Area, buffer, description

							Area, buffer, description
  Area of a polygon, center of mass,
    buffer of a polygon / polygonal
    line, and descriptive statistics
                 Area of a polygon

• Polygon without holes
• Polygon with holes


             (6,8)
                           (12,7)           Given: cyclic list
 (3,5)            (8,6)                     of points with
                                   (14,6)   their coordinates.
                  (7,4)
                          (11,3)
         (5,1)
Idea: area under edge

(6,8)                 (6,8)
        (12,7)                  (12,7)




          Area =
          width times average height
          = (12 - 6) * (8 + 7)/2 = 45
            Area, continued
Edges at “upper side”
give positive contribution




Edges at “lower side” give
negative contribution
           Area formula

                      Area(P) =

                      (X1 - Xn)(Y1 + Yn)/2 +
                      n-1
                       (Xi+1 - Xi)(Yi + Yi+1)/2
                      i=1


Assume points given
clockwise
   Center of mass of a polygon
• Take arbitrary point p
• Make triangles: each side of the polygon and
  point p
• Determine weight (area) and center of mass per
  triangle
• Compute weighted average of the centers of mass

    p
                       Counterclockwise triangles:
                       positive weight; clockwise
                       triangles: negative weight
  Center of mass of a polygon

• Center of mass of weighted points: weighted
  average of the x-coordinates and of the y-
  coordinates
                                   5

                            -8         -12
                                             21
                              11
                                       20
         Buffer of a polyline

• Buffer = Minkowski sum with a disk
• A polyline gives a polygon with holes
          Buffer computation:
           divide & conquer

1. Split polyline in two
2. Compute the buffer of
   the halves recursively
3. Merge the buffers
   to one buffer
            Buffer complexity

•   Two non-intersecting line segments S1 and S2
•   Consider the buffers of S1 and S2
•   They intersect at most twice
•   So a set of buffers of non-intersecting line
    segments is a set of pseudo-discs
          Buffer complexity

Theorem (13.9 of computational geometry book):
Let S be a collection of [polygonal] pseudodiscs
with n edges in total. Then the complexity of their
union is O(n)

Corollary: The buffer of a polygonal line consisting
of n edges (or n+1 points) has complexity O(n)
        Merge buffers to one
Assume we merge the buffers of
polylines having m1 and m2 edges
Do an “ordinary” line
sweep over the O(m1)
+ O(m2) segments
and circular arcs of
the buffers

Cost:
O((m1+m2+k) log (m1+m2+k))
time for the merge
How many intersection points?
• In general: m1 and m2 line segments can
  intersect m1m2 times, so k  m1m2
• Here: Every intersection point is a vertex of the
  buffer after the merge, so of a polyline with
  m1+m2 edges
• This new buffer has complexity O(m1+m2)
  according to the corollary, so k = O(m1+m2)
• Hence, the merge takes
  O((m1+m2) log (m1+m2)) time
              The algorithm

• If n >1, split polyline with n edges into two
  parts with n/2 edges each
• Compute the buffer of each part recursively
• Merge the buffers of the parts into one using
  plane sweep, in O(n log n) time


Recurrence: T(n) = 2 T(n/2) + O(n log n)

Gives T(n) = O(n log2 n) time (e.g. by induction)
Description: measures and use
• Mathematical description  statistics
• Sets of numbers (1-dimensional point set)
• Describe with:
  - average
  - range [min,max]
  - variance, standard deviation
  - histogram (by fixed-interval classification)

                                1 n
                            2   ( xi  xmean ) 2
                                n i 1
       Geometric description
• Description of a point, polyline, polygon
  - location               - size
  - orientation            - shape

• Description of a point set
  - clustering      - density

• Description of distance
  - two points             - two polylines
  - two polygons

• Description of similarity in shape
  - two polylines          - two polygons
      Description of location
• Description of object as two coordinates (point)
• For a point: trivial
• Center of mass: for polyline or polygon;
  can lie outside polygon
• Center of largest inscribed circle:
  for polygon
      Applications of location
• Symbolizing a city as point object during map
  generalization
• Place to put information (name) of polygon
• Preparation for cluster analysis type that only
  applies to point (set)
          Description of size
• For point: not applicable
• For polyline: length
• For polygon: area or diameter or width




Diameter: largest        Width: smallest distance
distance between two     between two parallel lines
points in the object     that contain the object
         Applications of size
• Length polyline for network analysis (shortest
  routes)
• Area preservation during map generalisation
• Cartograms
    Description of orientation
• For polyline:
  - direction of vector
    between endpoints
• For polygon:
  - direction of diagonal
  - direction of parallel lines that define width
  - direction of line that minimizes the average
  distance of all points in polygon to that line
   Description of shape by a
            number
• Elongatedness: 1.27 * area / diameter2

