In geometry, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of a point or other geometric element on a manifold such as Euclidean space.^[1][2] The order of the coordinates is significant and they are sometimes identified by their position in an ordered tuple and sometimes by a letter, as in 'the x-coordinate'. In elementary mathematics the coordinates are taken to be real numbers, but may be complex numbers or elements of a more abstract system such as a commutative ring. The use of a coordinate system allows problems in geometry to be translated into problems about numbers and vice versa; this is the basis of analytic geometry.^[3]

An example in everyday use is the system of assigning longitude and latitude to geographical locations. In physics, a coordinate system used to describe points in space is called a frame of reference.

Number line[link]

The simplest example of a coordinate system is the identification of points on a line with real numbers using the number line. In this system, an arbitrary point O (the origin) is chosen on a given line. The coordinate of a point P is defined as the signed distance from O to P, where the signed distance is the distance taken as positive or negative depending on which side of the line P lies. Each point is given a unique coordinate and each real number is the coordinate of a unique point.^[4]

Cartesian coordinate system[link]

The Cartesian coordinate system in the plane.

The prototypical example of a coordinate system is the Cartesian coordinate system. In the plane, two perpendicular lines are chosen and the coordinates of a point are taken to be the signed distances to the lines.

In three dimensions, three perpendicular planes are chosen and the three coordinates of a point are the signed distances to each of the planes. This can be generalized to create n coordinates for any point in n-dimensional Euclidean space.

Polar coordinate system[link]

The Polar coordinate system in the plane.

Another common coordinate system for the plane is the Polar coordinate system. A point is chosen as the pole and a ray from this point is taken as the polar axis. For a given angle θ, there is a single line through the pole whose angle with the polar axis is θ (measured counterclockwise from the axis to the line). Then there is a unique point on this line whose signed distance from the origin is r for given number r. For a given pair of coordinates (r, θ) there is a single point, but any point is represented by many pairs of coordinates. For example (r, θ), (r, θ+2π) and (−r, θ+π) are all polar coordinates for the same point. The pole is represented by (0, θ) for any value of θ.

Cylindrical and spherical coordinate systems[link]

There are two common methods for extending the polar coordinate system to three dimensions. In the cylindrical coordinate system, a z-coordinate with the same meaning as in Cartesian coordinates is added to the r and θ polar coordinates. Spherical coordinates take this a step further by converting the pair of cylindrical coordinates (r, z) to polar coordinates (ρ, φ) giving a triple (ρ, θ, φ)

Homogeneous coordinate system[link]

A point in the plane may be represented in homogeneous coordinates by a triple (x, y, z) where x/z and y/z are the Cartesian coordinates of the point. This introduces an "extra" coordinate since only two are needed to specify a point on the plane, but this system is useful in that it represents any point on the projective plane without the use of infinity. In general, a homogeneous coordinate system is one where only the ratios of the coordinates are significant and not the actual values.

Coordinates of other elements[link]

Coordinates systems are often used to specify the position of a point, but they may also be used to specify the position of more complex figures such as lines, planes, circles or spheres. For example Plücker coordinates are used to determine the position of a line in space. When there is a need, the type of figure being described is used to distinguish the type of coordinate system, for example the term line coordinates is used for any coordinate system that specifies the position of a line.

It may occur that systems of coordinates for two different sets of geometric figures are equivalent in terms of their analysis. An example of this is the systems of homogeneous coordinates for points and lines in the projective plane. The two systems in a case like this are said to be dualistic. Dualistic systems have the property that results from one system can be carried over to the other since these results are only different interpretations of the same analytical result; this is known as the principle of duality.^[5]

Transformations between coordinate systems[link]

Because there are often many different possible coordinate systems for describing geometrical figures, it is important to understand how they are related. Such relations are described by coordinate transformations which give formulas for the coordinates in one system in terms of the coordinates in another system. For example, in the plane, if Cartesian coordinates (x, y) and polar coordinates (r, θ) have the same origin, and the polar axis is the positive x axis, then the coordinate transformation from polar to Cartesian coordinates is given by x = r cosθ and y = r sinθ.

