An isothetic polygon is a polygon whose alternate sides belong to two parametric families of straight lines which are pencils of lines with centers at two points (possibly the point at

infinity Infinity is that which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol . Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions amo ...

). The most well-known example of isothetic polygons are rectilinear polygons, and the former term is commonly used as a

synonym A synonym is a word, morpheme, or phrase that means exactly or nearly the same as another word, morpheme, or phrase in a given language. For example, in the English language, the words ''begin'', ''start'', ''commence'', and ''initiate'' are all ...

for the latter one.

Etymology and history

The term is produced from Greek roots: '' iso-'' for "equal, same, similar" and (position, placement), i.e., the term is supposed to mean "polygon with similarly placed sides". The term was suggested during the early years of the

computational geometry Computational geometry is a branch of computer science devoted to the study of algorithms which can be stated in terms of geometry. Some purely geometrical problems arise out of the study of computational geometric algorithms, and such problems ar ...

. Much emphasis was placed on the development of efficient algorithms for operations with orthogonal polygons, since the latter ones had an important application: representation of shapes in

integrated circuit An integrated circuit or monolithic integrated circuit (also referred to as an IC, a chip, or a microchip) is a set of electronic circuits on one small flat piece (or "chip") of semiconductor material, usually silicon. Large numbers of tiny ...

mask layouts due to their simplicity for design and manufacturing. It was observed that the efficiency of many geometric algorithms for orthogonal polygons does not really depend on the fact that their sides meet at right angles, but rather on the fact that their sides are naturally split into two alternating sets (of vertical and horizontal segments). ---- ----

Sets of isothetic polygons

In many applications of computational geometry, when a problem is stated for a set of rectilinear polygons, it is very often implicitly assumed that these polygons have the same alignment (in fact, aligned to the same orthogonal coordinate axes), and hence the term "isothetic polygons" would be less ambiguous. In the context of digital geometry, isothetic polygons are practically axis-parallel and have integer coordinates of their vertices.

References

*, chapter 8: "The Geometry of Rectangles" * * *{{cite journal , author1=Arindam Biswas , author2=Partha Bhowmick , author3=Bhargab B. Bhattacharya , title = Construction of Isothetic Covers of a Digital Object: A Combinatorial Approach , journal = Journal of Visual Communication and Image Representation , year = 2010 , volume = 21 , issue=4 , pages = 295–310, doi=10.1016/j.jvcir.2010.02.001 Types of polygons