• Compactness: 0.32 * area / (radius smallest
  enclosing circle)2
• Circularity: 12.6 * area / perimeter2



            All between 0 and 1
      Application of shape by a
               number
• Analysis of voting districts:
  when making districts for
  voting and representatives,
  the areas must have “nice”
  shape to group shared
  interests of the population
  (zoning = making districts)
     Description of shape by
            skeleton
• Internal skeleton, also: medial axis, or
  centerline
• Voronoi diagram of the edges of the polygon
     Applications of shape by
             skeleton
• Replacement of lakes in a river by a linear
  object during map generalisation
• Also: centers of roads, for network analysis
       Computation skeleton
• Voronoi diagram of line segments; adaptation of
  sweep-line algorithm for Voronoi diagram of
  points
• Voronoi edges and vertices outside polygon can
  be removed later
• O(n log n) time for polygon with n vertices

• Note: the largest enclosed circle of the polygon
  has its center on a Voronoi vertex
Description of shape of polyline
• Also: boundary of polygon
• Sinuosity: total angular change
• Curviness: average change of angle
  (per unit of length)
• Number of inflection points
 Applications of shape polyline
• Generalisation of a road with hairpin turns, or a
  meandering river
• Suitability of a river segment for label placement
  along it

                              simplify
                     reduce

                                typify
     Description of point sets
• Clustering
• Density

• Examples of point sets to be analyzed
  – epicenters of earthquakes
  – occurrences of road
    accidents in a city
  – burglary locations




           proportional
           symbol map
    Description of a point set
• Clustering: even, random, clustered distribution




 E.g., compare actual nearest neighbor distance
 with nearest neighbor distance of random set
    Description of a point set
• Density: scale-dependent, locally defined




                             point count inside a
                             square or circle;
                             size determines the
                             scale of interest
         Example of density

• Population
  density
Distance between two objects
• For two arbitrary subsets of the plane: smallest
  Euclidean distance
• Average smallest Euclidean distance
    Applications of distance
• Smallest distance: overlap when drawing lines
  with thickness
• Average distance: part of a measure for
  cultural influence received by one area from
  another area
     Description of similarity
• Hausdorff distance: for any two subsets of the
  plane

           A
          B


Max ( max min dist(a,b) , max min dist(b,a) )
       aA bB                bB aA

               AB                   BA
     Description of similarity
• Area of symmetric difference: for polygons
  = complement of area of intersection
     Application of similarity
• Detection of change between two maps with
  same theme but different date
  For example:
  - land use in Zeeland in 1975 and 2005
  - area of lost forest and newly grown forest

• Joint occurrence of two values in different
  themes (co-location)
  For example, forest type and geological soil type
     Computation Hausdorff
           distance
• Where can largest distance from A to B occur?

      A                A

           B                   B
    Vertex of A     Point internal to edge of A

                           
            In this case, the minimum distance
            must be attained from that point on
            A to two places on B
      Computation Hausdorff
            distance
• Vertex of A that minimizes distance to B:
  - Compute Voronoi diagram of edges of B
  - Preprocess for planar point location
  - Query with ever vertex of A to find the closest
    point to B and the distance to it

                                                  A
|A| = n
            O(m log m + n log m) time                  B
|B| = m
                                                Vertex of A
      Computation Hausdorff
            distance
  A           Point internal to
              edge of A
                        Smallest distance to B
          B             attained at two places

• Compute Voronoi diagram of the edges of B
• Compute intersection points of the edges of A
  with the Voronoi edges of B
• Compute intersection point on A with
  maximum smallest distance to B
      Computation Hausdorff
            distance
• Worst case: (nm) intersection points
  between A and the Voronoi diagram of B,
  then O(nm log (nm)) time
• Typical: O(n+m) intersection points, then the
  algorithm takes O((n+m) log (n+m)) time

• Inclusive of distance B to A and taking
  maximum: O((n+m) log (n+m)) time
      Computation area of
      symmetric difference
• Perform map overlay (boolean operation) on
  the two polygons
• Compute area of symmetric difference of the
  polygons and add up

• Worst case: O(nm log (nm)) time
• Typical case: O((n+m) log (n+m)) time
                  Summary

• There are many possible size and shape
  descriptors (measures) for geometric objects
• Different descriptors have different applications;
  the options should always be studied
• Computation of some measures/descriptors
  requires advanced algorithms