Interpolation with Specified Error of a Point Series Belonging to a Monotone Curve

Abstract

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.

Representation of a Monotone Curve by a Contour with Regular Change in Curvature

“…Figure 4 shows a cubic non-periodic B-spline interpolating a point series consisting of 10 points created in the CAD system SolidWorks. The sequence of points that was used in [ 12 ] was taken as the reference point series to determine the area of location of the monotone curve.…”

“…When forming a contour representing a monotone curve, it is appropriate to assign positions of the tangents at the reference points where a point series can be interpolated by a monotone curve [ 12 ].…”

“…The contour was formed in the CAD system SolidWorks with the use of a specially developed software module. The given data are represented by a point series belonging to a monotone curve interpolation which was studied in the work [ 12 ]. The graph of curvature values along the contour formed with the CAD system demonstrates the regularity of joining of the sections.…”

“…The work [ 12 ] proposes a method for forming a contour by sections representing a monotone curve with specified accuracy. The problem is solved in the following stages: Analysis of the reference point series is carried out, which results in identifying its sections that can be interpolated by a monotone curve; At the reference points, the positions of the tangent and the radii of the curvature are assigned, at which the problem of forming a monotone curve has a solution; The area of the possible location of the monotone curve interpolating sections of the point series is determined; Within the obtained area, a contour is formed, representing a monotone curve with specified accuracy.…”

“…Work [ 12 ] proposes a method for forming arcs of circles whose radii increase in the same direction as the radii of the curvature of the monotone curve interpolating a point series. The disadvantage of this method is that the regularity of curvature values at the points at which the arcs of circles are conjoined is disturbed.…”