A common class type for all bidirectional iterators, c++

I have a bunch of classes which implement geometric objects based on vertices, such as TriangleMesh, PointClouds or Edgesets. They all inherit from VertexBasedGeometry. I now want all of them to return a bidirectional iterator on references of its vertices. This would allow to access the vertices of any VertexBasedGeometry in a non-templated way. Now, me not being very familiar with iterators, this turns out to be rather difficult. My try looks as follows:

class VertexBasedGeometry : public Geometry
{
public:

typedef std::iterator<std::bidirectional_iterator_tag,defines::Vector3 >  VertexIterator;

VertexBasedGeometry(){};

virtual VertexIterator begin()=0;
virtual VertexIterator end()=0;
};

In TraingleMesh, which inherits from VertexBasedGeometry I now try to implement the function begin by returning the begin iterator of the std::vector which contains its vertices. This results in the following compiler error on gcc 4.2 (apple):

Mesh.cpp:25: error: conversion from '__gnu_cxx::__normal_iterator<defines::Vector<double, 3>*, std::vector<defines::Vector<double, 3>, std::allocator<defines::Vector<double, 3> > > >' to non-scalar type 'std::iterator<std::bidirectional_iterator_tag, defines::Vector<double, 3>, long int, defines::Vector<double, 3>*, defines::Vector<double, 3>&>' requested

My question are now: Why does this not work, and how should I change this to get it work? Reading more about iterators, I got the slight feeling that I won't be able to find a common type for any bidirectional iterator, is that right? Some of the classes might have their vertices stored in containers other than std::vector, others already provide (non-stl-conforming ) iterators which I want to adapt to my common type. I am open for any recommendations how to implement this.

std::iterator is only a tag class that you derive from, and the type of your iterator must be known at compile-time to the implementing class. If you want to hide the iterator implementation from the iterating code, you must add an indirection level, e.g. Thomas Becker's any_iterator.