Construction and Shape Optimization of Quartic Convexity Preserving G² Interpolating Curves

In order to meet the demands of low degree and convexity preservation, a method is proposed for the construction and shape optimization of quartic convexity preserving G² interpolating curves. When the two prescribed end tangents are non-parallel, a quartic Bézier curve interpolating the G² data is expressed with two parameters that denote the lengths of the curve’s end tangents, and the parameters must be restricted within a feasible region for ensuring the convexity preserving property of the curve. When the two prescribed end tangents are parallel, a modified curve construction method is proposed. For determining the optimal parameter values, the objective function is defined as a weighted sum of bending energy and curvature variation energy, and the curves after shape optimization are obtained by finally solving a constrained optimization problem. Some examples for the construction of complex shapes and fonts demonstrate that the proposed methods have wider applications in geometric design and can generate curve shapes with more satisfactory curvature profiles.