Two-dimensional space

In physics and mathematics, two-dimensional space or bi-dimensional space is a geometric model of the planar projection of the physical universe. The two dimensions are commonly called length and width. Both directions lie in the same plane.

A sequence of n real numbers can be understood as a location in n-dimensional space. When n = 2, the set of all such locations is called two-dimensional space or bi-dimensional space, and usually is thought of as a Euclidean space.

History

Books I through IV and VI of Euclid's Elements dealt with two-dimensional geometry, developing such notions as similarity of shapes, the Pythagorean theorem (Proposition 47), equality of angles and areas, parallelism, the sum of the angles in a triangle, and the three cases in which triangles are "equal" (have the same area), among many other topics.

Later, the plane was described in a so-called Cartesian coordinate system, a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length. Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin, usually at ordered pair (0, 0). The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin.

Geometry

Geometry (from the Ancient Greek: γεωμετρία; geo- "earth", -metron "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer. Geometry arose independently in a number of early cultures as a body of practical knowledge concerning lengths, areas, and volumes, with elements of formal mathematical science emerging in the West as early as Thales (6th century BC). By the 3rd century BC, geometry was put into an axiomatic form by Euclid, whose treatment—Euclidean geometry—set a standard for many centuries to follow.Archimedes developed ingenious techniques for calculating areas and volumes, in many ways anticipating modern integral calculus. The field of astronomy, especially as it relates to mapping the positions of stars and planets on the celestial sphere and describing the relationship between movements of celestial bodies, served as an important source of geometric problems during the next one and a half millennia. In the classical world, both geometry and astronomy were considered to be part of the Quadrivium, a subset of the seven liberal arts considered essential for a free citizen to master.

Geometry (Ivo Perelman album)

Geometry is an album by Brazilian jazz saxophonist Ivo Perelman featuring American pianist Borah Bergman, which was recorded in 1996 and released on the English Leo label.

Reception

In his review for AllMusic, Alex Henderson says that "this CD doesn't quite fall into the 'essential' category... Nonetheless, Geometry is an enjoyable release that Perelman's more-devoted followers will want."

The Penguin Guide to Jazz notes that "Bergman is wily enough to find ways of both supporting and undercutting the mighty sound of the tenor."

Track listing

Personnel

References

Geometric group action

In mathematics, specifically geometric group theory, a geometric group action is a certain type of action of a discrete group on a metric space.

Definition

In geometric group theory, a geometry is any proper, geodesic metric space. An action of a finitely-generated group G on a geometry X is geometric if it satisfies the following conditions:

Uniqueness

If a group G acts geometrically upon two geometries X and Y, then X and Y are quasi-isometric. Since any group acts geometrically on its own Cayley graph, any space on which G acts geometrically is quasi-isometric to the Cayley graph of G.

Examples

Cannon's conjecture states that any hyperbolic group with a 2-sphere at infinity acts geometrically on hyperbolic 3-space.

References

Planar

Planar may refer to:

Science and technology

Other

See also

Planar Systems

Planar Systems, Inc. is a U.S. digital display manufacturing corporation based in Hillsboro, Oregon. Founded in 1983 as a spin-off from Tektronix, it was the first U.S. manufacturer of electroluminescent (EL) digital displays. Planar currently makes a variety of other specialty displays. The company, with $166.8 million in revenue (2013), is headed by chief executive officer and president Gerald K. Perkel. It is a subsidiary of Leyard Optoelectronic Co., a Chinese company.

History

1980s

Planar was founded in 1983 by Jim Hurd, Chris King, John Laney and others as a spin-off from the Solid State Research and Development Group of the Beaverton, Oregon, based Tektronix. In 1986, a division spun off from Planar to work on projection technology, InFocus.

1990s

In 1991, FinLux, a competitor in Espoo, Finland was purchased and is now the company's European headquarters. Planar's executives took the company public in 1993, listing the stock on the NASDAQ boards Planar acquired Tektronix's avionics display business, creating the short-lived Planar Advance in 1994. Standish Industries, a manufacturer of flat panel LCDs in Lake Mills, Wisconsin, was sold to Planar in 1997. This plant was closed in 2002 as worldwide LCD manufacturing shifted to East Asian countries.

Planar (computer graphics)

In computer graphics, planar is the method of representing pixel colours with several bitplanes of RAM. Each bit in a bitplane is related to one pixel on the screen. Unlike Chunky, Highcolour or Truecolour graphics, the whole data for an individual pixel isn't in one specific location in RAM, but spread across the bitplanes that make up the display.

This scheme originated in the early days of home computing, when cheaper memory chips could not supply data fast enough on their own to generate a picture on a TV screen or monitor. By splitting the data up into multiple planes, each plane could be stored on a separate memory chip. These chips could then be read in parallel at a slower rate, allowing graphical display on modest hardware. The CGA and EGA video adapters on early IBM PC computers used planar arrangement in colour graphical modes for this reason. When the later VGA was introduced, memory speeds had increased to the point where planar video modes were no longer a necessity, and aside from backwards-compatibility, using planar mode or not was left as a choice for the programmer.