Uncrossing polygon functions.

This adds some utility functions, the most significant of which is
`uncrossPolygon`.  This can handle polygons with or without holes.
Without holes, it takes a polygon in the form of a list of vertices with
each vertex a list or tuple of at least two elements.  For a polygon
with holes, it takes a list of such polygons, where the first element in
the list is the outside polygon and all other elements are the holes.

If a polygon self-crosses, additional vertices will be added and part of
the sequence of vertices will be revered to uncross the polygon.  If the
polygon contains holes, each hole is checked to make sure it doesn't
cross other holes or the outline polygon.

The resultant polygon will have consistently clockwise vertices (though
if the original polygon has multiple loops that only have a vertex in
common, not all of the polygon may be consistently oriented).

Examples:

- `uncrossPolygon([[0, 4], [0, 2], [2, 2], [2, 0], [0, 0], [2, 4]])`
  results in `[[0, 4], [2, 4], [1, 2], [2, 2], [2, 0], [0, 0], [1, 2],
  [0, 2]]`

- `uncrossPolygon([[[0, 0], [0, 2], [1.0, 2.0], [1, 1], [2.0, 1.0], [2,
  2], [1.0, 2.0], [1, 3], [3, 3], [3, 1], [2.0, 1.0], [2, 0]]])` results
  in `[[[0, 0], [0, 2], [1, 2], [1, 1], [2, 1], [2, 2], [1, 2], [1, 3],
  [3, 3], [3, 1], [2, 1], [2, 0]]]`

This still needs tests.

large_image/__init__.py

@@ -20,7 +20,8 @@
 import server
 from server import tilesource
 from server import cache_util
+from server import util
 
 getTileSource = tilesource.getTileSource  # noqa
 
-__all__ = ['server', 'tilesource', 'getTileSource', 'cache_util']
+__all__ = ['server', 'tilesource', 'getTileSource', 'cache_util', 'util']

