Degree Reduction of Q-Bézier Curves via Squirrel Search Algorithm

Q-Bézier curves find extensive applications in shape design owing to their excellent geometric properties and good shape adjustability. In this article, a new method for the multiple-degree reduction of Q-Bézier curves by incorporating the swarm intelligence-based squirrel search algorithm (SSA) is proposed. We formulate the degree reduction as an optimization problem, in which the objective function is defined as the distance between the original curve and the approximate curve. By using the squirrel search algorithm, we search within a reasonable range for the optimal set of control points of the approximate curve to minimize the objective function. As a result, the optimal approximating Q-Bézier curve of lower degree can be found. The feasibility of the method is verified by several examples, which show that the method is easy to implement, and good degree reduction effect can be achieved using it.

Design of an Intent Recognition System for Dynamic, Rapid Motions in Unstructured Environments

In this study, we developed an offline, hierarchical intent recognition system for inferring the timing and direction of motion intent of a human operator when operating in an unstructured environment. There has been an increasing demand for robot agents to assist in these dynamic, rapid motions that are constantly evolving and require quick, accurate estimation of a user's direction of travel.An experiment was conducted in a motion capture space with six subjects performing threat-evasion in 8 directions, and their mechanical and neuromuscular signals were recorded for use in our intent recognition system (XGBoost). Investigated against current, analytical methods, our system demonstrated superior performance with quicker direction of travel estimation occurring 140 ms earlier in the movement and a 11.6 degree reduction of error. The results showed that we could even predict movement start 100 ms prior to the actual, thus allowing any physical systems to start up. Our direction estimation had an optimal performance of 8.8 degrees, or 2.4% of the 360 degrees range of travel, using 3-axis kinetic data. The performance of other sensors and their combinations indicate that there are additional possibilities to obtain low estimation error. These findings are promising as they can be used to inform the design of a wearable robot aimed at assisting users in dynamic motions, while in environments with oncoming threats.

Combinatorial QFT and the Jacobian Conjecture

The Jacobian Conjecture states that any complex n-dimensional locally invertible polynomial system is globally invertible with polynomial inverse. In 1982, Bass et al. proved an important reduction theorem stating that the conjecture is true for any degree of the polynomial system if it is true in degree three. This degree reduction is obtained with the price of increasing the dimension n. We show in this chapter a result concerning partial elimination of variables, which implies a reduction of the generic case to the quadratic one. The price to pay is the introduction of a supplementary parameter 0≤n′≤n, parameter which represents the dimension of a linear subspace where some particular conditions on the system must hold. We exhibit a proof, in a QFT formulation, using the intermediate field method exposed in Chapter 3.

Quadratic Approximation of Cubic Curves

We present a simple degree reduction technique for piecewise cubic polynomial splines, converting them into piecewise quadratic splines that maintain the parameterization and C1 continuity. Our method forms identical tangent directions at the interpolated data points of the piecewise cubic spline by replacing each cubic piece with a pair of quadratic pieces. The resulting representation can lead to substantial performance improvements for rendering geometrically complex spline models like hair and fiber-level cloth. Such models are typically represented using cubic splines that are C1-continuous, a property that is preserved with our degree reduction. Therefore, our method can also be considered a new quadratic curve construction approach for high-performance rendering. We prove that it is possible to construct a pair of quadratic curves with C1 continuity that passes through any desired point on the input cubic curve. Moreover, we prove that when the pair of quadratic pieces corresponding to a cubic piece have equal parametric lengths, they join exactly at the parametric center of the cubic piece, and the deviation in positions due to degree reduction is minimized.