Section: Scientific Foundations

Geometry continuity and $\epsilon$ Geometry Continuity

The mathematical background of parametric surfaces is Differential Geometry. In differential geometry, Riemann (1826 1866), Shiing-Shen Chern (1911 2004), continuities play a very important kernel role. In 1980s, more and more engineering design using geometry modeling softwares found the problems of the parametric continuities. And the order of the parametric continuity depends on how the curve is parameterized. To day, engineers and scientists try to find a kind of continuities, which are the intuitive intrinsic properties of curves and surfaces, and the orders of the continuities are independent of the parameterization.

$G$-Continuity could be defined as the smoothness properties of a curve or a surface that are more than its order of differentiability. This problem is complex and progress in this domain is very slow. We proposed new ways to make through the bottleneck. Furthermore, we also wanted to fill the gap between the traditional mathematics and modern computer science. Hence, we developed the theories of epsilon-geometry continuities to accommodate the representation and the rounding errors of float-point arithmetic, and design new geometric modeling operators under the constraints of epsilon-geometry continuities.