outline a method to construct a fundamental set of local invariants for actions of G that are only semi-regular, meaning that all orbits have the same dimension. In particular Theorem 1.6 in [ HK07b ] states that for a semi-regular x 1 x 2 x 3 x 4 Remark 2.15. The algebraic actions that we define in the next section are automatically semi-regular on a Zariski-open subset of the target space (Proposition 2.16 (c)), and hence a local cross-section exists around any point in this subset. Since orbits are algebraic subsets, a local coordinate cross-section is a submanifold of complementary dimension (to the dimension of orbits) intersecting each orbit about z transversally, and hence in finitely-many points. If every sufficiently small neighborhood about z does not have path-wise connected intersection with each orbit, a local cross-section about z necessarily intersects some orbit at infinitely-many points, and hence a free algebraic group action necessarily satisfies condition (˚) from Theorem 2.11 .

Algebraic groups and Invariants.
In this work, we will be in the setting of an algebraic group G acting rationally on a variety X. In other words G is an algebraic variety equipped with a group structure, and the action of G on X is given by a rational map Φ : GˆX Ñ X. Here we outline some key facts and results about algebraic group actions and the invariants of such actions, following [ PV94 ] for much of our exposition. Unless specified otherwise, both G and X are both varieties over the algebraically closed field C.
The orbit G¨p of a point p P X under G is the image of Gˆtpu under the rational map Φ defining the action, and hence is open in its closure G¨p under the Zariski topology. 5 5 The Zariski topology on an affine space k d is the topology where closed sets are given by subsets of the form V pf 1 , . . . , fsq " The following proposition summarizes a few basic results on orbits of algebraic groups that can be found in [ PV94 , Section 1.3].
Proposition 2.16. For any point p P X, the stabilizer G p is an algebraic subgroup of G and G¨p satisfies the following: For an arbitrary field k, we denote the ring of polynomial functions on the variety X as krXs, i.e. if IpXq is the ideal generated by the polynomials defining the variety X Ă C d , then krXs " krx 1 , x 2 , . . . , x d s{IpXq. If X is irreducible, then the field kpXq of rational functions on X is defined similarly. The polynomial invariants (for the action of G on the variety X) form a subring of krXs defined by krXs G " tf P krXs | f pg¨pq " f ppq, for all g P G, p P Xu and the rational invariants form a subfield of kpXq given by respectively. Constructing invariant functions and finding generating 6 sets for CrXs G is the subject of classical invariant theory [ LGRS65 ,Olv99 ,Stu08 ]. In [ Hil90 ] Hilbert proved his finiteness theorem, showing that for linearly reductive groups acting on a vector space V the polynomial ring CrV s G is finitely generated leading him to conjecture in his fourteenth problem that CrXs G is always finitely generated. In [ Nag59 ] Nagata constructed a counter-example to this conjecture. For CpXq G , however, a finite generating set always exists and can be explicitly constructed (see for instance [ DK15 ,HK07a ] .
One way to understand the structure of invariant rings is by considering subsets of X that intersect a general orbit.
Definition 2.20. Let N Ă G be a subgroup. A subvariety S of X is a relative N -section for the action of G on X if the following hold: ‚ There exists a non-empty, G-invariant, and Zariski-open subset U Ă X, such that S intersects each orbit that is contained in U . In other words, we have that ΦpGˆSq " X, where closure is taken in the Zariski topology.
‚ One has N " tn P G | nS " Su.
We call the subgroup N the normalizer subgroup of S with respect to G. The following proposition summarizes a discussion in [ PV94 , Sec. 2.8].
Example 2.21. For the action of SO 2 pCq on the Zariski-open subset of C 2 defined by x 2`y2 ‰ 0, the variety S defined by x " 0 is a relative N -section for the action where N is the 2-element subgroup generated by the reflection about the y-axis. Then S intersects each orbit of the action in precisely two points.
Proposition 2.22. Let S be a relative N -section for the action of G on X. Then the restriction map restricts to a field isomorphism between CpXq G and CpSq N .
Corollary 2.23. Let S be a relative N -section for the action of G on X and I Ă CpXq G a set such that R XÑS pIq generates CpSq N where R XÑS is the restriction map from Proposition 2.22 . Then I is a generating set for CpXq G .
