Generalized evolutes of planar curves

The evolutes of regular curves in the Euclidean plane are given by the caustics of regular curves. In this paper, we define the generalized evolutes of planar curves which are spatial curves, and the projection of generalized evolutes along a fixed direction are the evolutes. We also prove that the generalized evolutes are the locus of centers of slant circles of the curvature of planar curves. Moreover, we define the generalized parallels of planar curves and show that the singular points of generalized parallels sweep out the generalized evolute. In general, we cannot define the generalized evolutes at the singular points of planar curves, but we can define the generalized evolutes of fronts by using moving frames along fronts and curvatures of the Legendre immersion. Then we study the behaviors of generalized evolutes at the singular points of fronts. Finally, we give some examples to show the generalized evolutes.

Principles of spatial design for haulage roads

The purpose of the study is to review and analyze the experience of specialists in the field of spatial design of haulage roads for further improving the methodological foundations by providing clarity and visually acceptable degree of spatial curves. Drivers perception of the haulage road as well as overspeeding lead to critical situations or road accidents. A decrease in speed at seemingly abrupt bends on the road affects the efficiency of the log trucking transport. Therefore, the perspective view of the road should clearly orient the driver, that is, be visually clear, clearly changing, ensuring the constancy or modulated reduction of the traffic condition. The need for an optimal spatial solution of the road increases. It is required to determine the conditions under which the visual smoothness and clarity of the most common simple spatial curves when looking from points corresponding to the normal position of the eyes of car drivers is ensured.

THE MOTION OF A PARTICLE ON A WAVY SURFACE DURING ITS TRANSLATIONAL CIRCULAR OSCILLATIONS IN HORIZONTAL PLANES

The rough plane is a universal structural element of many machines and devices for sifting and separation of parts of technological material. The motion of particles on the horizontal plane, which performs oscillating rectilinear or circular motion, is the most studied. A wavy surface with a sinusoidal cross-sectional line as a working surface will significantly change the trajectories of their motion. The mathematical description of such a motion will change accordingly. The sliding of a particle in a plane will be a partial case of sliding on a wavy surface when the amplitude of the sinusoid is equal to zero. When the wavy surface oscillates and all its points describe circles, the motion of the technological material changes significantly. The regularities of the motion of material particles on such a surface during its circular translational oscillations in the horizontal planes are investigated in the article. Differential equations of relative particle displacement are compiled and solved by numerical methods. The trajectories of the particle sliding on the surface and the graphs of its reaction are constructed. A partial case of a surface is a plane, and the sliding trajectory of a particle is a circle. An analytical expression to determine its radius is found. During circular oscillations of a wavy linear surface with a cross section in the form of a sinusoid relative trajectory of a particle after stabilization of the motion can be closed or periodic spatial curves. To avoid the breakaway of the particle from the surface, the oscillation mode should be set, which takes into account the shape of the surface and the kinematic parameters of oscillations. With the diameter of the circle, which is described by all points of the surface during its oscillation, is equal to the period of the sinusoid, the trajectory of the relative motion of the particle can be a periodic curve. In this case, the particle moves in a direction close to the transverse, overcoming depressions and ridges. In other cases, the trajectory is a closed spatial curve, the horizontal projection of which is close to a circle. The found analytical dependencies allow determining the influence of structural and technological parameters of the surface on the trajectory of the particle.

Synthesizing Mechanical Chains for Morphing Between Spatial Curves

This paper extends the kinematic synthesis methodology for designing a chain of bodies to match a set of arbitrary curves to the spatial case. The methodology initiates with an arbitrary set of spatial curves, and concludes with a set of bodies defined by their spatial features. The bodies synthesized can be one of three types: a rigid segment, a helical segment with constant curvature and torsion but varying length, and a growth segment that maintains its geometry but may be scaled to become larger or smaller. To realize mechanical chains for mechanisms that achieve spatial shape change, only rigid and helical segments are used. After designing the segments, they may be aligned with the original spatial curves with their ends connected via an optimization. For two curves, these connections may be made with revolute joints to obtain high accuracy. For three or more curves, spherical joint connections allow for best accuracy. To compare curves as is useful in morphometry, all three segment types may be employed. In this case, an accurate description of the changes between curves is important, and optimizing to connect the segments is not needed. The procedure for redefining the curves in a way that the techniques in this paper may be applied, as well as the methodologies for synthesizing the three segment types are presented. Examples include a continuum robot problem and the morphometric analyses of cochlear curves and the lambdoidal suture located on a human skull. This work extends the established planar techniques for synthesizing mechanisms and addressing morphometric issues that are motivated with curves in two-dimensions.

Mathematics and practice of color space invariants by the example of determining the gray balance for a digital printing system

In modern printing, a large number of tasks are associated with the mutual transformation of color spaces. In particular, the most common pair of hardware-dependent color spaces is RGB and CMYK, the mutual transformation of colors in which is ambiguous, which creates significant problems in color reproduction. To solve this problem, we propose using color space invariants — gradation trajectories and gradation surfaces, which are analogs of gradation curves for initial colorants and their binary overlays, constructed in the absolute color space of the CIE Lab. Invariants are introduced on the basis of the mathematical apparatus of the differential geometry of spatial curves and surfaces. Practical application of color space invariants involves certain difficulties associated with their complex analytical description; moreover, for most practical problems, the high accuracy of the model is redundant. For the practical application of invariants, we propose a simpler approach using natural color sampling in digital printing systems. As an example, the procedure for determining the gray balance for an electrophotographic printing press is given.

Synthesizing Mechanical Chains for Morphing Between Spatial Curves

This work investigates the kinematic synthesis methodology for designing a chain of three-dimensional bodies to match a set of arbitrary spatial curves. The bodies synthesized can be one of three types: a rigid segment, a helical segment with constant curvature and torsion but varying length, and a growth segment that maintains its geometry but may be scaled to become larger or smaller. To realize mechanical chains, only rigid and helical segments are used. After designing the segments, they may be aligned with the original spatial curves with their ends connected via an optimization. For two curves, these connections may be made with revolute joints to obtain high accuracy. For three or more curves, spherical joint connections allow for the best accuracy. To compare curves as is useful in morphometry, all three segment types may be employed. In this case, an accurate description of the changes between curves is important, and optimizing to connect the segments is not needed. The procedure for redefining the curves in a way that the techniques in this paper may be applied, as well as the methodologies for synthesizing the three segment types are presented. Examples include a continuum robot problem and the morphometric analyses of chochlear curves and the lambdoidal suture. This work extends the established planar techniques for synthesizing mechanisms and addressing morphometric issues that are motivated with curves in two-dimensions.