A2Ddigital picture consists of a finite number of pixels, each of which is defined by a location and a value at that location. The term pixel is short for “picture element”; this acronym was introduced in the late 1960s by a group at Jet Propulsion Laboratory in Pasadena, California, that was processing pictures taken by space vehicles [640]. The analogous 3D term is “voxel,” which is short for “volume element.”
We usually assume that objects are represented by connected sets of pixels or voxels and that the quantitative information involves quantities studied in Euclidean or similarity geometry (see Section 1.2.2); we can then attempt to ensure that the properties computed in digital geometry adequately approximate these quantities. In this sense, we can regard digital geometry as digitized similarity geometry. As we will see in Section 1.2.2, Euclidean geometry is a special case of similarity geometry. Digital geometry also often attempts to obtain topologic characterizations of pictures or to transform pictures into “simpler” topologically equivalent pictures. Due to the discrete nature of digital geometry, these topologic problems belong to the field of combinatorial topology (the topology of cell complexes).
The concept of defining the locations of points in a plane by their distances from two straight lines (“axes”) was used by Archimedes and Apollonius more than 2000 years ago. A Cartesian coordinate system makes use of a set of axes as introduced by R. Descartes (in Latin: Cartesius, 1596–1650) in [264] to define nonnegative coordinates in the plane.
For example, barycentric coordinates (homogeneous or triangular coordinates) in the plane were introduced by Möbius in 1827 [807] as a way of representing points in the plane relative to a given triple of noncollinear points.
Geometria perspectiva, aparuta in secolul 15, permite cartografierea elemetelor dintr-o multime in alta.
Teoria grafurilor se poate aplica pe voxeli asa cum gridul se aplica pe pixeli.
J.B. Listing (1802–1882) was the first to use the word “topology” in his correspondence, beginning in 1837. The term, which replaced Leibniz’s “geometria situs” or “analysis situs,” was introduced to distinguish “qualitative geometry” from geometric topics that emphasized quantitative measurements and relations. Topology can be informally viewed as “rubber-sheet geometry”: the study of properties of objects that remain the same when the objects are (continuously) deformed.
Computational geometry deals with finite collections of simple geometric objects (e.g., points, lines, circles) in Euclidean space. It studies algorithms for solving problems about such collections and the complexity of applying the algorithms as the number of objects increases. The phrase “computational geometry” was first used in the title of a 1969 book [733] about property computation, then in the early 1970s for geometric modeling by means of spline curves and surfaces [378], and finally in the mid-1970s [974] with the meaning that it has today.
In the last third of the 19th century, H. Poincaré and others established topology as a branch of modern mathematics. Point-set topology studies topologic spaces. In early publications about topology, the underlying set S of a topologic space was a Euclidean space, but, in modern topology, it can be an abstract set.
Concepte topologice :
Combinatorial topology studies partitions of objects into “complexes.” Apolyhedronis a finite union of simplexes. […] Apartition into simplexes or convex sets defines a geometric complex. The study of such partitions is of central interest in combinatorial topology.
A Euclidean complex is called simplicial iff all of its cells are simplexes.
A finite Euclidean complex that contains only triangles, line segments, and points is called a triangulation; evidently, such a complex is simplicial.
Diagrams are defined in metric spaces for countable sets S of “simple” geometric objects such as points, line segments, polygons, polyhedra, and so on. One type of diagram divides the metric space into cells such that each element of S is contained in exactly one cell.
(on voronoi) This is named after the Ukrainian mathematician G.F. Voronoi (1868–1908). An n-dimensional generalization was studied by the German mathematician J.P.G.L. Dirichlet (1805–1859) [273]. The U.S. officer A.H. Thiessen also used such polygonal cells in the discussion of meteorologic data; see, for example, [1050]. Voronoi cells are therefore also called Thiessen polygons and are the 2D case of Dirichlet polyhedra or Dirichlet cells.
Pag 443 – algoritm pentru delauney.
Nota : D(p,q,r) = x1y2+x3y1+x2y3−x3y2−x2y1−x1y3 – arata ordinea punctelor – daca plus sau minus.
A wide variety of geometries have been developed, motivated by a wide variety of applications: Euclidean (Thales of Miletus, Hippocrates of Chios, the secret society of the Pythagoreans, Euclid, Archimedes); analytic (Descartes, also known as Cartesius); perspective (Alberti, da Vinci, della Francesca, Dürer); projective (Desargues, Pascal); descriptive (Monge); non-Euclidean, such as elliptic and hyperbolic (Lobachevsky, Bólyai, Riemann); and combinatorial (Helly, Borsuk, Erdõs).