Relative sections can be used to construct generating sets of rational invariants for algebraic actions as in [ GHP18 ], which the authors refer to as the slice method. Similar in spirit to the approach in [ HK07b ], considerations can be restricted to an algebraic subset of X. When the intersection of S with each orbit is zero-dimensional, a relative N -section can be thought of as the algebraic analog to a local cross-section for an action.
We end the section by considering algebraic actions on varieties defined over R, where the issue is more delicate. For instance, in this setting Proposition 2.18 no longer holds meaning that generating sets of invariants are not necessarily separating and vice versa (see [ KRV20 , Rem. 2.7]). Suppose that XpRq and GpRq are real varieties with action given by Φ : GpRqˆXpRq Ñ XpRq and that X and G are the associated complex varieties. Then Φ defines an action of G on X.
Proof. If f P RpXpRqq GpRq , then the rational function f pg¨pq´f ppq is identically zero on GpRqˆXpRq, and hence is identically zero on GˆX. Thus f P CpXq G .
Proof. Suppose that I generates CpXq G and that f P RpXpRqq GpRq . Then there exists a rational function g P Cpy 1 , . . . , y s q such that f " gpI 1 , . . . , I s q. We can decompose g as g " Repgq`i¨Impgq where Repgq,¨Impgq P Rpy 1 , . . . , y s q. Since f is a real rational function 2f " rRepgq`i¨Impgqs`rRepgq´i¨Impgqs " 2Repgq.
Thus g must lie in Rpy 1 , . . . , y s q proving the result. Proof. Suppose that I " tI 1 , I 2 , . . .u generates RpXpRqq GpRq and that RpXpRqq GpRq separates orbits. Then for any two points p 1 , p 2 P XpRq if I 1 pp 1 q " I 1 pp 2 q, I 2 pp 1 q " I 2 pp 2 q, . . . for all invariants in I, then we also have Ipp 1 q " Ipp 2 q for any invariant I P RpXpRqq GpRq as I generates RpXpRqq GpRq . Thus p 1 and p 2 lie in the same orbit under GpRq.

Orthogonal invariants on g ď2 ppR d qq
In this section we take a closer look at the action of O d pRq on g ď2 ppR d qq -R d ' so d pRq. In particular we construct an explicit linear space, of complementary dimension to the orbits, intersecting each orbit in a large open subset of this space. To achieve this, we consider the associated action of the complex group O d pCq on the As described in Section 2.5 , we can consider O d pRq and R d ' so d pRq as the real points of the varieties O d pCq and C d ' so d pCq.
Remark 3.1. The real Lie group can be considered as a subgroup of the Lie group We note that O d pCq is a complex Lie group, in contradistinction to the Lie group of unitary matrices where A˚is the conjugate transpose of A. Even though U d contains matrices with complex entries, it is a real Lie group.
By investigating the associated complex action, we can utilize tools such as the relative sections described in Definition 2.20 , and then pass these results down to the real points. As before in ( 2.7 ) the action of O d pCq on C d ' so d pCq is given by A¨pv, M q " pAv, AM A T q.
where w is an element of C that satisfies w 2 " c 2 i`c 2 j . The transformation A is the complex analogue to a Givens Rotation which only rotates two coordinates. Then for Av " v we have that c k " c k for k R ti, ju, c i " 0, and c j " w ‰ 0. This process can be repeated until v is of the desired form.
We define a sequence of linear subspaces of C d ' so d pCq as (3.2) In particular the subspace L  Note again that all so d pCq matrices are skew-symmetric and thus have zero diagonal.
We will show that L we must have g jd " g dj " 0, j " 1, . . . , d´1. This proves the claim for i " 1.
Let the statement be true for some 1 ď i ď d´2. First, the normalizer of L We now show that L pd´1q d is a relative W d pCq-section, by constructing a sequence of relative sections for the action, drawing inspiration from recursive moving frame algorithms (see [ Kog03 ] for instance).
More precisely, there exists a set of rational invariants such that if we define the invariant, non-empty, Zariski-open subset we have that L pd´1q d intersects each orbit that is contained in U d pCq. Furthermore we can restrict each invariant to obtain Proof. By Proposition 3.2 , outside of f 1 " ||v|| 2 " 0, there exists a rotation Remark 3.7. Denoting ς 1 :" σ 1 , ς i`1 :" σ i`1˝σ´1 i , we have the following chain of O d pRq transformations A i and field isomorphisms ς i : Note though that while the ς i are uniquely determined, the A i are not. The composition A d´1 A d´2¨¨¨A2 A 1 however is unique up to a multiplication of a W d pCq matrix from the left.
