Generalized Helicoidal Surfaces in Euclidean 5-space

In this paper, we study generalized helicoidal surfaces in Euclidean 5-space. We obtain the necessary and sufficient conditions for generalized helicoidal surfaces in Euclidean 5-space to be minimal, flat or of zero normal curvature tensor, which are ordinary differential equations. We solve those equations and discuss the completeness of the surfaces.

Analytical solution to contact characteristics for a variable hyperbolic circular-arc-tooth-trace cylindrical gear

The variable hyperbolic circular-arc-tooth-trace (VH-CATT) cylindrical gear is a new type of gear. In order to research the contact characteristics of the VH-CATT cylindrical gear, tooth surface mathematical models of this kind of gear pair are derived based on the forming principle of the rotating double-edged cutting method with great cutter head in this regard. Then, according to the differential geometry theory and Hertz theory, the formula of the induced normal curvature and equation of the contact ellipse are derived based on the condition of continuous tangency of two meshing surfaces, which proves that the contact form of VH-CATT cylindrical gear is point contact. The present work establishes analytical solutions to research the effect of different parameters for the contact characteristic of the VH-CATT cylindrical gear by incorporating elastic deformation on the tooth surface, which have shown that the module, tooth number and cutter radius have a crucial effect on the induced normal curvature and contact ellipse of the VH-CATT cylindrical gear in the direction of tooth trace and tooth profile. Moreover, a theoretical analysis solution, a finite element analysis and the gear tooth contact pattern are carried out to verify the correctness of the computerized model and to investigate the contact type of the gear; it is verified that the contact form on the concave surface of the driving VH-CATT cylindrical gear rotates from dedendum at the heel to the addendum at toe and is an instantaneous oblique ellipse due to elastic deformation of the contact tooth profile, and the connecting line of the ellipse center is the contact trace path. It is indicated that the research results are beneficial for research on tooth break reduction, pitting, wear resistance and fatigue life improvement of the VH-CATT cylindrical gear. The results also have a certain reference value for development of the VH-CATT cylindrical gear.

Propagation of curved folding: the folded annulus with multiple creases exists

In this paper we consider developable surfaces which are isometric to planar domains and which are piecewise differentiable, exhibiting folds along curves. The paper revolves around the longstanding problem of existence of the so-called folded annulus with multiple creases, which we partially settle by building upon a deeper understanding of how a curved fold propagates to additional prescribed foldlines. After recalling some crucial properties of developables, we describe the local behaviour of curved folding employing normal curvature and relative torsion as parameters and then compute the very general relation between such geometric descriptors at consecutive folds, obtaining novel formulae enjoying a nice degree of symmetry. We make use of these formulae to prove that any proper fold can be propagated to an arbitrary finite number of rescaled copies of the first foldline and to give reasons why problems involving infinitely many foldlines are harder to solve.

On k-type spacelike slant helices lying on lightlike surfaces

In this paper, we define k-type spacelike slant helices lying on a lightlike surface in Minkowski space E31 according to their Darboux frame for k ? {0,1,2}. We obtain the necessary and the sufficient conditions for spacelike curves with non-null and null principal normal lying on lightlike surface to be the k-type spacelike slant helices in terms of their geodesic curvature, normal curvature and geodesic torsion. Additionally, we determine their axes and show that the Darboux frame of a spacelike curve lying on a lightlike surface coincides with its Bishop frame if and only if it has zero geodesic torsion. Finally, we give some examples.

Gauss Map Based Curved Origami Discretization

This paper addresses the problem of discretizing the curved developable surfaces that are satisfying the equivalent surface curvature change discretizations. Solving basic folding units occurs in such tasks as simulating the behavior of Gauss mapping. The Gauss spherical curves of different developable surfaces are setup under the Gauss map. Gauss map is utilized to investigate the normal curvature change of the curved surface. In this way, spatial curved surfaces are mapped to spherical curves. Each point on the spherical curve represents a normal direction of a ruling line on the curved surface. This leads to the curvature discretization of curved surface being transferred to the normal direction discretization of spherical curves. These developable curved surfaces are then discretized into planar patches to acquire the geometric properties of curved folding such as fold angle, folding direction, folding shape, foldability, and geometric constraints of adjacent ruling lines. It acts as a connection of curved and straight folding knowledge. The approach is illustrated in the context of the Gauss map strategy and the utility of the technique is demonstrated with the proposed principles of Gauss spherical curves. It is applicable to any generic developable surfaces.