S2LaxClosedPolylineShape is like S2LaxPolylineShape except that the last vertex is implicitly joined to the first. It is also like S2LaxLoopShape except that it does not have an interior (which makes it more efficient to index).
S2LaxLoopShape represents a closed loop of edges surrounding an interior region. It is similar to S2Loop::Shape except that this class allows duplicate vertices and edges. Loops may have any number of vertices, including 0, 1, or 2. (A one-vertex loop defines a degenerate edge consisting of a single point.)
S2VertexIdLaxLoopShape is just like S2LaxLoopShape, except that vertices are specified as indices into a vertex array. This representation can be more compact when many loops are arranged in a mesh structure.