Bézier and B-spline curves are foundational tools for curve representation in computer graphics and computer-aided geometric design, with their intersection computation presenting a fundamental challenge in geometric modeling. This study introduces an innovative algorithm that quickly and effectively resolves intersections between Bézier and B-spline curves. The number of intersections between the two input curves within a specified region is initially determined by applying the resultant of a polynomial system and Sturm’s theorem. Subsequently, the potential region of the intersection is established through the utilization of the pseudo-curvature-based subdivision scheme and the bounding box detection technique. The projected Gauss-Newton method is ultimately employed to efficiently converge to the intersection. The robustness and efficiency of the proposed algorithm are demonstrated through numerical experiments, demonstrating a speedup of 3 to 150 times over traditional methods@en

Cubic Bézier curves are widely used in computer graphics and geometric modeling, favored for their intuitive design and ease of implementation. However, self-intersections within these curves can pose significant challenges in both geometric modeling and analysis. This paper presents a comprehensive approach to detecting and computing self-intersections of cubic Bézier curves. We introduce an efficient algorithm that leverages both the geometric properties of Bézier curves and numerical methods to accurately identify intersection points. The self-intersection problem of cubic Bézier curves is firstly transformed into a quadratic problem by eliminating trivial solutions. Subsequently, this quadratic system is converted into a linear system that may be easily analyzed and solved. Finally, the parameter values corresponding to the self-intersection points are computed through the solution of the linear system. The proposed method is designed to be robust and computationally efficient, making it suitable for real-time applications.@en

In weaving areas, vehicles frequently carry out conflicting lane-changing manoeuvres. The frequent lane change in this area results in rapid changes in vehicles’ speed, which in turn reduces traffic efficiency and create traffic bottlenecks at weaving areas. This research proposes a cooperative weaving motion planner for connected and automated vehicles to reduce traffic oscillation. The proposed motion planner is based on model predictive control method and solved by Chang-Hu’s method. Paper presented at the Intelligent Transportation Systems (ITSC), 2018 IEEE). The motion planner only requires longitudinally automation which is accessible for most commercialized luxury vehicles. Simulation evaluation was conducted to quantify the performance of the proposed motion planner. The results show that the proposed motion planner is able to reduce traffic oscillation by 2.7% to 28.0%. Furthermore, the computation time of the proposed planner is fewer than 20 milliseconds indicating readiness to real-time application.

@en