Determining how two intervals overlap

IntervalOverlap.cs

/// <summary>
/// Provides tools to detect the kind of the overlap between two intervals
/// </summary>
public static class IntervalOverlap
{
    /// <summary>
    /// Detects kind of the overlap that the specified two ranges have.
    /// </summary>
    /// <param name="start1">The inclusive start of the first interval.</param>
    /// <param name="end1">The exclusive end of the first interval.</param>
    /// <param name="start2">The inclusive start of the second interval.</param>
    /// <param name="end2">The exclusive end of the second interval.</param>
    /// <returns></returns>
    public static IntervalOverlapKinds Detect(int start1, int end1, int start2, int end2)
    {
        IntervalOverlapKinds relation = IntervalOverlapKinds.FirstAfterSecond;

        if (start2 <= start1 && end1 <= end2)
        {
            // (--)
            relation = IntervalOverlapKinds.FirstInsideSecond;
        }
        else if (start1 <= start2 && end2 <= end1)
        {
            // -()-
            relation = IntervalOverlapKinds.FirstIncludesSecond;
        }
        else if (start1 < start2 && start2 < end1 && end1 <= end2)
        {
            // -(-)
            relation = IntervalOverlapKinds.FirstBeforeAndInsideSecond;
        }
        else if (start2 <= start1 && start1 < end2 && end1 > end2)
        {
            // (-)-
            relation = IntervalOverlapKinds.FirstInsideAndAfterSecond;
        }
        else if (start1 < start2 && end1 <= start2)
        {
            // --()
            relation = IntervalOverlapKinds.FirstBeforeSecond;
        }
        else if (start1 >= end2)
        {
            // ()--
            relation = IntervalOverlapKinds.FirstAfterSecond;
        }
        else
        {
            throw new Exception("Could not detect interval overlapping kinds");
        }

        return relation;
    }

    /// <summary>
    /// Gets the sign corresponding to the given overlap kind.
    /// -- means the first interval, and
    /// () means the second interval.
    /// </summary>
    /// <param name="overlapKind">Kind of the overlap.</param>
    /// <returns></returns>
    public static string GetSign(IntervalOverlapKinds overlapKind)
    {
        switch (overlapKind)
        {
            case IntervalOverlapKinds.FirstInsideSecond:
                return "(--)";
            case IntervalOverlapKinds.FirstIncludesSecond:
                return "-()-";
            case IntervalOverlapKinds.FirstBeforeAndInsideSecond:
                return "-(-)";
            case IntervalOverlapKinds.FirstInsideAndAfterSecond:
                return "(-)-";
            case IntervalOverlapKinds.FirstBeforeSecond:
                return "--()";
            case IntervalOverlapKinds.FirstAfterSecond:
                return "()--";
            default:
                throw new Exception("Unknown interval overlap type!");
        }
    }
}

IntervalOverlapKinds.cs

/// <summary>
/// Enumerates all possible ways two intervals may overlap
/// </summary>
public enum IntervalOverlapKinds
{
    /// <summary>
    /// (--)
    /// </summary>
    FirstInsideSecond, 

    /// <summary>
    /// -()-
    /// </summary>
    FirstIncludesSecond,

    /// <summary>
    /// -(-)
    /// </summary>
    FirstBeforeAndInsideSecond, 

    /// <summary>
    /// (-)-
    /// </summary>
    FirstInsideAndAfterSecond,

    /// <summary>
    /// --()
    /// </summary>
    FirstBeforeSecond, 

    /// <summary>
    /// ()--
    /// </summary>
    FirstAfterSecond
}