Misplaced Pages

#428571

54-609: The Leeds Arena (currently known as the First Direct Arena for sponsorship reasons) is an entertainment-focused indoor arena located in the Arena Quarter of Leeds , West Yorkshire, England. It is the first in the United Kingdom to have a fan-shaped orientation. The arena officially opened on 4 September 2013, with Sir Elton John , playing to an audience of 12,000. Bruce Springsteen had, however, held

108-704: A ) ∣ a ∈ A } {\textstyle d(x,\,A)=\inf\{d(x,\,a)\mid a\in A\}} denotes the distance between the point x {\textstyle x} and the subset A {\textstyle A} , then R k = { x ∈ X ∣ d ( x , P k ) ≤ d ( x , P j ) for all j ≠ k } {\displaystyle R_{k}=\{x\in X\mid d(x,P_{k})\leq d(x,P_{j})\;{\text{for all}}\;j\neq k\}} The Voronoi diagram

162-511: A Dirichlet tessellation (after Peter Gustav Lejeune Dirichlet ). Voronoi cells are also known as Thiessen polygons , after Alfred H. Thiessen . Voronoi diagrams have practical and theoretical applications in many fields, mainly in science and technology , but also in visual art . In the simplest case, shown in the first picture, we are given a finite set of points { p 1 , … p n } {\displaystyle \{p_{1},\dots p_{n}\}} in

216-410: A closed half-space , whose boundary is the perpendicular bisector of line segment p j p k {\displaystyle p_{j}p_{k}} . Cell R k {\displaystyle R_{k}} is the intersection of all of these n − 1 {\displaystyle n-1} half-spaces, and hence it is a convex polygon . When two cells in

270-415: A Voronoi diagram generated on the set  S  −  X . For a set of n points the ( n  − 1) -order Voronoi diagram is called a farthest-point Voronoi diagram. For a given set of points S  = { p 1 ,  p 2 , ...,  p n } the farthest-point Voronoi diagram divides the plane into cells in which the same point of P is the farthest point. A point of P has

324-545: A cell in the farthest-point Voronoi diagram if and only if it is a vertex of the convex hull of P . Let H  = { h 1 ,  h 2 , ...,  h k } be the convex hull of P ; then the farthest-point Voronoi diagram is a subdivision of the plane into k cells, one for each point in H , with the property that a point q lies in the cell corresponding to a site h i if and only if d( q , h i ) > d( q , p j ) for each p j  ∈  S with h i ≠ p j , where d( p , q )

378-488: A huge floor area holding thousands of standing spectators. The interior design is flexible and the Leeds Arena accommodates a range of events including: music concerts in various configurations, comedy events, family shows, such as "Walking with Dinosaurs", ice dance shows and family dance shows, basketball, tennis, boxing, wrestling, UFC and shows ranging in capacity from 1,500 to a maximum of 13,781. The external design

432-524: A new skyscraper in the Arena Quarter that will become the tallest building in Yorkshire upon completion. Altus House will occupy the site of Hume House, which has been demolished as part of the development. Voronoi diagram The Voronoi diagram is named after mathematician Georgy Voronoy , and is also called a Voronoi tessellation , a Voronoi decomposition , a Voronoi partition , or

486-405: A particular set of n points in S as its n nearest neighbors. Higher-order Voronoi diagrams also subdivide space. Higher-order Voronoi diagrams can be generated recursively. To generate the n -order Voronoi diagram from set  S , start with the ( n  − 1) -order diagram and replace each cell generated by X  = { x 1 ,  x 2 , ...,  x n −1 } with

540-448: A point that generates its Voronoi region R 1 {\textstyle R_{1}} , P 2 {\textstyle P_{2}} that generates R 2 {\textstyle R_{2}} , and P 3 {\textstyle P_{3}} that generates R 3 {\textstyle R_{3}} , and so on. Then, as expressed by Tran et al , "all locations in

594-484: A set of indices and let ( P k ) k ∈ K {\textstyle (P_{k})_{k\in K}} be a tuple (indexed collection) of nonempty subsets (the sites) in the space X {\textstyle X} . The Voronoi cell, or Voronoi region, R k {\textstyle R_{k}} , associated with the site P k {\textstyle P_{k}}

SECTION 10

#1732786756429

648-403: A tessellation of space with rhombo-hexagonal dodecahedra . For the set of points ( x ,  y ) with x in a discrete set X and y in a discrete set Y , we get rectangular tiles with the points not necessarily at their centers. Although a normal Voronoi cell is defined as the set of points closest to a single point in S , an n th-order Voronoi cell is defined as the set of points having

702-479: Is common on most recent Arena developments as seen in the Manchester, Nottingham and Liverpool developments. Each arena gained public funding and this was not protested by other established arenas. However, there was some opposition to the Leeds Arena from the owners of Sheffield Arena and Clive Betts , MP for Sheffield Attercliffe , who believe that the arena would adversely affect Sheffield Arena especially as

