Boolean operations on polygons are a set of Boolean operations (AND, OR, NOT, XOR, ...) operating on one or more sets of polygons in computer graphics. These sets of operations are widely used in computer graphics, CAD, and in EDA (in integrated circuit physical design and verification software).
Early algorithms for Boolean operations on polygons were based on the use of bitmaps. Using bitmaps in modeling polygon shapes has many drawbacks. One of the drawbacks is that the memory usage can be very large, since the resolution of polygons is proportional to the number of bits used to represent polygons. The higher the resolution is desired, the more the number of bits is required.
Modern implementations for Boolean operations on polygons tend to use plane sweep algorithms (or Sweep line algorithms). A list of papers using plane sweep algorithms for Boolean operations on polygons can be found in References below.
Boolean operations on convex polygons and monotone polygons of the same direction may be performed in linear time. [1]
In computer science, binary space partitioning (BSP) is a method for space partitioning which recursively subdivides a Euclidean space into two convex sets by using hyperplanes as partitions. This process of subdividing gives rise to a representation of objects within the space in the form of a tree data structure known as a BSP tree.
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 are also considered to be part of computational geometry. While modern computational geometry is a recent development, it is one of the oldest fields of computing with a history stretching back to antiquity.
Constructive solid geometry is a technique used in solid modeling. Constructive solid geometry allows a modeler to create a complex surface or object by using Boolean operators to combine simpler objects, potentially generating visually complex objects by combining a few primitive ones.
Solid modeling is a consistent set of principles for mathematical and computer modeling of three-dimensional shapes (solids). Solid modeling is distinguished from related areas of geometric modeling and computer graphics, such as 3D modeling, by its emphasis on physical fidelity. Together, the principles of geometric and solid modeling form the foundation of 3D-computer-aided design and in general support the creation, exchange, visualization, animation, interrogation, and annotation of digital models of physical objects.
In 3D computer graphics, solid objects are usually modeled by polyhedra. A face of a polyhedron is a planar polygon bounded by straight line segments, called edges. Curved surfaces are usually approximated by a polygon mesh. Computer programs for line drawings of opaque objects must be able to decide which edges or which parts of the edges are hidden by an object itself or by other objects, so that those edges can be clipped during rendering. This problem is known as hidden-line removal.
In computational geometry, polygon triangulation is the partition of a polygonal area P into a set of triangles, i.e., finding a set of triangles with pairwise non-intersecting interiors whose union is P.
In geometry, a simple polygon is a polygon that does not intersect itself and has no holes. That is, it is a flat shape consisting of straight, non-intersecting line segments or "sides" that are joined pairwise to form a single closed path. If the sides intersect then the polygon is not simple. The qualifier "simple" is frequently omitted, with the above definition then being understood to define a polygon in general.
Clipping, in the context of computer graphics, is a method to selectively enable or disable rendering operations within a defined region of interest. Mathematically, clipping can be described using the terminology of constructive geometry. A rendering algorithm only draws pixels in the intersection between the clip region and the scene model. Lines and surfaces outside the view volume are removed.
In geometry, a bitangent to a curve C is a line L that touches C in two distinct points P and Q and that has the same direction as C at these points. That is, L is a tangent line at P and at Q.
In computational geometry, the multiple line segment intersection problem supplies a list of line segments in the Euclidean plane and asks whether any two of them intersect (cross).
In geometry, a set K ⊂ Rd is defined to be orthogonally convex if, for every line L that is parallel to one of standard basis vectors, the intersection of K with L is empty, a point, or a single segment. The term "orthogonal" refers to corresponding Cartesian basis and coordinates in Euclidean space, where different basis vectors are perpendicular, as well as corresponding lines. Unlike ordinary convex sets, an orthogonally convex set is not necessarily connected.
In computational geometry, a sweep line algorithm or plane sweep algorithm is an algorithmic paradigm that uses a conceptual sweep line or sweep surface to solve various problems in Euclidean space. It is one of the key techniques in computational geometry.
In geometry, visibility is a mathematical abstraction of the real-life notion of visibility.
The Sutherland–Hodgman algorithm is an algorithm used for clipping polygons. It works by extending each line of the convex clip polygon in turn and selecting only vertices from the subject polygon that are on the visible side.
In computational geometry, the Bentley–Ottmann algorithm is a sweep line algorithm for listing all crossings in a set of line segments, i.e. it finds the intersection points of line segments. It extends the Shamos–Hoey algorithm, a similar previous algorithm for testing whether or not a set of line segments has any crossings. For an input consisting of line segments with crossings, the Bentley–Ottmann algorithm takes time . In cases where , this is an improvement on a naïve algorithm that tests every pair of segments, which takes .
The Vatti clipping algorithm is used in computer graphics. It allows clipping of any number of arbitrarily shaped subject polygons by any number of arbitrarily shaped clip polygons. Unlike the Sutherland–Hodgman and Weiler–Atherton polygon clipping algorithms, the Vatti algorithm does not restrict the types of polygons that can be used as subjects or clips. Even complex (self-intersecting) polygons, and polygons with holes can be processed. The algorithm is generally applicable only in 2D space.
The Greiner-Hormann algorithm is used in computer graphics for polygon clipping. It performs better than the Vatti clipping algorithm, but cannot handle degeneracies. It can process both self-intersecting and non-convex polygons. It can be trivially generalized to compute other Boolean operations on polygons, such as union and difference.
In computer graphics, A-buffer, also known as anti-aliased, area-averaged or accumulation buffer, is a general hidden surface mechanism suited to medium scale virtual memory computers. It resolves visibility among an arbitrary collection of opaque, transparent, and intersecting objects. Using an easy to compute Fourier window, it increases the effective image resolution many times over the Z-buffer, with a moderate increase in cost.
This is a glossary of terms relating to computer graphics.