A construction is given for a planar rational Pythagorean hodograph spiral which interpolates any two-point G Hermite data that a spiral can match. When the curvature at one of the points is zero, the construction gives the unique interpolant that is the involute of a rational Pythagorean hodograph curve of the form cubic over linear. Otherwise, the spiral comprises an involute of a Tschirnhaus...

We construct explicit left invariant quaternionic contact structures on Lie groups with zero and non-zero torsion, and with non-vanishing quaternionic contact conformal curvature tensor, thus showing the existence of quaternionic contact manifolds not locally quaternionic contact conformal to the quaternionic sphere. We present a left invariant quaternionic contact structure on a seven dimensio...

We develop quaternionic analysis using as a guiding principle representation theory of various real forms of the conformal group. We first review the Cauchy-Fueter and Poisson formulas and explain their representation theoretic meaning. The requirement of unitarity of representations leads us to the extensions of these formulas in the Minkowski space, which can be viewed as another real form of...

Motivated by the analogies between the projective and the almost quaternionic geometries, we first study the generalized planar curves and mappings. We follow, recover, and extend the classical approach, see e.g. [9, 10]. Then we exploit the impact of the general results in the almost quaternionic geometry. In particular we show, that the natural class of H–planar curves coincides with the clas...

BGG–sequences offer a uniform construction for invariant differential operators for a large class of geometric structures called parabolic geometries. For locally flat geometries, the resulting sequences are complexes, but in general the compositions of the operators in such a sequence are nonzero. In this paper, we show that under appropriate torsion freeness and/or semi–flatness assumptions c...

Quaternionic linear algebra and plurisubharmonic functions of quaternionic variables. Abstract We remind known and establish new properties of the Dieudonné and Moore determinants of quaternionic matrices. Using these linear algebraic results we develop a basic theory of plurisubharmonic functions of quaternionic variables. The main point of this paper is that in quaternionic algebra and analys...

Distinguished with some basic theory and approaches of traditional tooth surface design, the spiral bevel gear with new tooth surface, namely spherical involute tooth surface, is investigated. Firstly, it improved its formational principle based on the spherical involute theory. As a key part, the parametric equation of the generating line was made a detailed derivation. To this end, fast and a...

This paper describes the geometric properties required to form the walls of a functional scroll compressor, and a practical design procedure that has been used at the University of Maryland to generate more general scroll geometries and model their performance. It presents several examples of this technique, showing how it can be used to meet practical design challenges, optimize operating para...

The notion of a quaternionic gerbe is presented as a new way of bundling algebraic structures over a four manifold. The structure groupoid of this fibration is described in some detail. The Euclidean conformal group RSO(4) appears naturally as a (non-commutative) monoidal structure on this groupoid. Using this monoidal structure we indicate the existence of a canonical quaternionic gerbe associ...