Mathematical Sciences: Generating Higher Dimensional Paths

这位研究人员和他的同事们研究了高维曲线的计算机生成，这些曲线插入给定的点序列，没有不必要的振荡。从该序列中，在所有的插补曲线上都找到了简单的几何约束，使得在一般插补约束下的振荡最小化，并且相关的理论允许去除多余的约束。结合几何约束的计算机算法被用来寻找分段有理曲线族，从这些曲线族中选择特定的曲线受附加的应用特定约束。这项研究的动机是涉及物体移动路径设计的应用程序，如机器人和动画。在这些应用中，希望产生平滑且没有不必要的振荡的运动。例如，这对于减少控制机器人所使用的能量或根据测量结果创建准确的图像动画非常重要，例如产生医学成像数据的图像。对于这种应用，通过在高维曲线上施加适当的数学属性来控制决定运动的某些重要变量的变化，从而实现最大限度地减少不必要的振荡次数。计算机自动生成这些无振荡的高维曲线的算法可以用作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.