Mathematical Sciences: Generating Higher Dimensional Paths

研究者和他的同事们研究了高维曲线的计算机生成，这些曲线可以插值给定的点序列，并且没有不必要的振荡。从序列中，在所有插值曲线上发现了简单的几何约束，这些约束使受一般插值约束的振荡最小化，并且相关理论允许去除冗余约束。结合几何约束的计算机算法用于查找整个分段有理曲线族，从中选择受附加应用特定约束的特定曲线。这项研究的动机是涉及物体移动路径设计的应用，比如机器人和动画。在这些应用中，希望创建平滑且没有不必要振荡的运动。这是很重要的，例如，减少在控制机器人时使用的能量，或者从产生医学成像数据的测量中创建精确的图像动画。对于这样的应用，通过在高维曲线上施加适当的数学性质来实现最小化不必要的振荡的数量，高维曲线控制着决定运动的某些重要变量的变化。计算机自动生成这些无振荡高维曲线的算法可以作为许多不同应用的基础。

The investigator and his colleagues study the computer generation of higher dimensional curves that interpolate a given sequence of points and are free from unnecessary oscillations. From the sequence, simple geometric constraints are found on all interpolating curves that minimize oscillations subject to the general interpolation constraints and the associated theory allows for the removal of redundant constraints. Computer algorithms incorporating the geometric constraints are used to find entire families of piecewise rational curves from which particular curves are selected subject to additional application specific constraints. The study is motivated by applications that involve the design of paths along which objects are to move, such as robotics and animation. In these applications it is desirable to create motions that are smooth and free from unnecessary oscillations. This is important, for example, to reduce the energy used in controlling robots or to create accurate animations of images from measurements such as those taken to produce medical imaging data. For such applications, minimizing the number of unnecessary oscillations is accomplished by imposing the appropriate mathematical properties on a higher dimensional curve that governs the changes in certain important variables that dictate the motion. Algorithms for the automatic computer generation of these oscillation free higher dimensional curves can be used as a basis for many different applications.