[edit] Coordinate curves and surfaces

Coordinate surfaces in the Spherical coordinate system

In two dimensions if all but one coordinate in a point coordinate system is held constant and the remaining coordinate is allowed to vary, then the resulting curve is called a coordinate curve (some authors use the phrase "coordinate line"). This procedure does not always make sense, for example there are no coordinate curves in a homogeneous coordinate system. In the Cartesian coordinate system the coordinate curves are, in fact, lines. Specifically, they are the lines parallel to one of the coordinate axes. For other coordinate systems the coordinates curves may be general curves. For example the coordinate curves in polar coordinates obtained by holding r constant are the circles with center at the origin. Coordinates systems for Euclidean space other than the Cartesian coordinate system is called curvilinear coordinate systems.^[6]

In three dimensional space, if one coordinate is held constant and the remaining coordinates are allowed to vary, then the resulting surface is called a coordinate surface. For example the coordinate surfaces obtained by holding ρ constant in the spherical coordinate system are the spheres with center at the origin. In three dimensional space the intersection of two coordinate surfaces is a coordinate curve. Coordinate hypersurfaces are defined similarly in higher dimensions.^[7]

Coordinate maps[link]

The concept of a coordinate map, or chart is central to the theory of manifolds. A coordinate map is essentially a coordinate system for a subset of a given space with the property that each point has exactly one set of coordinates. More precisely, a coordinate map is a homeomorphism from an open subset of a space X to an open subset of Rⁿ. It is often not possible to provide one consistent coordinate system for an entire space. In this case, a collection of coordinate maps are put together to form an atlas covering the space. A space equipped with such an atlas is called a manifold and additional structure can be defined on a manifold if the structure is consistent where the coordinate maps overlap. For example a differentiable manifold is a manifold where the change of coordinates from one coordinate map to another is always a differentiable function.

Change of coordinates[link]

In geometry and kinematics, coordinate systems are used not only to describe the (linear) position of points, but also to describe the angular position of axes, planes, and rigid bodies. In the latter case, the orientation of a second (typically referred to as "local") coordinate system, fixed to the node, is defined based on the first (typically referred to as "global" or "world" coordinate system). For instance, the orientation of a rigid body can be represented by an orientation matrix, which includes, in its three columns, the Cartesian coordinates of three points. These points are used to define the orientation of the axes of the local system; they are the tips of three unit vectors aligned with those axes.

Transformations[link]

A coordinate transformation is a conversion from one system to another, to describe the same space.

With every bijection from the space to itself two coordinate transformations can be associated:

• such that the new coordinates of the image of each point are the same as the old coordinates of the original point (the formulas for the mapping are the inverse of those for the coordinate transformation)
• such that the old coordinates of the image of each point are the same as the new coordinates of the original point (the formulas for the mapping are the same as those for the coordinate transformation)

For example, in 1D, if the mapping is a translation of 3 to the right, the first moves the origin from 0 to 3, so that the coordinate of each point becomes 3 less, while the second moves the origin from 0 to -3, so that the coordinate of each point becomes 3 more.

Systems commonly used[link]

Some coordinate systems are the following:

• The Cartesian coordinate system (also called the "rectangular coordinate system"), which, for two- and three-dimensional spaces, uses two and three numbers (respectively) representing distances from the origin in three mutually perpendicular directions.
• Curvilinear coordinates are a generalization of coordinate systems generally; the system is based on the intersection of curves.
• Polar coordinate system represents a point in the plane by a distance from the origin and an angle measured from a reference line intersecting the origin.
• Log-polar coordinate system represents a point in the plane by the logarithm of the distance from the origin and an angle measured from a reference line intersecting the origin.
• Cylindrical coordinate system represents a point in three-space using two perpendicular axes; distance is measured along one axis, while the other axis formes the reference line for a polar coordinate representation of the remaining two components.
• Spherical coordinate system represents a point in three space by the distance from the origin and two angles measured from two reference lines which intersect the origin.
• Plücker coordinates are a way of representing lines in 3D Euclidean space using a six-tuple of numbers as homogeneous coordinates.
• Generalized coordinates are used in the Lagrangian treatment of mechanics.
• Canonical coordinates are used in the Hamiltonian treatment of mechanics.
• Parallel coordinates visualise a point in n-dimensional space as a polyline connecting points on n vertical lines.
• Barycentric coordinates (mathematics) as used for Ternary_plot

There are ways of describing curves without coordinates, using intrinsic equations that use invariant quantities such as curvature and arc length. These include:

• Whewell equation relates arc length and tangential angle.
• Cesàro equation relates arc length and curvature.

