Design and connection of parametric surfaces through given regular curves

Abstract

This study focused on the connections between two parametric surfaces through the given regular curves. Accordingly, we analyzed a $C^0$-continuous connected surface in terms of the marching-scale functions of these surfaces. It should be noted that, in general, differentiability along a common curve for a $C^0$-continuously connected surface is not guaranteed. To solve this problem, we introduced a $C^1$-continuous connection and proved its existence for such continuous connections.
These connections are improved versions of the $G^1$-connection of developable surfaces introduced by Paluszny.
Moreover, we suggested applications to illustrate the $C^1$-continuous connection using B\'{e}zier curves and
some marching-scale functions of the parametric surfaces.