geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, visibility is a mathematical

abstraction Abstraction in its main sense is a conceptual process wherein general rules and concepts are derived from the usage and classification of specific examples, literal ("real" or "concrete") signifiers, first principles, or other methods. "An abstr ...

of the real-life notion of visibility. Given a set of obstacles in the

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...

, two points in the space are said to be visible to each other, if the

line segment In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between ...

that joins them does not intersect any obstacles. (In the

Earth's atmosphere The atmosphere of Earth is the layer of gases, known collectively as air, retained by Earth's gravity that surrounds the planet and forms its planetary atmosphere. The atmosphere of Earth protects life on Earth by creating pressure allowing for ...

light follows a slightly curved path that is not perfectly predictable, complicating the calculation of actual visibility.) Computation of visibility is among the basic problems in

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 ...

and has applications in

computer graphics Computer graphics deals with generating images with the aid of computers. Today, computer graphics is a core technology in digital photography, film, video games, cell phone and computer displays, and many specialized applications. A great de ...

motion planning Motion planning, also path planning (also known as the navigation problem or the piano mover's problem) is a computational problem to find a sequence of valid configurations that moves the object from the source to destination. The term is used ...

, and other areas.

Concepts and problems

* Point visibility * Edge visibilityE. Roth, G. Panin and A. Knoll,
Sampling feature points for contour tracking with graphics hardware
, "In International Workshop on Vision, Modeling and Visualization (VMV)", Konstanz, Germany, October 2008. * Visibility polygon * Weak visibility *

Art gallery problem The art gallery problem or museum problem is a well-studied visibility problem in computational geometry. It originates from the following real-world problem: In the geometric version of the problem, the layout of the art gallery is represente ...

or museum problem *

Visibility graph In computational geometry and robot motion planning, a visibility graph is a graph of intervisible locations, typically for a set of points and obstacles in the Euclidean plane. Each node in the graph represents a point location, and each edge repr ...

** Visibility graph of vertical line segments * Watchman route problem *Computer graphics applications: **

Hidden surface determination In 3D computer graphics, hidden-surface determination (also known as shown-surface determination, hidden-surface removal (HSR), occlusion culling (OC) or visible-surface determination (VSD)) is the process of identifying what surfaces and parts o ...

Hidden line removal 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 ...

z-buffering A depth buffer, also known as a z-buffer, is a type of data buffer used in computer graphics to represent depth information of objects in 3D space from a particular perspective. Depth buffers are an aid to rendering a scene to ensure that the ...

portal engine In computer-generated imagery and real-time 3D computer graphics, portal rendering is an algorithm for visibility determination. For example, consider a 3D computer game environment, which may contain many polygons, only a few of which may be vi ...

Star-shaped polygon In geometry, a star-shaped polygon is a polygonal region in the plane that is a star domain, that is, a polygon that contains a point from which the entire polygon boundary is visible. Formally, a polygon is star-shaped if there exists a poi ...

**Kernel of a polygon *

Isovist A single isovist is the volume of space visible from a given point in space, together with a specification of the location of that point. It is a geometric concept coined by Clifford Tandy in 1967 and further refined by the architect Michael Bene ...

Viewshed A viewshed is the geographical area that is visible from a location. It includes all surrounding points that are in line-of-sight with that location and excludes points that are beyond the horizon or obstructed by terrain and other features (e.g. ...

* Zone of Visual Influence *

Painter's algorithm The painter’s algorithm (also depth-sort algorithm and priority fill) is an algorithm for visible surface determination in 3D computer graphics that works on a polygon-by-polygon basis rather than a pixel-by-pixel, row by row, or area by are ...

References

* * * Chapter 15: "Visibility graphs"

External links

Software

VisiLibity: A free open source C++ library of floating-point visibility algorithms and supporting data types
Geometry Geometric algorithms {{geometry-stub