In particular the above proposition implies that L (3.8) Proof. Suppose that the action is not free. Then there exists D P W d pCq such that D¨pv, M q " pv, M q and D is not the identity. Necessarily we have that for some 1 ď i ď d´1, w i "´1. Since w i w i`1 c ipi`1q " c ipi`1q and c ipi`1q ‰ 0, then w i`1 "´1. Using a similar argument, w i`2 "´1 and so forth. However w d c d " c d , where c d ‰ 0, implying that w d " 1 which is a contradiction. . We show that every fiber of this map is exactly an orbit of W d pCq. Consider any pv, M q P L pd´1q d X U d pCq; then set of points in the fiber of its image is given by c 12 "˘c 12 , . . . ,c pd´1qd "˘c pd´1qd u.
We can individually change the sign for any coordinate of pv, M q. To change the sign of only c d one can act by the matrix D P W d pCq such that w i "´1 for all 1 ď i ď d. Similarly for c ipi`1q we can act by the matrix D P W d pCq such that w k "´1 for 1 ď k ď i and w k " 1 otherwise. This implies that the above set is exactly the orbit of pv, M q under W d pCq, and hence I W d pCq is separating on L Proof. In the proof of Proposition 3.6 , each function f i is obtaining by taking the inverse image of a real invariant function under the field isomorphism σ i : Then for the action of A¨pv, M q we have that where 1 ă i ă d´1. Note that M k pi, dq is linear combination of M k´1 pi´1, dq and M k´1 pi`1, dq. By the induction hypothesis we know that M k´1 pi, dq " 0 if i ă d´k`1, and hence M k pi, dq " 0 when i`1 ă d´k`1, or equivalently when i ă d´k. This proves (b).
Suppose that i ą d´k. Then M k pi, dq is linear in the terms where c i´1,i and c i,i`1 are of the form c pd´jqpd´j`1q for 1 ď j ă k. By the induction hypothesis, the latter two terms are polynomials in c pd´jqpd´j`1q where 1 ď j ă k´1, proving (c). Finally suppose that i " d´k. We have that By the induction hypothesis we know that M k´1 pd´k`1, dq " Thus we can rewrite the above equality to By the induction hypothesis each c 2 pd´iqpd´i`1q for 1 ď i ă k is a rational function of v¨v| L pd´1q

O d pRq-invariant iterated-integral signature
4.1. Moving frame on g ďn ppR 2 qq. In this section, we construct a moving frame map for the action of O 2 pRq on g ďn ppR 2 qq, and show how this can be used to construct O 2 pRq-invariants in g ďn ppR 2 qq and hence in the coefficients of the iterated-integralas signature of a curve Z. First consider the action on g ď2 ppR 2 qq " R 2 ' rR 2 , R 2 s. We can denote any element of g ď2 ppR 2 qq as c ď2 with coordinates c 1 , c 2 , and c 12 . Through the isomorphism in ( 2.4 ) we can consider c ď2 as an element of R 2 ' sop2, Rq, and with action as in ( 2.7 ). We will now show that O 2 pRq is free on g ď2 ppR 2 qq and the following submanifold K 2,ď2 :" c ď2 P g ď2 ppR 2 qq | c 1 " 0; c 2 , c 12 ą 0 ( is a cross-section for the action. Similarly to Example 2.14 , we start by defining the group element which is defined outside of tc 1 " c 2 " 0u. For any such element c ď2 P g ď2 ppR 2 qq, we have that Unlike in Example 2.14 , the action is not free on R 2 , the submanifold defined by c 1 " 0, c 2 ą 0 is not a cross-section, and Apc ď2 q does not define a moving frame map. This is due to the fact that a reflection about the y-axis will fix v, but change the sign of M . Thus to define a moving frame map we must consider the diagonal action of O 2 pRq on all of g ď2 ppR 2 qq, not just the action on g ď1 ppR 2 qq " R 2 . The mapρ 2 : U 2;ď2 Ñ O 2 pRq given bỹ ρ 2 pc ď2 q " 1 a c 2 1`c 2 2 " sgnpc 12 qc 2´s gnpc 12 qc 1 c 1 c 2  defines the group elementρ 2 pc ď2 q such thatρ 2 pc ď2 q¨c ď2 P K where Note that K 2;ď2 is a subset of L p1q;R 2 and and U 2;ď2 is equal to U d pRq, both defined in Proposition 3.12 . The (unique) intersection point of the orbit O 2 pRq¨c ď2 with K 2;ď2 is given byρ 2 pc ď2 q¨c ď2 . Since the action is free on g ď2 ppR 2 qq (Corollary 3.13 ), the mapρ 2 defines a moving frame with cross-section K. This immediately implies that the coordinates ofρ 2 pc ď2 q¨c ď2 are invariants for the action of O 2 pRq on g ď2 ppR 2 qq 8 : and |c 12 | " |c 12 |.