List of orthogonal coordinate systems[link]

In mathematics, two vectors are orthogonal if they are perpendicular. The following coordinate systems all have the properties of being orthogonal coordinate systems, that is the coordinate surfaces meet at right angles.

See also[link]

• Active and passive transformation
• Alpha-numeric grid
• Frame of reference
• Galilean transformation
• Coordinate-free
• Nomogram, graphical representations of different coordinate systems
• Geographic coordinate system
• Astronomical coordinate systems
• Axes conventions in engineering

References[link]

1. ^ Woods p. 1
2. ^ Weisstein, Eric W., "Coordinate System" from MathWorld.
3. ^ Weisstein, Eric W., "Coordinates" from MathWorld.
4. ^ Woods p. 8
5. ^ Woods p. 2
6. ^ Tang, K. T. (2006). Mathematical Methods for Engineers and Scientists. 2. Springer. p. 13. ISBN 3-540-30268-9. 
7. ^ Liseikin, Vladimir D. (2007). A Computational Differential Geometry Approach to Grid Generation. Springer. p. 38. ISBN 3-540-34235-4. 

• Voitsekhovskii, M.I.; Ivanov, A.B. (2001), "Coordinates", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4, http://www.encyclopediaofmath.org/index.php?title=C/c026470 
• Woods, Frederick S. (1922). Higher Geometry. Ginn and Co.. pp. 1ff. http://books.google.com/books?id=3ZULAAAAYAAJ&pg=PA1#v=onepage&q&f=false. 
• Shigeyuki Morita, Teruko Nagase, Katsumi Nomizu (2001). Geometry of Differential Forms. AMS Bookstore. p. 12. ISBN 0-8218-1045-6. http://books.google.com/books?id=5N33Of2RzjsC&pg=PA12. 

External links[link]

Look up coordinate in Wiktionary, the free dictionary.

• Hexagonal Coordinate System

This biographical article needs additional citations for verification. Please help by adding reliable sources. Contentious material about living persons that is unsourced or poorly sourced must be removed immediately, especially if potentially libelous or harmful. (May 2010)

The Basics
The Basics at the Oxford Art Factory, 2010
Background information
Origin	Melbourne, Australia
Genres	Rock
Years active	2002–present
Labels	Independent, Albert Records
Associated acts	Gotye, Down Hills Home, Dog With Wheels, Blood Red Bird
Website	The Basics Official Website
Members
Wally De Backer Kris Schroeder Tim Heath
Past members
Michael Hubbard
Notable instruments
Lead guitar, drums, bass guitar.

The Basics ( /ˈbæzɪtʃ/ BAZZ-ich) are a three-piece band from Melbourne, Australia, formed by Wally De Backer and Kris Schroeder in 2002. Their style has been described as anything from 'indie-pop' to 'rock'n'roll' to 'pop-rock', though their records show they span a wide range of genres, including reggae, ska, country, and electro-pop. They are "recognised as one of Australia's hardest-working bands".^[1]

With close to 1000 shows in their ten-year career, their live performances are well-known for their sense of humour and energy.^[2]

Formation[link]

The Basics were formed after Kris met Wally at a party in Melbourne.^[3] The event was to see off the young aspiring producer, who was about to depart for Los Angeles, and together with a number of musician friends, the two jammed on some blues-rock standards. Later that night, they bonded over a mutual love of The Beatles, 70s and 80s cartoon theme songs and old Sierra adventure games.