server/util/__init__.py

@@ -0,0 +1,236 @@
+# General utility functions
+
+from six.moves import range
+
+
+def distance2dSquared(pt1, pt2):
+    """
+    Get the square of the Euclidean 2D distance between two points.
+
+    :param pt1: The first point.  This is an array with at least two numeric
+        elements.
+    :param pt2: The second point.
+    :returns: The distance squared.
+    """
+    dx = pt1[0] - pt2[0]
+    dy = pt1[1] - pt2[1]
+    return dx * dx + dy * dy
+
+
+def distance2dToLineSquared(pt, line1, line2):
+    """
+    Get the square of the Euclidean 2D distance between a point and a line
+    segment.
+
+    :param pt: The point.
+    :param line1: One end of the line.
+    :param line2: The other end of the line.
+    :returns: The distance squared.
+    """
+    dx = line2[0] - line1[0]
+    dy = line2[1] - line1[1]
+    lengthSquared = dx * dx + dy * dy
+    t = 0
+    if lengthSquared:
+        t = float((pt[0] - line1[0]) * dx + (pt[1] - line1[1]) * dy) / lengthSquared
+        t = max(0, min(1, t))
+    return distance2dSquared(pt, [line1[0] + t * dx, line1[1] + t * dy])
+
+
+def triangleTwiceSignedArea2d(pt1, pt2, pt3):
+    """
+    Get twice the signed area of a 2d triangle.
+
+    :param pt1: A vertex.  This is an array with at least two numeric elements.
+    :param pt2: A vertex.
+    :param pt3: A vertex.
+    :returns: Twice the signed area.
+    """
+    return (pt2[1] - pt1[1]) * (pt3[0] - pt2[0]) - (pt2[0] - pt1[0]) * (pt3[1] - pt2[1])
+
+
+def crossLineSegments2d(seg1pt1, seg1pt2, seg2pt1, seg2pt2):
+    """
+    Determine if two line segments cross.  They are not considered crossing if
+    they share a vertex.  They are crossing if either of one segment's
+    vertices are colinear with the other segment.
+
+    :param line1pt1: one endpoint on the first segment.
+    :param line1pt2: the other endpoint on the first segment.
+    :param line2pt1: one endpoint on the second segment.
+    :param line2pt2: the other endpoint on the second segment.
+    :returns: True uf the segments cross.
+    """
+    # If the segments don't have any overlap in x or y, they can't cross
+    if ((seg1pt1[0] > seg2pt1[0] and seg1pt1[0] > seg2pt2[0] and
+         seg1pt2[0] > seg2pt1[0] and seg1pt2[0] > seg2pt2[0]) or
+        (seg1pt1[0] < seg2pt1[0] and seg1pt1[0] < seg2pt2[0] and
+         seg1pt2[0] < seg2pt1[0] and seg1pt2[0] < seg2pt2[0]) or
+        (seg1pt1[1] > seg2pt1[1] and seg1pt1[1] > seg2pt2[1] and
+         seg1pt2[1] > seg2pt1[1] and seg1pt2[1] > seg2pt2[1]) or
+        (seg1pt1[1] < seg2pt1[1] and seg1pt1[1] < seg2pt2[1] and
+         seg1pt2[1] < seg2pt1[1] and seg1pt2[1] < seg2pt2[1])):
+        return False
+    # If any vertex is in common, it is not considered crossing
+    if (seg1pt1 == seg2pt1 or seg1pt1 == seg2pt2 or seg1pt2 == seg2pt1 or
+            seg1pt2 == seg2pt2):
+        return False
+    # If the lines cross, the signed area of the triangles formed between one
+    # segment and the other's vertices will have different signs.  By using
+    # > 0, colinear points are crossing.
+    if (triangleTwiceSignedArea2d(seg1pt1, seg1pt2, seg2pt1) *
+            triangleTwiceSignedArea2d(seg1pt1, seg1pt2, seg2pt2) > 0 or
+            triangleTwiceSignedArea2d(seg2pt1, seg2pt2, seg1pt1) *
+            triangleTwiceSignedArea2d(seg2pt1, seg2pt2, seg1pt2) > 0):
+        return False
+    return True
+
+
+def lineIntersection2d(line1pt1, line1pt2, line2pt1, line2pt2):
+    """
+    Given lines defined by pairs of points, find the point of intersection.
+
+    :param line1pt1: a point on the first line.
+    :param line1pt2: a second point on the first line.
+    :param line2pt1: a point on the second line.
+    :param line2pt2: a second point on the second line.
+    :returns: the point of intersection, or None if the lines are parallel.
+    """
+    line1dx, line1dy = line1pt1[0] - line1pt2[0], line1pt1[1] - line1pt2[1]
+    line2dx, line2dy = line2pt1[0] - line2pt2[0], line2pt1[1] - line2pt2[1]
+    det = float(line1dx * line2dy - line1dy * line2dx)
+    if not det:
+        return
+    return [
+        ((line1pt1[0] * line1pt2[1] - line1pt1[1] * line1pt2[0]) * line2dx -
+         (line2pt1[0] * line2pt2[1] - line2pt1[1] * line2pt2[0]) * line1dx) / det,
+        ((line1pt1[0] * line1pt2[1] - line1pt1[1] * line1pt2[0]) * line2dy -
+         (line2pt1[0] * line2pt2[1] - line2pt1[1] * line2pt2[0]) * line1dy) / det,
+    ]
+
+
+def uncrossPolygonWithoutHoles(vertices):
+    """
+    Given a list of vertices ensure that the polygon does not cross itself.
+    Repeated vertices are removed.  If resulting polygon has 2 or less vertices
+    (this can only happen if so specified or there are duplicate vertices), an
+    empty list is returned.
+
+    :param vertices: a list of vertices of the polygon.  Each vertex is a list
+        of at least 2 coordinates (only the first two values are considered).
+    :returns: vertices: a list of vertices.
+    """
+    if len(vertices) <= 2:
+        return []
+    # Add the first point at the end so we can skip some modulo work
+    pts = vertices[:] + vertices[0:1]
+    idx1 = 0
+    # Iterate through all segments but the last