For any path Z in R 2 , let c ď2 pZq denote the element of g ď2 ppR 2 qq given by proj ď2 plogpIISpZqqq. Then we can define the "invariantized" path Y :"ρ 2 pc ď2 pZqq¨Z. The above statement implies that for any two paths Z, Z 1 , we have that c ď2 pY q " c ď2 pY 1 q if and only if there exists some g P O 2 pRq such that g¨c ď2 pZq " c ď2 pg¨Zq " c ď2 pZ 1 q.
In particular, since the log map is an equivariant bijection, the same holds true for the IIS of a path under the projection proj ď2 .
Given a path Z starting at the origin, the values of c 1 pZq, c 2 pZq correspond to x and y values of Zp1q. Similarly the value of c 12 pZq corresponds to the so-called Lévy area traced by Z (see [ DR19 , Section 3.2] in the context of classical invariant theory). Thus the moving frame map applied to such a path Z, rotates the end point Zp1q to the y-axis (and reflects about the y-axis if the Lévy area is negative).
The resulting invariants on g ď2 ppR 2 qq are perhaps unsurprising, but the above method also yields O 2 pRqinvariants on g ďn ppR 2 qq for an arbitrary truncation order n, as we now show.
Then the resulting coordinate functions of ρ 2 pc ďn q¨c ďn P g ďn ppR 2 qq are O 2 pRq invariants for the action on g ďn ppR 2 qq (see Section 5 for a more detailed investigation of these invariants), and hence O 2 pRq invariants for paths in R 2 . Furthermore, for any truncation order n and paths Z, Z 1 P R 2 , we have that c ďn pY q " c ďn pY 1 q if and only if there exists some element of O 2 pRq such that g¨c ďn pZq " c ďn pZ 1 q. The following is then true by induction and the fact that the log map is an equivariant bijection.
Then IISpY q " IISpY 1 q if and only if there exists g P O 2 pRq such that IISpg¨Zq " IISpZ 1 q.
Therefore two paths, starting at the origin are equivalent up to tree-like extensions and action of O 2 pRq if and only if IISpY q " IISpY 1 q. In this sense, the moving frame mapρ 2 yields a method to invariantize a path Z. In the following section, we show that this construction extends to paths in R d .

4.2.
Moving Frame on g ďn ppR d qq. As for O 2 pRq on R 2 , the action of O d pRq on paths in R d induces an action on its (truncated) signature that coincides with the diagonal action on the ambient space T ďn pR d q. The induced action on the log-signature coincides with this diagonal action as well, when considering g ďn ppR d qq as a subspace of T ďn pR d q.
Let c ďn be an element of g ďn ppR d qq with coordinates given by c i1i2¨¨¨im for m ď n. We define the following submanifold of g ďn ppR d qq: Let proj ď2 : g ďn ppR d qq Ñ g ď2 ppR d qq be the projection onto the first two levels (Section 2.2 ). The projection of this submanifold onto g ď2 ppR d qq, proj ď2 pK d;ďn q, is equal (up to the identification g ď2 ppR d qq -..) to the real, positive points of L Similarly we can define the analogue to U d pCq in ( 3.7 ). Consider the rational functions on g ďn ppR d qq given by F i pc ďn q :" f i pv, M q| vj "cj m k "c k for 1 ď i ď d where f i pv, M q is given in Proposition 3.6 . By Proposition 3.12 , the functions F i are rational functions on g ď2 ppR d qq with real coefficients. Then the following is a Zariski-open subset of g ďn ppR d qq, where proj ď2 pU d;ďn q " U d pCq if we identify c ď2 with pv, M q as above. In particular, both U d;ďn and K d;ďn are completely characterized by proj ď2 pc ďn q, i.e. U d;ďn " proj´1 ďnÑď2`p roj ď2 pU d;ďn q˘Ă g ďn ppR d qq K d;ďn " proj´1 ďnÑď2`p roj ď2 pK d;ďn q˘Ă g ďn ppR d qq, with proj ďnÑď2 denoting the canonical projection from g ďn ppR d qq onto g ď2 ppR d qq.
