Mathematical Sciences: Generating Higher Dimensional Paths

研究人员和他的同事们研究计算机生成的高维曲线，插值一个给定的点序列，是免费的不必要的振荡。 从序列中，简单的几何约束被发现在所有的插值曲线，最大限度地减少振荡的一般插值约束和相关的理论允许删除冗余的约束。 结合几何约束的计算机算法被用来找到整个家庭的分段有理曲线，从特定的曲线被选择受到额外的应用特定的约束。 这项研究的动机是应用程序，涉及的路径设计沿着移动的对象，如机器人和动画。在这些应用中，期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从测量（例如用于产生医学成像数据的测量）创建精确的图像动画是重要的。对于这样的应用，最小化不必要的振荡的数量是通过在更高维的曲线上施加适当的数学性质来实现的，该曲线控制着决定运动的某些重要变量的变化。这些无振荡的高维曲线的自动计算机生成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.