Rectilinear polygon

Some examples of rectilinear polygons

A rectilinear polygon is a polygon all of whose edges meet at right angles. Thus the interior angle at each vertex is either 90° or 270°. Rectilinear polygons are a special case of isothetic polygons.

In many cases another definition is preferable: a rectilinear polygon is a polygon with sides parallel to the axes of Cartesian coordinates. The distinction becomes crucial when spoken about sets of polygons: the latter definition would imply that sides of all polygons in the set are aligned with the same coordinate axes. Within the framework of the second definition it is natural to speak of horizontal edges and vertical edges of a rectilinear polygon.

Rectilinear polygons are also known as orthogonal polygons. Other terms in use are iso-oriented, axis-aligned, and axis-oriented polygons. These adjectives are less confusing when the polygons of this type are rectangles, and the term axis-aligned rectangle is preferred, although orthogonal rectangle and rectilinear rectangle are in use as well.

The importance of the class of rectilinear polygons comes from the following.

• They are convenient for the representation of shapes in integrated circuit mask layouts due to their simplicity for design and manufacturing. Many manufactured objects result in orthogonal polygons.
• Problems in computational geometry stated in terms polygons often allow for more efficient algorithms when restricted to orthogonal polygons. An example is provided by the art gallery theorem for orthogonal polygons, which leads to more efficient guard coverage than is possible for arbitrary polygons.

Properties

• The numbers of vertical and horizontal edges of a rectilinear polygon are equal.
  • Corollary: Orthogonal polygons have an even number of edges.
• The number of 270° interior angles in a simple orthogonal polygon is four less than the number of 90° interior angles.
  • Corollary: any rectilinear polygon has at least four 90° interior angles.

Special cases and generalizations

• orthogonally convex rectilinear polygon
• Axis-aligned rectangle
  • Minimum bounding rectangle
• Monotone rectilinear polygon, a monotone polygon which is also rectilinear
• Rectilinear polygon with (rectilinear) holes
• Rectilinear polyhedron
• Rectilinearity [1]

See also orthogonal polyhedra (under polyhedron, "Other important families of polyhedra"), the natural generalization of orthogonal polygons to 3D.

Algorithmic problems involving rectilinear polygons

Most of them may be stated for general polygons as well, but expectation of more efficient algorithms warrants a separate consideration

• Orthogonal range searching
• Orthogonal convex hull construction
• Boolean operations/Boolean expressions for orthogonal polygons (e.g., intersection and union)
• Motion planning/path planning/routing among restilinear obstacles
• Visibility problems (Illumination problems)
  • Rectilinear art gallery problems
• Rectangular decomposition (partition/packing/covering with rectangles)
• Maximal empty rectangle

References

• Franco P. Preparata and Michael Ian Shamos (1985). Computational Geometry - An Introduction. Springer. 1st edition: ISBN 0-387-96131-3; 2nd printing, corrected and expanded, 1988: ISBN 3-540-96131-3. , chapter 8: "The Geometry of Rectangles"