The problem of modelling a smooth contour with a regular change in curvature representing a monotone curve with specified accuracy is solved in this article. The contour was formed within the area of the possible location of a convex curve, which can interpolate a point series. The assumption that if a sequence of points can be interpolated by a monotone curve, then the reference curve on which these points have been assigned is monotone, provides the opportunity to implement the proposed approach to estimate the interpolation error of a point series of arbitrary configuration. The proposed methods for forming a convex regular contour by arcs of ellipses and B-spline ensure the interpolation of any point series in parts that can be interpolated by a monotone curve. At the same time, the deflection of the contour from the boundaries of the area of the possible location of the monotone curve can be controlled. The possibilities of the developed methods are tested while solving problems of the interpolation of a point series belonging to monotone curves. The problems are solved in the CAD system of SolidWorks with the use of software application created based on the methods developed in the research work.

The paper addresses the problem of modeling a smooth contour interpolating a point series belonging to a curve containing no special points, which represents the original curve with specified accuracy. The contour is formed within the area of possible location of the parts of the interpolated curve along which the curvature values are monotonously increased or decreased. The absolute interpolation error of the point series is estimated by the width of the area of possible location of the curve. As a result of assigning each intermediate point, the location of two new sections of the curve that lie within the area of the corresponding output section is obtained. When the interpolation error becomes less than the given value, the area of location of the curve is considered to be formed, and the resulting point series is interpolated by a contour that lies within the area. The possibility to shape the contours with arcs of circles specified by characteristics is investigated.

The purpose of the study is to develop a method for the formation of a one-dimensional contour with provision of a given accuracy of interpolation. Determination of the accuracy of interpolation relies on the formation of a curve based on known geometric properties. We build the geometric model with the assumption: if there is a curve line without special points that interpolates the point series, then there are no special points in the original object. Such points include: points of inflection, changes in the direction of growth along the curve of values of the curvature, the rounding, etc.We build the interpolating curve in the form of a condensed point series consisting of arbitrarily large numbers of nodes, which are determined based on the possibility of interpolating their curve with a given line with the given characteristics. The error, with which the discrete representation of the curve represents the original curve, is evaluated as an area of the possible arrangement of all curves that interpolate an output point with properties identical to the properties of the original curve. We evaluate the error of formation of the interpolating curve line as the area of a possible location of the curve line that interpolates the condensed point series. In the study, we propose the solution of the problem for a flat curve based on the condition of the absence of oscillations and the conditions for the monotonous change of the curvature. The area of a location of the curve determined from the condition of the convexity of the curve is maximal and is the output one. Overlaying the following conditions: monotonous curvature change along the curve and the appointment of fixed positions of tangents and values of the curvature at the output points, localizes an area of a possible solution. One can use the developed method for solving problems requiring determination of the maximum absolute error with which a model represents the original object. These are approximate calculations, construction of graphs that describe processes and phenomena, formation of surface models representing existing physical samples.

The method of designing elements of the surfaces' frames based on array of points is suggested in the work. Elements of frames are contours that are received via interpolation of sets of points, which are selected from the initial array...

The common properties of images on a plane and a sphere are considered in the scientific works by scientists-designers of spherical mechanisms. This is due to the fact that the plane and the sphere share common geometric parameters. They include constancy at all points of the Gaussian curve, which has a zero value for a plane and a positive value for a sphere. Figures belonging to them can slide freely on both surfaces. With unlimited growth of the radius of the sphere, its limited section approaches the plane, and the spherical shape transforms into a plane. Thus, a loxodrome that crosses all meridians at a constant angle is transformed into a logarithmic spiral that intersects at a constant angle the radius vectors that come from the pole. The tooth profile of cylindrical gears is outlined by the involute of a circle. A spherical involute is used for the corresponding bevel gears. Other spherical curves are also known, which are analogs of flat ones. The formation of a cycloid and an involute of a circle are associated with the mutual rolling of a line segment with each of these figures. If the segment is fixed and the circle rolls along it, then the point of the circle describes the cycloid. In the case of a stationary circle along which a segment is rolled, the point of the segment will execute the involute. To move to the spherical analogs of these curves, it is necessary to replace the circle with a cone, and the straight line with a plane. The spherical prototype of the cycloid will be the trajectory of the point of the base of the cone, which rolls along the plane, that is, along the sweep of the cone. The sweep of a cone is a sector, the radius of the limiting circle of which is equal to the generating cone. If this sweep, like a section of a plane, is rolled around a fixed cone, when its top coincides with the center of the sector, then the point of the limiting radius of the sector will execute a spherical involute. This paper analytically implements these two motions and reports the parametric equations of the spherical analogs of the circle involute and the cycloid