Planar G3 Hermite interpolation by quintic Bezier curves

To achieve G(3) Hermite interpolation with a lower degree curve, this paper studies planar G(3) Hermite interpolation using a quintic Bdzier curve. First, the first and second derivatives of the quintic Bdzier curve satisfying G(2) condition are constructed according to the interpolation conditions. Four parameters are introduced into the construction. Two of them are set as free design parameters, which represent the tangent vector module length of the quintic Bdzier curve at the two endpoints, and the other two parameters are derived from G(3) condition. Then, to match G(3) condition, it is necessary to ensure that the first derivative of curvature with respect to arc length is equal. Nevertheless, the direct calculation of the derivative of curvature involves the calculation of square root. Alternatively, an equivalent condition is derived by investigating the first derivative of curvature square. Based on this condition, the two parameters can be computed as the solutions of linear systems. Finally, the control points of the quintic Bdzier curve are obtained. Several comparative examples are provided to demonstrate the effectiveness of the proposed method. A variety of complex shape curves can be obtained by adjusting the two free design parameters. Applications to shape design are also shown.

Automated summary

This paper investigates planar G(3) Hermite interpolation using a quintic Bdzier curve, constructing first and second derivatives satisfying G(2) condition and deriving two parameters satisfying G(3) condition. By ensuring equality of the first derivative of curvature with respect to arc length, the two parameters can be computed as solutions of linear systems, obtaining the control points of the quintic Bdzier curve.