Initially, they played around Melbourne as an acoustic guitar/drums combo, starting at The Opposition in Frankston and the House of Fools in Footscray, where in late 2002 they met Michael Hubbard and later invited him to join them on electric guitar.^[4] Though lacking any real experience with the instrument, Kris willingly purchased his first bass guitar, and the trio was born. During this time, the group start performing songs with more complicated three-part harmony.

During late 2002 and early 2003, The Basics recorded and released an album called Get Back through MGM Distribution. They began what was to become regular tours of the East and West coasts of Australia.

2004-2005: Current lineup[link]

Tim Heath joined the group in June 2004, after Michael decided to pursue his solo career, which has included being one half of Down Hills Home.^[4] Tim was reportedly the only person to reply to an advertisement in Melbourne's Beat Magazine looking for a new guitarist. Tim apparently answered the advertisement - which said the applicant "must have a love of 50s and 60s rock'n'roll" - thinking that he would be meeting "a couple of old guys to play with on the weekends over some beers". They soon recorded For Girls Like You.

The Basics in 2005 continued to tour and play small festivals, and after releasing For Girls Like You through MGM Distribution, they departed for a national tour of Australia, which lasted for two months and took them to every Australian state and territory. Stuart Padbury, a young Melbourne sound engineer, joined them on much of this tour. On returning, the band began demonstrating songs for a new album.

2006-2009: Touring and soundtracks[link]

In 2006, The Basics began recording what was to become known as Stand Out/Fit In. Taking several months, it was not completed until the end of the year, despite intentions to have it released in June. Instead, they held a successful 'album preview', where each of the tracks was demonstrated in 'rough form' accompanied by a music video for each (produced by Melbourne photographic artist James Bryans). The band also talked about the origins of the songs. This event brought a lot of attention to the band for the idea's originality. They pre-released a radio single 'Just Hold On' from the album. The band were invited to play the hugely popular Homebake Festival in Sydney, where they received a great response, after being introduced onto the stage by a short stocky Asian guy with big calves ("one of the highlights" wrote The Daily Telegraph).

In 2007, the band departed on a tri-state residency which saw them play Melbourne, Sydney and Brisbane every week for the month of April - a bold move which brought further attention to the band for its original idea.^[5] Riding on the wave of this tour, Schroeder, De Backer and Heath departed on their first overseas tour which saw them performing 26 shows around Japan, Norway and the United Kingdom.^[6] They also toured twice more up the East Coast, one of which was with popular Japanese band The Bawdies. Their song 'Rattle My Chain' was used in a series of commercials for Volvo Australia. The song 'Hey There!' featured in the British film The Bank Job - starring Jason Statham - and was used as the backing for a February 2007 short film directed by Tim Longhurst called The Rip. Their song 'Better' was featured in a surfing documentary about the invention of the shortboard in 1967.

In 2008, the band appeared on Australia's Channel 9 for Australia Day celebrations, performing two of their songs - 'Just Hold On' and 'Hey There!' - to an estimated audience of 2 million. Securing an Australian Government grant, they embarked on a two-month tour of Australia, specifically targeting "culturally underprivileged groups" in rural and regional schoolchildren and Indigenous communities. The programme had them holding masterclasses and performing alongside Indigenous groups in the remote Northern Territory and Queensland.^[7][8] Their initiative also had them raising money for the charity Lifeline^{[disambiguation needed ]}, their contribution to which was recognised with a plaque at the end of the tour. Season two of Californication starring David Duchovny featured their cover of the classic 'Have Love, Will Travel', most famously covered by 'The Sonics'.

In 2009, The Basics for the first time enjoyed the support of Australian radio network Triple J, which had reportedly snubbed the band's previous material.^[9] The singles 'With This Ship' and 'Like A Brother' were added to the station's playlist, and momentum carried The Basics overseas for a second tour of the United Kingdom and Norway, and also a series of shows in Dublin, Ireland. They were joined by the longtime front-of-house engineer Stuart Padbury, who had been mixing them since 2005. The band was often seen in the Grafton Street Mall in Dublin, busking to promote their concerts there. Their second show in Oslo, Norway, saw the band perform with Hot Tub, one of the members of the Norwegian boy band "Boyzvoice", in his first performance for several years. Season Eight of Scrubs featured the song 'Lookin' Over My Shoulder'.