+    while idx1 < len(pts) - 2:
+        seg1pt1 = pts[idx1]
+        seg1pt2 = pts[idx1 + 1]
+        idx2 = idx1 + 1
+        # Iterate through the remaining segments
+        while idx2 < len(pts) - 1:  # - 1 sicne we duplicated the first point
+            seg2pt1 = pts[idx2]
+            seg2pt2 = pts[idx2 + 1]
+            # Check if the two segments cross
+            if crossLineSegments2d(seg1pt1, seg1pt2, seg2pt1, seg2pt2):
+                # If crossing, add the crossing point and reverse the loop
+                crossPt = lineIntersection2d(seg1pt1, seg1pt2, seg2pt1, seg2pt2)
+                pts = (pts[:idx1 + 1] + [crossPt] + pts[idx2:idx1:-1] +
+                       [crossPt] + pts[idx2 + 1:])
+                break
+            idx2 += 1
+        else:
+            idx1 += 1
+    # Get rid of duplicates except the duplicated first point
+    pts = [pt for idx, pt in enumerate(pts) if not idx or pt != pts[idx - 1]]
+    # Ensure clockwiseness (if counterclockwise, reverse points).
+    #   This still leaves the possibility that the original polygon contained
+    # two separable loops that are in different orientations.  For instance, if
+    # the polygon is two triangles that share a single vertex and no edges, the
+    # two triangles would ideally be in opposite orientations if one is inside
+    # the other and the same orientation if they are not.  Adjusting this is
+    # currently beyond the scope of this function.
+    if sum((pts[idx + 1][0] - pt[0]) * (pts[idx + 1][1] + pt[1])
+           for idx, pt in enumerate(pts[:-1])) < 0:
+        pts = pts[::-1]
+    # Remove duplicated first point
+    pts = pts[:-1]
+    return pts
+
+
+def mergeCrossingPolygons(poly1, poly2):
+    """
+    Given two polygons, check if any edge from the first polygon crosses any
+    edge in the second polygon.  If so, merge the two polygons by adding the
+    crossing point, combining the second polygon at this point, and passing the
+    result through `uncrossPolygonWithoutHoles`.
+
+    :param poly1: a list of vertices forming an uncrossed polygon without
+        holes.
+    :param poly2: a list of vertices forming an uncrossed polygon without
+        holes.
+    :returns: None if the polygons do not cross, or a new polygon if they do.
+    """
+    for idx1, seg1pt1 in enumerate(poly1):
+        seg1pt2 = poly1[(idx1 + 1) % len(poly1)]
+        for idx2, seg2pt1 in enumerate(poly2):
+            seg2pt2 = poly2[(idx2 + 1) % len(poly2)]
+            if crossLineSegments2d(seg1pt1, seg1pt2, seg2pt1, seg2pt2):
+                # If crossing, combine the polygons at the crossing point
+                crossPt = lineIntersection2d(seg1pt1, seg1pt2, seg2pt1, seg2pt2)
+                poly = (poly1[:idx1 + 1] + [crossPt] +
+                        (poly2[idx2 + 1:] + poly2[:idx2 + 1])[::-1] +
+                        [crossPt] + poly1[idx1 + 1:])
+                return uncrossPolygonWithoutHoles(poly)
+
+
+def uncrossPolygon(vertices):
+    """
+    Given a list of vertices, or a list of lists of vertices where the first
+    entry if the polygon and subsequent entries are holes, ensure that the
+    polygon does not cross itself.  Each hole is uncrossed on its own.  If two
+    holes (or a hole and a polygon) cross each other, they are
+    joined into a single more complicated polygon.  Repeated vertices and
+    degenerate polygons (2 or less vertices) are removed.
+
+    :param vertices: a list of vertices of the polygon.  Each vertex is a list
+        of at least 2 coordinates (only the first two values are considered).
+        Alternately, a list of lists of vertices, where the first entry is the
+        polygon and subsequent entries are holes.
+    :returns: vertices: a list of vertices or a list of lists of vertices.
+        This has the same depth as the original argument.
+    """
+    if not len(vertices) or not len(vertices[0]):
+        return vertices
+    if not isinstance(vertices[0][0], list):
+        return uncrossPolygonWithoutHoles(vertices)
+    # uncross the outer polygon and all holes
+    polygons = [uncrossPolygonWithoutHoles(pts) for pts in vertices]
+    # If the containing polygon is degenerate, return an empty set
+    if not len(polygons[0]):
+        return [[]]
+    # discard degenerate holes
+    polygons = [polygon for polygon in polygons if len(polygon)]
+    # For each polygon, check if it crosses any other polygon.  If it does,
+    # join them together at the first crossing point and uncross the result.
+    pidx1 = 0
+    while pidx1 < len(polygons) - 1:
+        poly1 = polygons[pidx1]
+        pidx2 = pidx1 + 1
+        while pidx2 < len(polygons):
+            poly2 = polygons[pidx2]
+            crossed = mergeCrossingPolygons(poly1, poly2)
+            if crossed:
+                polygons[pidx1] = crossed
+                del polygons[pidx2]
+                break
+            pidx2 += 1
+        else:
+            pidx1 += 1
+    # reverse all holes so that they are opposite direction as the main polygon
+    for idx in range(1, len(polygons)):
+        polygons[idx] = polygons[idx][::-1]
+    return polygons