Geometric Continuity
There is another notion of continuity called geometric continuity. Although the idea existed in differential geometry, the concept was introduced for geometric modelling and graphics applications in mid 80s. In one variable it essentially means matching of the directions of the derivatives of the adjacent curves. In other words, this means matching the derivatives when the two curves have been defined with respect to arc length as the parameter. This provides flexibility in the shape modelling. Lot of research has been done in this direction.