We now show that on the subset U d;ďn Ă g ďn ppR d qq the submanifold K d;ďn is a cross-section, which induces a moving frame. Proof. First, we recall that, by definition, O d pRq¨c ď2 and K d;ď2 intersect transversally if and only if, at every point q in the intersection, the tangent spaces T q pO d pRq¨c ď2 q and T q K d;ď2 generate the whole ambient space g ď2 ppR d qq, that is T q pO d pRq¨c ď2 q`T q K d;ď2 " g ď2 ppR d qq.
Now, at a point q " A¨c ď2 " pAv, AM A J q in the orbit, the tangent space has the form T q´Od pRq¨c ď2¯" tpHAv, rH, AM A J sq : H P so d pRqu. (4.2) Indeed, recall that for a manifold M , its tangent space at a point q is the linear space T q M :" tγ 1 p0q : γ curve s.t. γp0q " qu. A curve γ on O d pRq¨c ď2 such that γp0q " q has the form γptq " pLptqAq¨c ď2 for some curve t Þ Ñ Lptq in O d pRq such that Lp0q " I. Hence, The tangent space to the cross section is We note that where the second equality since the action of O d pRq is free on U d;ď2 by Corollary 3.13 . Thus we have that dim T q K d;ď2`d im T q´Od pRq¨c ď2¯" dim g ď2 ppR d qq. Therefore, we only need to show that T q K d;ď2 XT q´Od pRqc ď2¯" t0u for all q P K d;ď2 X pO d pRq¨c ď2 q.
Let pΓ i,j : 1 ď i ă j ď dq be the standard basis of so d pRq, that is, pΓ i,j q k,l " δ i,k δ j,l´δj,k δ i,l . It is not hard to show that the commutation relations hold for all 1 ď k ă d and 1 ď i ă j ď d. By eq. ( 4.2 ), a generic element p P T q´Od pRq¨c ď2¯h as the form But since q " pAv, AM A J q P K d;ď2 , with α ą 0, and β k ą 0 for all k P t1, . . . , d´1u. If p also belongs to T q K d;ď2 , then we have in particular that for some α 1 P R, thus h i,d " 0 for all i P t1, . . . , d´1u. Now we show that h i,j " 0 for all 1 ď i ă j ď d´1 by induction on r -d´1´j. By eq. ( 4.3 ), we see that . . , d´2u. Therefore, h i,d´1 " 0 for all i P t1, . . . , d´2u, and the claim is proven when r " 0. Suppose it is true for all r 1 ă r. Then rH, AM A J s i,d´1´r " h i,d´1´r β d´1´r " 0 for i P t1, . . . , d´2´ru, hence h i,d´1´r " 0 for all i P t1, . . . , d´2´ru. Finally, we have H " 0 thus p " pHAv, rH, AM A J sq " 0.
We have shown that if q P O d pRq¨c ď2 X K d;ď2 then dim T q´Od pRq¨c ď2¯`d im T q K d;ď2 " dim g ď2 ppR d qq and T q´Od pRq¨c ď2¯X T q K d;ď2 is trivial. It follows that if q P pO d pRq¨c ď2 q X K d;ď2 , then g ď2 ppR d qq " T q´Od pRq¨c ď2¯' T q K d;ď2 , and in particular O d pRq¨c ď2 and K d;ď2 intersect transversally. Proof. We first claim that K d;ďn intersects each orbit in U d;ďn at a unique point. Denote the linear span of Note that the action on proj ď2 g ďn ppR d qq " g ď2 ppR d qq is isomorphic to the action on R d ' so d pRq given in ( 2.7 ). Thus for any c ďn P U d;ďn , by Proposition 3.12 and the diagonality of the action (see Section 2.3 ), there exists an element of g P O d pRq such that g¨c ďn "c ďn P K.
