Planar $${G^{3}}$$ Hermite interpolation by quintic Bézier curves

To achieve $$G^3$$ Hermite interpolation with a lower degree curve, this paper studies planar $$G^3$$ Hermite interpolation using a quintic Bezier curve. First, the first and second derivatives of the quintic Bezier 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 Bezier 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 Bezier 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.