Planar quartic G2 Hermite interpolation for curve modeling

Abstract

We study planar quartic G2 Hermite interpolation, that is, a quartic polynomial curve interpolating two planar data points along with the associated tangent directions and curvatures. When the two specified tangent directions are non-parallel, a quartic Bézier curve interpolating such G2 data is constructed using two geometrically meaningful shape parameters which denote the magnitudes of end tangent vectors. We then determine the two parameters by minimizing a quadratic energy functional or curvature variation energy. When the two specified tangent directions are parallel, a quartic G2 interpolating curve exists only when an additional condition on G2 data is satisfied, and we propose a modified optimization approach. Finally, we demonstrate the achievable quality with a range of examples and the application to curve modeling, and it allows to locally create G2 smooth complex shapes. Compared with the existing quartic interpolation scheme, our method can generate more satisfactory results in terms of approximation accuracy and curvature profiles.