Related papers: Introducing the Classical Method of Moving Frames
Invertible image representation methods (transforms) are routinely employed as low-level image processing operations based on which feature extraction and recognition algorithms are developed. Most transforms in current use (e.g. Fourier,…
We show that if a Lagrangian is invariant under a transformation (with the invariance defined in the standard manner), then the equations of motion obtained from it maintain their form under the transformation. We also show that the…
Interferometric closure invariants encode calibration-independent details of an object's morphology. Excepting simple cases, a direct backward transformation from closure invariants to morphologies is not well established. We demonstrate…
Planning trajectories for nonholonomic systems is difficult and computationally expensive. When facing unexpected events, it may therefore be preferable to deform in some way the initially planned trajectory rather than to re-plan entirely…
Affine transformations in Euclidean space generates a correspondence between integrable systems on cotangent bundles to the sphere, ellipsoid and hyperboloid embedded in $R^n$. Using this correspondence and the suitable coupling constant…
We compare the Serret-Frenet frame with a {\em relatively parallel adapted frame} (RPAF) introduced by Bishop to parametrize $W^{2,2}$-curves. Next, we derive the geometric invariants, curvature and torsion, with the RPAF associated to the…
Dynamic patterns are characterized by complex spatial and motion patterns. Understanding dynamic patterns requires a disentangled representational model that separates the factorial components. A commonly used model for dynamic patterns is…
Around 1923, Elie Cartan introduced affine connections on manifolds and definedthe main related concepts: torsion, curvature, holonomy groups. He discussed applications of these concepts in Classical and Relativistic Mechanics; in…
We examine the relationships between the differential invariants of objects and of their images under a surjective map. We analyze both the case when the underlying transformation group is projectable and hence induces an action on the…
Generalizing a definition by Kalra \cite{Kalra}, the purpose of this paper is to analyze cyclic frames in finite-dimensional Hilbert spaces. Cyclic frames form a subclass of the dynamical frames introduced and analyzed in detail by Aldroubi…
We present a method for constructing all bounded rational motions that frame a space curve $\mathbf{r}(t)$. This means that the motion guides an orthogonal frame along the curve such that one frame axis is in direction of the curve tangent.…
Recently the general linear transformation for point rotation coordinate frames was considered. A distinguishing feature of the frame, in contrast to the Cartesian one, is the existence of the rotation axis at every point. The frame…
Within the frames of the analytical mechanics the method of the description of dynamics of nonequilibrium systems of potentially interacting elements is develops. The method is based on an opportunity of representation of nonequilibrium…
This paper investigates scalable frame in ${\mathbb R}^n$. We define the reduced diagram matrix of a frame and use it to classify scalability of the frame under some conditions. We give a new approach to the scaling problem by breaking the…
The concept of dynamical movement primitives (DMPs) has become popular for modeling of motion, commonly applied to robots. This paper presents a framework that allows a robot operator to adjust DMPs in an intuitive way. Given a generated…
This article proposes a novel approach for determining exact solutions to nonlinear ordinary differential equations. The recommended iterative method provides the solution via a rapidly converging series that readily approaches a closed…
Frames for $\R^n$ can be thought of as redundant or linearly dependent coordinate systems, and have important applications in such areas as signal processing, data compression, and sampling theory. The word "frame" has a different meaning…
The f-invariant is a higher version of the e-invariant that takes values in the divided congruences between modular forms; it can be formulated as an elliptic genus of manifolds with corners of codimension two. In this thesis, we develop a…
We introduce a notion of rough paths on embedded submanifolds and demonstrate that this class of rough paths is natural. On the way we develop a notion of rough integration and an efficient and intrinsic theory of rough differential…
Forecasting vehicular motions in autonomous driving requires a deep understanding of agent interactions and the preservation of motion equivariance under Euclidean geometric transformations. Traditional models often lack the sophistication…