Consider the subgroup W R Ă O d pRq of diagonal matrices w with diagonal entries w jj P t´1, 1u, 1 ď j ď d. By Proposition 3.5 any element of W R sends a point in K to K. For anyc ďn P K, the action of W R on the coordinates proj ď2 pc ďn q " c ď2 is given by the following (see ( 3.8 )): The element w P W R such that w jj "´1 for 1 ď j ď d changes only the sign on c d . The element w P W R where w jj "´1 for 1 ď j ď i and w jj " 1 for i ă j ď d changes only the sign of c ipi`1q . Thus there exists g P W R such that g¨c ďn P K d;ďn , implying K d;ďn intersects each orbit in U d;ďn . Now suppose that for some c ďn P K d;ďn , g P O d pRq we have g¨c ďn P K d;ďn . We show that this implies g " id. Since the action of O d pRq on T 1 pR d q is isomorphic to the canonical action on R d , g P O d´1 d pRq (recall the notation after ( 3.4 )). By Proposition 3.4 , the action of O d´1 d pRq on the coordinates c 1d , c 2,d , . . . , c pd´1qd of c ďn is isomorphic to the canonical action on R d´1 . Thus we deduce that g must be in O d´2 d pRq. Iterating, we obtain that g must be the identity, as claimed, implying that K d;ďn intersects each orbit in U d;ďn exactly once.
We now show that the intersection with each orbit is transverse. By Corollary 3.13 the action is free on U d;ď2 , and thus on U d;ďn . Since the action is free on U d;ďn , each orbit O d pRq¨c ďn is smooth and of dimension npn´1q{2 (see Proposition 2.16 ). Let c ďn be a point in K d;ďn . Since Since O d pRq acts diagonally we have that proj ď2 pT cďn K d;ďn`Tcďn pO d pRq¨c ďn qq " proj ď2 pT cďn K d;ďn q`proj ď2 pT cďn pO d pRq¨c ďn qq " T proj ď2 pcďnq proj ď2 pK d;ďn q`T proj ď2 pcďnq`Od pRq¨proj ď2 pc ďn q˘, where V`W denotes the span of two subspaces V, W . Then by Lemma 4.2 proj ď2 pT cďn K d;ďn`Tcďn pO d pRq¨c ďn qq " g ď2 ppR d qq.
Since for any vector v P T proj ď2 pcďnq proj ď2 pK d;ďn q, xvy ' g ď3 ppR d qq is a subspace of T cďn K d;ďn , we have that T cďn K d;ďn`Tcďn pO d pRq¨c ďn q " g ďn ppR d qq. Thus to find a unique representative of the orbit of c ďn P U d;ďn we can "invariantize" c ďn by computing ρ d pc ďn q¨c ďn , and the smooth functions defining the non-zero coordinates of K d;ďn X O d pRq¨c ďn are invariant functions which characterize the orbit. Note that the cross-section K d;ďn and the moving frame only depend on the g ď2 ppR d qq coordinates. In particular we have that for any path Z such that c ďn pZq " proj ďn plogpIISpZqqq P U d n ρ d pc ďn pZqq " ρ d pproj ď2 pc ďn pZqqq ":ρ d pc ď2 pZqq which implies that the "invariantization" of a path Y :"ρ d pc ď2 pZqq¨Z is well-defined. This is due to the diagonal nature of the action of O d pRq on g ďn ppR d qq, and the fact that dimpO d pRqq ă dimpg ď2 ppR d qqq. Since the action of the coordinates on g ď2 ppR d qq is not affected by the higher level coordinates, we can define a cross-section on g ď2 ppR d qq that extends naturally to g ďn ppR d qq. For higher-dimensional groups one may have to consider a cross-section on g ď3 ppR d qq or higher.
As a consequence, the infinite log signature (and thus the iterated-integrals signature) of a path Z under the action of O d pRq is characterized by its value on the cross-section. In fact, there is even the following conjecture [ DR19 , Conjecture 7.2] that polynomial invariants seperate orbits of paths up to tree like equivalence for any subgroup of SL d pRq.
Conjecture 4.6. (Diehl-Reizenstein) Let Z, Z 1 : r0, T s Ñ R d be two curves such that xIISpZq, ϕy " xISSpZ 1 q, ϕy for any ϕ P T pR d q such thatφ A J pϕq " ϕ for any A P G. Then, there is A P G and a curveZ which is tree-like equivalent to Z such that AZ " Z 1 .
While a proof of this conjecture for any compact group G is finished and part of work in progress by J.D., Terry Lyons, Hao Ni and R.P., we are here interested in a 'constructive' proof which leads to an algorithm for the computation of a minimal set of polynomial seperating orbits. The following definition and proposition now provide a sufficient condition for a moving frame to lead to a fundamental set of invariants consisting only of polynomial invariants.
