Geometric Hermite interpolation by a family of intrinsically defined planar curves

• A family of planar curves defined by prescribed curvature radius functions are studied.
• The arc lengths and offsets of the obtained curves can be represented in non-rational forms.
• G1G1 Hermite interpolation by curves that have linear curvature radius functions or have prescribed arc lengths is presented.
• G2G2 Hermite interpolation by curves with cubic curvature radius functions is given.

This paper proposes techniques of interpolation of intrinsically defined planar curves to Hermite data. In particular, a family of planar curves corresponding to which the curvature radius functions are polynomials in terms of the tangent angle are used for the purpose. The Cartesian coordinates, the arc lengths and the offsets of this type of curves can be explicitly obtained provided that the curvature functions are known. For given G1G1 or G2G2 boundary data with or without prescribed arc lengths the free parameters within the curvature functions can be obtained just by solving a linear system. By choosing low order polynomials for representing the curvature radius functions, the interpolating curves can be spirals that have monotone curvatures or fair curves with small numbers of curvature extremes. Several examples of shape design or curve approximation using the proposed method are presented.