SmoothenDistance is an option to ConnectionList. The value 0. is default. If the distance between two point is smaller or equal than the SmoothenDistance value, the two points are not connected.
SmoothenDistance is also an option to Interpolation when Method -> "ObtuseAngle" is specified. Close to the affected control points, the interpolated value will be an average of the corresponding function values. That implies that the interpolated function will not take exactly the function values in the control points.