static inline bool point_is_spike_or_equal(Point1 const& last_point,
            Point2 const& segment_a,
            Point3 const& segment_b,
            RobustPolicy const& robust_policy)
{
    if (point_is_spike_or_equal(last_point, segment_a, segment_b))
    {
        return true;
    }

    if (! RobustPolicy::enabled)
    {
        return false;
    }

    // Try using specified robust policy
    typedef typename geometry::robust_point_type
    <
        Point1,
        RobustPolicy
    >::type robust_point_type;

    robust_point_type last_point_rob, segment_a_rob, segment_b_rob;
    geometry::recalculate(last_point_rob, last_point, robust_policy);
    geometry::recalculate(segment_a_rob, segment_a, robust_policy);
    geometry::recalculate(segment_b_rob, segment_b, robust_policy);

    return point_is_spike_or_equal
        (
            last_point_rob,
            segment_a_rob,
            segment_b_rob
        );
}
inline void append_no_dups_or_spikes(Range& range, Point const& point,
        RobustPolicy const& robust_policy)
{
#ifdef BOOST_GEOMETRY_DEBUG_INTERSECTION
    std::cout << "  add: ("
        << geometry::get<0>(point) << ", " << geometry::get<1>(point) << ")"
        << std::endl;
#endif
    // The code below thies condition checks all spikes/dups
    // for geometries >= 3 points.
    // So we have to check the first potential duplicate differently
    if (boost::size(range) == 1
        && points_equal_or_close(*(boost::begin(range)), point, robust_policy))
    {
        return;
    }

    traits::push_back<Range>::apply(range, point);

    // If a point is equal, or forming a spike, remove the pen-ultimate point
    // because this one caused the spike.
    // If so, the now-new-pen-ultimate point can again cause a spike
    // (possibly at a corner). So keep doing this.
    // Besides spikes it will also avoid adding duplicates.
    while(boost::size(range) >= 3
            && point_is_spike_or_equal(point,
                *(boost::end(range) - 3),
                *(boost::end(range) - 2),
                robust_policy))
    {
        // Use the Concept/traits, so resize and append again
        traits::resize<Range>::apply(range, boost::size(range) - 2);
        traits::push_back<Range>::apply(range, point);
    }
}
inline bool is_spike_or_equal(Point1 const& a,
            Point2 const& b,
            Point3 const& c,
            SideStrategy const& strategy)
{
    return point_is_spike_or_equal(c, a, b, strategy);
}
inline bool point_is_spike_or_equal(Point1 const& last_point,
            Point2 const& segment_a,
            Point3 const& segment_b,
            SideStrategy const& strategy,
            RobustPolicy const& robust_policy)
{
    if (point_is_spike_or_equal(last_point, segment_a, segment_b, strategy))
    {
        return true;
    }

    if (BOOST_GEOMETRY_CONDITION(! RobustPolicy::enabled))
    {
        return false;
    }

    // Try using specified robust policy
    typedef typename geometry::robust_point_type
    <
        Point1,
        RobustPolicy
    >::type robust_point_type;

    robust_point_type last_point_rob, segment_a_rob, segment_b_rob;
    geometry::recalculate(last_point_rob, last_point, robust_policy);
    geometry::recalculate(segment_a_rob, segment_a, robust_policy);
    geometry::recalculate(segment_b_rob, segment_b, robust_policy);

    return point_is_spike_or_equal
        (
            last_point_rob,
            segment_a_rob,
            segment_b_rob,
            strategy
        );
}
inline void clean_closing_dups_and_spikes(Range& range,
                RobustPolicy const& robust_policy)
{
    std::size_t const minsize
        = core_detail::closure::minimum_ring_size
            <
                geometry::closure<Range>::value
            >::value;

    if (boost::size(range) <= minsize)
    {
        return;
    }

    typedef typename boost::range_iterator<Range>::type iterator_type;
    static bool const closed = geometry::closure<Range>::value == geometry::closed;

// TODO: the following algorithm could be rewritten to first look for spikes
// and then erase some number of points from the beginning of the Range

    bool found = false;
    do
    {
        found = false;
        iterator_type first = boost::begin(range);
        iterator_type second = first + 1;
        iterator_type ultimate = boost::end(range) - 1;
        if (BOOST_GEOMETRY_CONDITION(closed))
        {
            ultimate--;
        }

        // Check if closing point is a spike (this is so if the second point is
        // considered as a spike w.r.t. the last segment)
        if (point_is_spike_or_equal(*second, *ultimate, *first, robust_policy))
        {
            range::erase(range, first);
            if (BOOST_GEOMETRY_CONDITION(closed))
            {
                // Remove closing last point
                range::resize(range, boost::size(range) - 1);
                // Add new closing point
                range::push_back(range, range::front(range));
            }
            found = true;
        }
    } while(found && boost::size(range) > minsize);
}