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Cartograms
Published in Terry A. Slocum, Robert B. McMaster, Fritz C. Kessler, Hugh H. Howard, Thematic Cartography and Geovisualization, 2022
Terry A. Slocum, Robert B. McMaster, Fritz C. Kessler, Hugh H. Howard
A rectilinear cartogram is a generalization of the rectangular cartogram in which enumeration units are represented by rectilinear polygons. A rectilinear polygon can be defined in two ways: (1) the line segments forming the polygon are all at right angles (the angle at each vertex is either 90° or 270°), or (2) the line segments forming the polygon are all parallel to either the x or y axis. If a rectilinear polygon is composed of only four line segments, then the resulting polygon is a rectangle.
Roof model recommendation for complex buildings based on combination rules and symmetry features in footprints
Published in International Journal of Digital Earth, 2018
Xuke Hu, Hongchao Fan, Alexey Noskov
We assume that building footprints consisting of rectangle structures are quite common in real environments. In order to achieve reasonable partitions from footprints, an advanced MNC partitioning algorithm is used. A reasonable partition means each rectangle in a partition correctly corresponds to a roof primitive of a true building roof. We assume that the horizontal and vertical edges of rectilinear polygons are parallel to x-axis and y-axis, respectively. The conventional MNC algorithm (Wu and Sahni 1994) consists of the following six steps: Identify the concave vertexes of a rectilinear polygon.Generate vertical and horizontal chords by connecting two concave vertexes that have an equal x and y coordinate, respectively. Due to the noise in footprints, we define that two x or y coordinates are equal when their difference is below a threshold of 0.3 m. The concave vertex whose x or y coordinate is unequal to the x or y coordinate of any other concave vertexes is called free concave vertex.Obtain a maximum matching (MM) by using the Hungarian algorithm (Kuhn 1956). We treat vertical and horizontal chords as the left and right parts of a bipartite graph, respectively, where each chord is denoted by an endpoint. Then, we add an edge to connect two chords if they are intersected. Therefore, a bipartite graph can be denoted by , where H, V, and E represent horizontal chords, vertical chords, and the edges among them, respectively. A matching in a bipartite graph is a set of edges chosen in such a way that no two edges share an endpoint. An MM is a matching of a maximum number of edges.Find a maximum independent set (MIS) from the bipartite graph (Wu and Sahni 1994). An independent set in a graph is a set of endpoints chosen in such a way that no two endpoints are connected by an edge. An MIS is an independent set of maximum number of endpoints.Draw the chords in MIS to partition the polygon.Draw a horizontal or vertical line segment from the free concave vertexes to the nearest edge.