Despite all this promise, the band's album Keep Your Friends Close - produced and part-recorded by Peter Cobbin at London's Abbey Road Studios - though enjoying many favourable reviews,^{[10][11][12][13]} failed to impress Richard Kingsmill, music director of Triple J. This effectively ended the band's apparent upward spiral and the resulting album launch tour was reportedly "trying".

2010: Hiatus and live album[link]

In 2010, following the band's fourth stint at the Tamworth Country Music Festival, and the recording of a yet-unreleased live album, an indefinite hiatus was announced on the band's websites. This was in part due to the impending release of a new album produced by Wally for his Gotye project, and also to reported tensions within the band following the previous year's disappointments.

However, this hiatus was soon broken with a two-month residency at the Northcote Social Club in Melbourne, which saw an apparent return to form.^[14] The band's EP "Wait For You" enjoys airplay on Triple J.^[15] The Basics also played a one-off show at Sydney's Oxford Art Factory.^[16]

On 20 August 2010, The Basics released their fourth LP /ðəˈbæzɪtʃ/, available free and exclusively from their website for digital download^[17]. The band had reportedly moved to expand their listenership through this free giveaway, though 300 "Deluxe" CD/vinyl versions have been printed. /ðəˈbæzɪtʃ/ was recorded at the Northcote Social Club on 6 February, "in front of a live studio audience", and like the previous record, was mixed by Peter Cobbin at Abbey Road Studios. Technically a self-titled album, the phonetics spell out the accurate pronunciation of the band’s name. Kris Schroeder: "For years we’ve worked at bringing our live energy into the studio and thus far it’s been a bit hit-and-miss. Our trick this time was to pull a switcheroo and bring the studio to us instead; the result is the best and closest to us we’ve sounded yet."

Melbourne thespian and Dog With Wheels bassist David Bramble - friends with Heath from university days - has occasionally joined the band on tour, playing keys, often while sporting a large moustache. Other guests have included Jake Mason (saxophone), also of Cookin' on 3 Burners and The Bamboos^{[disambiguation needed ]}, Gideon Brazil (saxophone, flute), Simon Imrei (guitar and vocals) and Monty MacKenzie (saxophone).

Covers have often made an appearance from various sources: to date, some of the bands covered have been Cream, The Kinks, The Rolling Stones, The Police, Neil Young, Bob Dylan, New Kids on the Block, Richard Berry, Sonny Curtis and The Crickets, JJ Cale, Harry Nilsson, Bill Scott, Ryan Adams and The Coasters (via The Beatles for their cover of Three Cool Cats).

Members[link]

Current

• Wally De Backer - drums, vocals (2002-present)
• Kris Schroeder - bass, vocals (2002-present)
• Tim Heath - guitar, vocals (2004-present)

Former

• Michael Hubbard - guitar, vocals (2002-2004)

Discography[link]

Albums[link]

• Get Back (15 January 2004)
• Stand Out/Fit In (5 May 2007)
• Keep Your Friends Close (25 September 2009)
• /ðəˈbæzɪtʃ/ (20 August 2010)

EP[link]

• For Girls Like You (11 July 2005)
• Lookin' Over My Shoulder (10 February 2007)
• Like A Brother (5 June 2009)
• Wait For You (21 May 2010)

Singles[link]

• Call It Rhythm & Blues (20 August 2004)
• Just Hold On
• With This Ship (22 February 2009)

Upcoming projects[link]

The band is taking a three year hiatus, with an announced return to the stage together in early 2014. Wally De Backer launched Making Mirrors under his Gotye act in August 2011, Tim Heath will be touring and recording with Blood Red Bird, and Kris Schroeder recorded a solo EP titled, Patience in the Face of Control, before leaving Australia to work with the Red Cross in Kenya for a two year sabbatical.

References[link]

External links[link]

• Official Website
• Myspace Website