相关论文: Paravectors and the Geometry of 3D Euclidean Space
We discuss how transformations in a three dimensional euclidean space can be described in terms of the Clifford algebra $\mathcal{C}\ell_{3,3}$ of the quadratic space $\mathbb{R}^{3,3}$. We show that this algebra describes in a unified way…
Paravectors just like integers have a ring structure. By introducing an integrated product we get geometric properties which make paravectors similar to vectors. The concepts of parallelism, perpendicularity and the angle are conceptually…
Objects' rigid motions in 3D space are described by rotations and translations of a highly-correlated set of points, each with associated $x,y,z$ coordinates that real-valued networks consider as separate entities, losing information.…
We introduce a linear algebraic object called a bidiagonal triad. A bidiagonal triad is a modification of the previously studied and similarly defined concept of bidiagonal triple. A bidiagonal triad and a bidiagonal triple both consist of…
Principal angles are used to define an angle bivector of subspaces, which fully describes their relative inclination. Its exponential is related to the Clifford geometric product of blades, gives rotors connecting subspaces via minimal…
We introduce a linear algebraic object called a bidiagonal triple. A bidiagonal triple consists of three diagonalizable linear transformations on a finite-dimensional vector space, each of which acts in a bidiagonal fashion on the…
This paper revisits the little-known Gibbs-Rodrigues representation of rotations in a three-dimensional space and demonstrates a set of algorithms for handling it. In this representation the rotation is itself represented as a…
Geometric algebra is an optimal frame work for calculating with vectors. The geometric algebra of a space includes elements that represent all the its subspaces (lines, planes, volumes, ...). Conformal geometric algebra expands this…
In geometric algebra, the rotation of a vector is described using rotors. Rotors are phasors where the imaginary number has been replaced by a oriented plane element of unit area called a unit bivector. The algebra in three dimensional…
The algebra of fourvectors is described. The fourvectors are more appropriate than the Hamilton quaternions for its use in Physics and the sciences in general. The fourvectors embrace the 3D vectors in a natural form. It is shown the…
Complementary idempotent paravectors and their ordered compositions, are used to represent multivector basis elements of geometric Clifford algebra for 3D Euclidean space as the states of a geometric byte in a given frame of reference. Two…
This paper is a modern exposition of old ideas. The setting is a Euclidian space $E$ of dimension $n$ with associated vector space $V$ of dimension $n$. A (non-zero) sliding vector is a vector in $V$ that is free to move, but only within a…
Projective geometry provides the preferred framework for most implementations of Euclidean space in graphics applications. Translations and rotations are both linear transformations in projective geometry, which helps when it comes to…
We present algebraic projective geometry definitions of 3D rotations so as to bridge a small gap between the applications and the definitions of 3D rotations in homogeneous matrix form. A general homogeneous matrix formulation to 3D…
In this part of the series I show how five-tensors can be used for describing in a coordinate-independent way finite and infinitesimal Poincare transformations in flat space-time. As an illustration, I reformulate the classical mechanics of…
A set of basic vectors locally describing metric properties of an arbitrary 2-dimensional (2D) surface is used for construction of fundamental algebraic objects having nilpotent and idempotent properties. It is shown that all possible…
A space curve in a Euclidean 3-space $\mathbb E^3$ is called a rectifying curve if its position vector field always lies in its rectifying plane. This notion of rectifying curves was introduced by the author in [Amer. Math. Monthly {\bf…
In three-dimensional Euclidean geometry, the scalar product produces a number associated to two vectors, while the vector product computes a vector perpendicular to them. These are key tools of physics, chemistry and engineering and…
An introductory overview of vector spaces, algebras, and linear geometries over an arbitrary commutative field is given. Quotient spaces are emphasized and used in constructing the exterior and the symmetric algebras of a vector space.…
The mechanics of an oriented point (point with "spin") based on 3D and 4D Frenet equations is considered. In such mechanics there is an opportunity to describe formally any physical trajectory of a particle with own rotation. We use…