Planar interpolation with a pair of rational spirals
\(G^2\) Hermite data is defined as follows: (i) a pair of points \(A_0\) and \(A_1\), (ii) unit tangent vectors \(T_{0},T_{1}\) at \(A_0\) and \(A_1\), and a (iii) pair of signed curvatures \(\kappa_0,\kappa_1\) at \(A_0,A_1\). This paper addresses the \(G^2\) Hermite interpolation problem of finding a curve that fits the Hermite data when the data is based on a finite set of points at which the curvature vanishes or when the curvature changes sign between consecutive points. In the case of zero curvature at the ends, one seeks a C-shaped solution by adjoining two spiral segments; in the case of curvature changing sign one seeks an S-shaped solution by adjoining a pair of spiral segments. This paper addresses the feasibility of finding a suitable pair of spirals depending on the Hermite data. A spiral is defined as a curve of monotone increasing or monotone decreasing curvature that does not change sign. In the case of vanishing curvature at \(A_0\) and \(A_1\), let \(B\) denote the point at which the spiral segments are joined and let \(\alpha_i\), \(i=0,1\) denote the external angle between the line segment joining \(A_0,A_1\) and the tangent line at \(A_i\). Finding a C-shaped pair of spiral segments that fits the data depends on the location of \(B\) and the sum of the angles \(\alpha_i\). Several diagrams are provided to describe precisely the conditions required on the location of \(B\) depending on whether \(\alpha_1+\alpha_2\) is less than \(\pi\), equal to \(\pi \) or greater than \(\pi\). In the case \(\alpha_1+\alpha_2<\pi\) the condition says, roughly, that \(B\) should be near the centroid of the triangle formed by the line segment joining \(A_0,A_1\) and the tangent lines at \(A_i\). In the case of curvature changing sign, the feasible region for \(B\) is described in terms of the circles \(C_i\) of curvature at the \(A_i\) and the tangent lines to the \(C_i\) that are perpendicular to the tangents \(T_i\). The equation generating the spirals from which the solution curve is built is taken from parallel work of the authors [\textit{T. N. T. Goodman, D. S. Meek}, and \textit{D. J. Walton}, An involute spiral that matches \(G^2\) Hermite data in the plane].
scientific article; zbMATH DE number 5131547