Interpolation properties of planar spiral curves

Some inequalities on the planar curves termed spirals (due to monotonous curvature function) are considered. For a spiral represented as a set of interpolation nodes, a region covering the parent curve is constructed. The width of the region provides an estimate of curve definiteness by the given discrete representation, this estimate being independent of any interpolation method. A similar problem is set up for any smooth curve, assuming that vertices are distanced “sufficiently far away” from one another. The problem originates from and can be applied to the practice of tolerance control in industry.