Definition 4.7. A moving frame : U Ñ G for the action of G on U , where U is a non-empty, G-invariant, semialgebraic subset of g ďn ppR d qq, is called almost-polynomial, if there are maps λ : U Ñ GL d pRq and κ : g ďn ppR d qq Ñ GL d pRq, where λ is G-invariant, such that λpc ďn q is diagonal for all c ďn P U , such that λ ii ij P Rrg ďn ppR d qqs for all i, j " 1, . . . , d and λpc ďn q " κ`λpc ďn q pc ďn q¨c ďn˘( 4.4) for all c ďn P U .
Remark 4.8. Equation ( 4.4 ) may seem odd as an asumption at first sight, however it is necessary for the coordinates of λpc ďn q pc ďn q¨c ďn to form a fundamental set of invariants: Since we assume λ to be an invariant function it must functionally depend on any fundamental set of invariants.
Example 4.9. Looking at the example ρ 2 , we introduce λ 2 pc ďn q " As we see in Section 5 , λ 2 is invariant under O 2 pRq. Then, λ 2 pc ďn qρ 2 pc ďn q " " c 12 c 2´c12 c 1 c 1 c 2  , whose entries are polynomial functions. In particular, the non-zero coordinates of λ 2 pc ďn qρ 2 pc ďn q¨c ďn are given byĉ 2 :" c 2 1`c 2 2 ,ĉ 12 :" c 2 12 pc 2 1`c 2 2 q We finally obtain λ 2 pc ďn q " κ 2 pλ 2 pc ďn q pc ďn q¨c ďn q via κ 2 pĉ ďn q " showing that ρ 2 is an almost-polynomial moving frame. Proof. Obviously c ďn Þ Ñ λpc ďn q pc ďn q¨c ďn is a polynomial map on U since λ ii µ ij is polynomial. The components of pc ďn q¨c ďn form a fundamental set of invariants since is a moving frame which implies that Ipc ďn q " Ip pc ďn q¨c ďn q for any invariant function I : U Ñ R. Since λ is G invariant by assumption, we have that the components of λpc ďn q pc ďn q¨c ďn are invariants. Furthermore pc ďn q¨c ďn " κpλpc ďn q pc ďn q¨c ďn q´1λpc ďn q pc ďn q¨c ďn , implies that the non-zero components of λpc ďn q pc ďn q¨c ďn form a fundamental set of invariants, too.
We formulate the following conjecture for the moving frame defined in the previous section.
Conjecture 4.11. For any d ě 2, the moving frame ρ d : We will see in the next section that this conjecture at least also holds true for d " 3, however, the conjecture remains open for d ą 3. This in particular means that we have our 'constructive' proof for Conjecture 4.6 restricted to paths Z such that c ďn pZq P U d;ďn in the special case of O d pRq for d " 2, 3. We hope to extend this result to all dimensions, all paths and to further compact groups in future work.

5.
Invariants of planar and spatial curves 5.1. Planar curves. In Section 4.1 we detailed a moving frame construction for g ďn ppR 2 qq under O 2 pRq for any truncation order n. In particular on the subset U 2;ďn " c ďn P g ďn ppR 2 qq | pc 1 , c 2 q ‰ p0, 0q, c 12 ‰ 0 ( , the map ρ 2 : U 2;ďn Ñ O 2 pRq defined by ρ 2 pc ďn q " µ 2 pc ďn qν 2 pc ďn q for c ďn P g ďn ppR 2 qq, where is a moving frame map, bringing any element of g ďn ppR 2 qq to the intersection of its orbit with the cross-section (see Figure 2 ) K 2;ďn " c ďn P g ďn ppR 2 qq | c 1 " 0, c 2 , c 12 ą 0 ( . Figure 2. Applying the moving-frame map for planar curves to two paths X and X 1 lying on the same O 2 pRq orbit Any path Z in R 2 defines an element c ďn pZq " proj ďn plogpIISpZqqq P g ďn ppR 2 qq. Since ρ 2 pc ďn q "ρ 2 pc ď2 q, we can define the invariantization of Z with respect to O 2 pRq as Y :"ρ 2 pc ď2 pZqq¨Z. The coordinates of logpIISpY qq as functions of the coordinates of logpIISpZqq are invariant functions for paths under O 2 pRq. A (Lyndon) basis for g ď4 ppR 2 qq corresponds to the coordinates (see Example 2.1 ) c 4 " pc 1 , c 2 , c 12 , c 112 , c 122 , c 1112 , c 1122 , c 1222 q.
