We show how to efficiently smooth a polygon with an approximating spline that stays to one side of the polygon.We also show how to find a smooth spline path between two polygons that form a channel.Problems of this type arise in many physical motion planning tasks where not only forbidden regions have to be avoided but also a smooth traversal of tbe motion path is required.Both algorithms are based on a new tight and efficiently computable bound on the distance of a spline from its control polygon and employ only standard linear and quadratic ptogmmming techniques.