G2 Continues Plane Curve Transition Method with Shape Reference

Curve transition is a quintessential challenge in the realm of freeform curve modeling, extensively utilized for intricate product design endeavors. Nevertheless, conventional approaches to curve transition predominantly concentrate on the smoothness between the transitioning curves, often overlooking the potential for further refining the curves' shape. In this study, we introduce an innovative plane curve transition technique that incorporates constraints related to reference shapes, with the objective of ensuring that the transitioning curves' shapes closely mimic the reference curve, all while adhering to the G2 continuity criterion. To achieve this, we have developed a G2 continuous transition curve model that incorporates four shape adjustment parameters, constructed from quintic Bézier curves. Subsequently, we have formulated and resolved a shape similarity function that quantifies the resemblance between the transition curve and the reference curve, grounded in the concept of quasi-curvature. Furthermore, we have devised a strategy for accurately determining the initial parameters of the reference curve, which addresses the common engineering dilemma of imprecise reference curve range specifications. The efficacy and applicability of our proposed method are substantiated through a series of numerical experiments and a blade restoration task, showcasing its robustness and practical utility in industry applications.