It is well known that Cayley’s ruled cubic surface carries a three-parameter family of twisted cubics sharing a common point, with the same tangent and the same osculating plane. We report on various results and open problems with respect to contact of higher order and dual contact of higher order for these curves.

We reprove in modern terms and extend to arbitrary dimension a classical result of Enrico Bompiani on algebraic surfaces X ⊂ P having very degenerate osculating spaces. Let X ⊂ P be an integral nondegenerate projective variety of dimension n defined over the field C. In order to introduce the osculating space of order m to a smooth point p ∈ X, fix a lifting U ⊆ C −→ C \ {0} t 7−→ p(t) of a loc...

Material particles interact with the working moving surfaces of machines in various technological processes. Mechanics considers a technique to describe movement point and decompose speed acceleration into single unit vectors accompanying trajectory trihedron for simple movement. The shape spatial curve uniquely sets Frenet trihedral as solid body. This paper has considered relative material pa...

Developable surfaces are of considerable importance to many industry applications, e.g. sheet metal forming processes. The objective of this paper is to provide algorithms on the approximation of developable surfaces with pieces of right circular cones. Special emphasis is devoted to practical choices of free parameters and to error estimation. Furthermore, a new algorithm for the approximation...

This paper presents some basics for the analysis of point clouds using the geometrically intuitive mathematical framework of conformal geometric algebra. In this framework it is easy to compute with osculating circles for the description of local curvature. Also methods for the fitting of spheres as well as bounding spheres are presented. In a nutshell, this paper provides a starting point for ...

The construction of planar conchoids can be carried over to the Euclidean unit sphere. We study the case of conchoids of (spherical) lines and circles. Some elementary constructions of tangents and osculating circles are stil valid on the sphere. Further, we aim at the illustration and a precise description of the algebraic properties of the principal views of spherical conchoids, i.e., the con...

The Euclidean Algorithm is the often forgotten key to rational approximation techniques, including Taylor, Lagrange, Hermite, osculating, cubic spline, Chebyshev, Padé and other interpolation schemes. A unified view of these various interpolation techniques is eloquently expressed in terms of the concept of the spectral basis of a factor ring of polynomials. When these methods are applied to th...