Before revision:  Let \delta>0, P be  a polygonal curve  with n vertices in \mathbb{R^d, and D(\cdot,\cdot) be a distance measure.
After revision:  Let \delta>0, P be is a long-studied problem in computational geometry and adjacent disciplines, such as graphics, geographical information science, etc. Given a polygonal curve with n vertices in \mathbb{R^d, and D(\cdot,\cdot) be a distance measure.