Conformal geometric algebra (CGA) is a framework that allows the representation of objects, such as points, planes and spheres, deformations, translations, rotations dilations uniform vectors, called multivectors. In this work, we demonstrate merits multivector usage with novel, integrated rigged character simulation based on CGA. framework, for first time, one may perform real-time cuts tears ...

This paper describes a Kalman filter for the real-time estimation of a rigid body orientation from measurements of acceleration, angular velocity and magnetic field strength. A quaternion representation of the orientation is computationally effective and avoids problems with singularities. The nonlinear relationship between estimated orientation and expected measurement prevent the usage of a c...

In SCARA robots, which are often used in industrial applications, all joint axes are parallel, covering three degrees of freedom in translation and one degree of freedom in rotation. Therefore, conventional approaches for the handeye calibration of articulated robots cannot be used for SCARA robots. In this paper, we present a new linear method that is based on dual quaternions and extends the ...

We express the probability distribution of the solution of a random (standard Gaussian) instance of a convex cone program in terms of the intrinsic volumes and curvature measures of the reference cone. We then compute the intrinsic volumes of the cone of positive semidefinite matrices over the real numbers, over the complex numbers, and over the quaternions in terms of integrals related to Meht...

Recent work investigates the combination of Computer Algebra Systems and Automated Theorem Provers We elaborate a concrete example the generic construction of complex numbers quaternions Cayley numbers on top of real numbers The implementation is done by a functor in the programming language SML which can be instantiated to perform the algebraic computations Another instantiation of the same fu...

A software system, called RTSIMU, was developed for analyzing worker movements in the textile industry using Inertial Measurement Units (IMUs). RTSIMU converts raw positioning data from IMUs into quaternions, which represent orientation relative to Earth, and further translates them angular values of movement. To assess relevant simulate various working (such as abduction, flexion, rotation, el...

In this paper we will present a new construction of any real geometric (Clifford) algebra $${\mathbb {G}}^{(p,q)}$$ with signature (p, q) where $$p+q=n$$ by defining product on the vector space {R}}^{(2^n)}$$ in manner similar to Gauss’ ordered pair complex numbers ( {C}}$$ ) and Hamilton’s quadruple quaternions {H}}$$ ). We motivate definition generalizing tuple multiplication each . Similar w...

In the problem of blind source separation from instantaneous linear mixtures, a unitary transformation remains unknown after second-order spatial whitening. We present a novel approach for the identification of the orthogonal matrix in the real-valued three-signal scenario which, in contrast to existing procedures, operates in a single closed-form step, with no iterations. The new approach is b...

Supersymmetry is deeply related to division algebras. For example, nonabelian Yang–Mills fields minimally coupled to massless spinors are supersymmetric if and only if the dimension of spacetime is 3, 4, 6 or 10. The same is true for the Green–Schwarz superstring. In both cases, supersymmetry relies on the vanishing of a certain trilinear expression involving a spinor field. The reason for this...

A Sylvester-Gallai (SG) configuration is a finite set S of points such that the line through any two points in S contains a third point of S. According to the Sylvester-Gallai Theorem, an SG configuration in real projective space must be collinear. A problem of Serre (1966) asks whether an SG configuration in a complex projective space must be coplanar. This was proved by Kelly (1986) using a d...