Rotation minimizing frames and spherical curves in simply isotropic and pseudo-isotropic 3-spaces

In this work, we are interested in the differential geometry of curves in the simply isotropic and pseudo-isotropic 3-spaces, which are examples of Cayley-Klein geometries whose absolute figure is given by a plane at infinity and a degenerate quadric. Motivated by the success of rotation minimizing (RM) frames in Euclidean and Lorentzian geometries, here we show how to build RM frames in isotropic geometries and apply them in the study of isotropic spherical curves. Indeed, through a convenient manipulation of osculating spheres described in terms of RM frames, we show that it is possible to characterize spherical curves via a linear equation involving the curvatures that dictate the RM frame motion. For the case of pseudo-isotropic space, we also discuss on the distinct choices for the absolute figure in the framework of a Cayley-Klein geometry and prove that they are all equivalent approaches through the use of Lorentz numbers (a complex-like system where the square of the imaginary unit is $+1$). Finally, we also show the possibility of obtaining an isotropic RM frame by rotation of the Frenet frame through the use of Galilean trigonometric functions and dual numbers (a complex-like system where the square of the imaginary unit vanishes).

Affine Reconstruction Based on Parallel Plane and Infinity Point

Affine reconstruction is to restore the affine shape of the object. Generally, there are two ways of achieving, one is to determine the plane at infinity, another is to determine the plane homography. Using the homography which had determined the plane at infinity achieve affine reconstruction. In this paper, firstly give out the homography of infinity plane and the algorithm of affine reconstruction, then proved: if the scene contains a set of parallel planes and a infinity point, the homography of infinity plane can be obtained and affine reconstruction can be linearly got in the scene. Computer simulation and real experiments show that the linear affine reconstruction algorithm is correct, and the approach has a good precision.

Affine cubic functions. IV. Functions on C 3 , nonsingular at infinity

The object of this paper is to classify cubic functions f on C 3 according to their singularities. A level surface of such a function extends to a cubic surface in projective 3-space. The intersections S^,Tœ of S and its Hessian quartic T with the plane at infinity are the same for all levels. We assume throughout that is a nonsingular cubic curve. In §3 we show how the equisingularity class of Tœ determines the number and multiplicities of critical points of f .In § 2 we investigate n T^, and show that the equisingularity class of the pair (S^Tœ) determines that of f. Next we study the case when some point of has polar quadric a plane-pair; complete enumerations are given in §5 for the case when contains a line, and in §6 for when it contains an Eckardt point of S. In the final section we give a detailed analysis of cases when f has just two critical values, and show how to obtain a complete list of types of functions f.

Stress Concentration Due to a Hemispherical Pit at a Free Surface

This paper contains a solution in series form for the stresses and displacements around a hemispherical pit at a free surface of an elastic body. The problem is idealized by considering a semi-infinite medium which otherwise is bounded by a plane. At infinity the body is assumed to be in a state of plane hydrostatic tension perpendicular to the axis of symmetry of the pit. The present method of solution may be generalized to loadings which are not rotationally symmetric. Numerical results are given for the variation along the axis of symmetry of the normal stress which is parallel to the tractions at infinity; these results are compared with the known corresponding numerical values appropriate to the two-dimensional analog of the present problem.