Good parametrisations of affine transformations are essential to interpolation, deformation, and analysis of shape, motion, and animation. It has been one of the central research topics in computer graphics. However, there is no single perfect method and each one has both advantages and disadvantages. In this paper, we propose a novel parametrisation of affine transformations, which is a generalisation to or an improvement of existing methods. Our method adds yet another choice to the existing toolbox and shows better performance in some applications. A C++ implementation is available to make our framework ready to use in various applications.
The group of rigid transformations @math can be parametrised with the dual quaternions, which is a generalisation of the parametrisation of @math by the quaternions. In @cite_12 , they used the dual quaternions to blend rigid transformations and their method works particularly well in skinning. However, for other applications we may be troubled by the complicated structure of the parameter space; the space of the unit dual quaternions is the semi-direct product of the group of the unit quaternions and @math .
@cite_12
2012294177 1487724026 2942026896 2208813460
This paper proposes a new class of unit quaternion curves in 3 . A general method is developed that transforms a curve in 3 (defined as a weighted sum of basis functions) into its unit quaternion analogue in 3 . Applying the method to well-known spline curves (such as B´ ezier, Hermite, and B-spline curves), we are able to construct various unit quaternion curves which share many important differential properties with their original curves. Many of our naive common beliefs in geometry break down even in the simple non-Euclidean space 3 or 3 . For example, the de Casteljau type construction of cubic B-spline quaternion curves does not preserve 2 -continuity [10]. Through the use of decomposition into simple primitive quaternion curves, our quaternion curves preserve most of the algebraic and differential properties of the original spline curves. Motivated by practical numerical issues in a number of modeling and simulation problems, we introduce the notion of a compatible dual complex to a primal triangulation, such that a simplicial mesh and its compatible dual complex (made out of convex cells) form what we call a primal-dual triangulation. Using algebraic and computational geometry results, we show that compatible dual complexes exist only for a particular type of triangulation known as weakly regular. We also demonstrate that the entire space of primal-dual triangulations, which extends the well known (weighted) Delaunay Voronoi duality, has a convenient, geometric parametrization. We finally discuss how this parametrization may play an important role in discrete optimization problems such as optimal mesh generation, as it allows us to easily explore the space of primal-dual structures along with some important subspaces. In this work, we move beyond the traditional complex-valued representations, introducing more expressive hypercomplex representations to model entities and relations for knowledge graph embeddings. More specifically, quaternion embeddings, hypercomplex-valued embeddings with three imaginary components, are utilized to represent entities. Relations are modelled as rotations in the quaternion space. The advantages of the proposed approach are: (1) Latent inter-dependencies (between all components) are aptly captured with Hamilton product, encouraging a more compact interaction between entities and relations; (2) Quaternions enable expressive rotation in four-dimensional space and have more degree of freedom than rotation in complex plane; (3) The proposed framework is a generalization of ComplEx on hypercomplex space while offering better geometrical interpretations, concurrently satisfying the key desiderata of relational representation learning (i.e., modeling symmetry, anti-symmetry and inversion). Experimental results demonstrate that our method achieves state-of-the-art performance on four well-established knowledge graph completion benchmarks. In this article we prove global rigidity results for hyperbolic actions of higher-rank lattices. Suppose @math is a lattice in semisimple Lie group, all of whose factors have rank @math or higher. Let @math be a smooth @math -action on a compact nilmanifold @math that lifts to an action on the universal cover. If the linear data @math of @math contains a hyperbolic element, then there is a continuous semiconjugacy intertwining the actions of @math and @math , on a finite-index subgroup of @math . If @math is a @math action and contains an Anosov element, then the semiconjugacy is a @math conjugacy. As a corollary, we obtain @math global rigidity for Anosov actions by cocompact lattices in semisimple Lie group with all factors rank @math or higher. We also obtain global rigidity of Anosov actions of @math on @math for @math and probability-preserving Anosov actions of arbitrary higher-rank lattices on nilmanifolds.