## Section: Overall Objectives

### -geometry continuities

Continuities are the basic and important properties of shapes. Continuities of a surface can be addressed by the continuities of curves on the surface in arbitrary directions. Our prior works have found that epsilon-geometric continuities have some advantages over traditional parametric continuities and geometric continuities. Thus, we focus on designing curves with epsilon-geometric continuities and completely lying on a free form surface. This work will play an important role in surface blending, surface-surface intersection, surface trimming, numerical control (NC) tool path generation for machining surfaces, and so on as well.

To ensure positional continuity between the blending and base surfaces for example, linkage curves are usually first computed in the parametric domain, and then represented as the mapping of the domain curves on the base surfaces. To compute the exact curve on a free form surface in control point representation, many algorithms have been presented. However, the degrees of exact curves are considerably high, which results in computationally demanding evaluations and introduces numerical instability.

To overcome the problems of the exact, explicit representation, many approximation algorithms have been presented. To our knowledge, all presented approximation algorithms generate curves not lying completely on the surface.

If such a curve is used as a boundary curve of another surface, gaps may occur between the two surfaces, which are not acceptable in many CAD applications. In surface blending for example, if the linkage curves are not completely on the base surfaces, the blending surface and the base surfaces are not even G^{0} continuous.

*Novelty and originality*:

We plan to study approximation algorithms that generate low degree curves lying completely on the free form surfaces. Until now, we know how to generate the initial approximate polyline, how to control the Hausdorff distance between the approximate curve and the user-specific tolerance and, finally, how to generate an -G^{1} continuous curve. Our goal is to climb-up a new step in the continuity problem. We plan to study approximation algorithms that generate low degree curves lying completely on the free-form surfaces and satisfying the epsilon-continuity condition.