Thus we have λ 3 pc ďn q " κ 3 pλ 3 pc ďn qρ 3 pc ďn q¨c ďn q with κ 3 pĉ ďn q as the diagonal matrix with entries aĉ 3ĉ12 , aĉ 23 , aĉ 3 . Hence, ρ 3 is an almost-polynomial moving frame, leading to a fundamental set of polynomial invariants.  In this sense, this two step process is similar to the iterative process outlined in the proof of Proposition 3.6 of bringing an element of g ď2 ppR d qq to successively smaller linear spaces. The transformation A 1 brings c ď2 pZq to L p1q 3 pRq Ă L p1q 3 , then finally to K 3;ď2 Ĺ L p2q 3 pRq Ă L p2q 3 by a transformation A 2 . In principle, given a procedure to rotate an element of R d to a particular axis, this iterative process is quite easy to perform to bring any c ď2 pZq for any path Z to K d;ďn , and hence invariantize any path.
Alternatively one can directly use the moving frame map in ( 5.1 ); note that this is equivalent to the single action by the matrixρ 3 pc ď2 pZqq " A 2 A 1 "

Discussion and open problems
We conclude with a discussion of some interesting questions arising from this work. We presented a method to construct O d pRq invariants for a path Z from the coordinates of the log signature (of the iterated-integrals signature) in a way that completely characterizes the orbit of proj n plogpIISpZqq (or proj n pIISpZqq) under O d pRq. This procedure also furnishes a quick method to compare equivalence classes of paths under O d pRq without computing the full set of invariants (see Example 5.3 ).
In particular Theorem 4.5 is similar in spirit to [ DR19 , Conjecture 7.2], where the authors characterize all linear SO d pRq-invariants in the coordinates of IISpZq and ask if these determine a path up to SO d pRq and tree-like extensions. The invariant sets we construct are smooth functions in the coordinates of logpIISpZqq, though in many cases we can, by inspection, find an equivalently generating polynomial set (see Section 5 ). Polynomials in coordinates of logpIISpZqq correspond to polynomial invariants in the coordinates of IISpZq, which yield linear O d pRq-invariants by the shuffle relations. Thus the conjecture remains open, and more broadly the connection between the two sets of invariants should be explored.
In Section 3 , we investigate sets of separating sets of rational and polynomial invariants for the action of O d pRq on g ď2 ppR d qq. An open question is whether the polynomial invariants we construct, generate the ring of polynomial invariants for this action. In even more generality questions remain about the relationship between the polynomial invariants we construct and the ring of polynomial invariants for the action of O d pRq on g ďn ppR d qq.
Additionally we only consider O d pRq-invariants (and to a lesser extend SO d pRq) in this work. The dimension of O d pRq implies that to construct a cross-section for the action, one only has to consider the action on g ď2 ppR d qq.
For larger groups like GL d pRq one may have to construct a cross-section using coordinates on g ď3 ppR d qq.
The cross section K in Section 4.2 can also be used as a starting point for groups containing O d pRq, since any element of g ď2 ppR d qq can be brought to K by an element of O d pRq. For instance if one considers scaling transformations in addition to orthogonal transformations, changing the conditions of c d , c ipi`1q ą 0 on K to c d " c ipi`1q " 0, for 1 ď i ă d, likely yields a cross-section.
In Section 4.3 we introduce Conjecture 4.11 , which we prove holds for d " 2 in Theorem 5.2 and for d " 3 in Section 5.2 . In general the invariants produced by the moving frame method are only guaranteed to be smooth, and hence proving this conjecture for d ą 3 is of interest. In particular, polynomial invariant functions of iterated-integral values are desired because they can be expressed as elements of T pR d q, they provide the simplest and most structured (graded) way of looking at invariants, with an immediate connection to polynomial algebraic geometry, and it is widely assumed and partially proven that they are sufficient for characterization of orbits, see the discussion of the Diehl-Reizenstein conjecture in Section 4.3 .
As mentioned in the introduction, there are many applications of the iterated-integrals signature of paths where finding O d pRq-invariant features could be advantageous. It would be interesting to see if the sets of integral invariants constructed, or "invariantization" procedure outlined can be useful for such applications.