756-520: Is constructed in a super theatre fan shaped format as opposed to the more conventional bowl or horse-shoe seating arrangement commonly used by most arenas in the United Kingdom. The arena promises 'perfect sightlines' from each seat and the longest distance from the stage is 68 metres as opposed to 95–110 metres at a traditionally designed arena. According to the City Council, flat floor seating and 15 rows of retractable seating can be removed to create

810-412: Is known as influence area of the station. The average precipitation is calculated by the formula P ¯ = ∑ A i P i ∑ A i {\displaystyle {\bar {P}}={\frac {\sum A_{i}P_{i}}{\sum A_{i}}}} Several efficient algorithms are known for constructing Voronoi diagrams, either directly (as

864-400: Is less than or equal to the minimum distance to any other site p j {\displaystyle p_{j}} . For one other site p j {\displaystyle p_{j}} , the points that are closer to p k {\displaystyle p_{k}} than to p j {\displaystyle p_{j}} , or equally distant, form

918-584: Is located in Leeds city centre and the area is best known for housing Leeds Arena . In recent years, a number of developments have been built in Arena Quarter. This includes Opal 3 and Sky Plaza , both of which are student accommodation. Yorkshire's tallest building began construction in the Quarter in October 2018. The area of Leeds near Leeds Arena , was often referred to as Little London , however, that

972-474: Is located in Leeds city centre , on Claypit Lane. There are over 7500 car parking spaces within a 15-minute walk of the arena, and Leeds railway station and Leeds City bus station are each a 15-minute walk away. There are bus stands nearby on Woodhouse Lane. Arena Quarter Arena Quarter (sometimes referred to as Arena Qtr ) is a mixed city centre development with residential, retail and office developments in Leeds , West Yorkshire , England . It

1026-461: Is mainly located to the north of Lovell Park . Following the construction of the city's first indoor arena, the area nearby was renamed by Leeds City Council as Arena Quarter. The rezoning of this area meant that shopping center, Merrion Centre became part of the Arena Quarter. Its location is directly north of Merrion Street in Leeds city centre. The inner ring road borders the district on both

1080-534: Is simply the tuple of cells ( R k ) k ∈ K {\textstyle (R_{k})_{k\in K}} . In principle, some of the sites can intersect and even coincide (an application is described below for sites representing shops), but usually they are assumed to be disjoint. In addition, infinitely many sites are allowed in the definition (this setting has applications in geometry of numbers and crystallography ), but again, in many cases only finitely many sites are considered. In

1134-476: Is the Euclidean distance between two points p and  q . The boundaries of the cells in the farthest-point Voronoi diagram have the structure of a topological tree , with infinite rays as its leaves. Every finite tree is isomorphic to the tree formed in this way from a farthest-point Voronoi diagram. As implied by the definition, Voronoi cells can be defined for metrics other than Euclidean, such as

SECTION 20

#1732786756429

1188-480: Is the set of all points in X {\textstyle X} whose distance to P k {\textstyle P_{k}} is not greater than their distance to the other sites P j {\textstyle P_{j}} , where j {\textstyle j} is any index different from k {\textstyle k} . In other words, if d ( x , A ) = inf { d ( x ,

1242-462: Is used in meteorology and engineering hydrology to find the weights for precipitation data of stations over an area (watershed). The points generating the polygons are the various station that record precipitation data. Perpendicular bisectors are drawn to the line joining any two stations. This results in the formation of polygons around the stations. The area ( A i ) {\displaystyle (A_{i})} touching station point

1296-609: The Yorkshire Evening Post lobbied for a new arena to be built in the city. This resulted in a widescale 'Leeds needs an arena' campaign that had endorsements from local artists including the Kaiser Chiefs as well as local businesses and residents. The campaign was well publicised in local media, and included popular Myspace and Facebook groups. Following this, competitions began to find an operator and developer for an arena. In March 2008, SMG were chosen as

1350-544: The 2013 BBC Sports Personality of the Year Award , on Thursday 20 March 2014, it hosted week 7 of the 2014 Premier League Darts and has hosted two evenings of boxing matches in 2014 including Leeds own Josh Warrington in a British and commonwealth featherweight title match. On 3 July 2014 it was the host for the Tour de France Grand Départ Team Presentation. It also hosted week 1 of the 2015 Premier League Darts due to

1404-505: The Euclidean plane . In this case each site p k {\displaystyle p_{k}} is one of these given points, and its corresponding Voronoi cell R k {\displaystyle R_{k}} consists of every point in the Euclidean plane for which p k {\displaystyle p_{k}} is the nearest site: the distance to p k {\displaystyle p_{k}}

1458-620: The Mahalanobis distance or Manhattan distance . However, in these cases the boundaries of the Voronoi cells may be more complicated than in the Euclidean case, since the equidistant locus for two points may fail to be subspace of codimension 1, even in the two-dimensional case. A weighted Voronoi diagram is the one in which the function of a pair of points to define a Voronoi cell is a distance function modified by multiplicative or additive weights assigned to generator points. In contrast to

1512-617: The Merrion Centre . The land the arena was built on was the Claypit Lane Council car park and land formerly owned by Leeds Metropolitan University . Demolition of the former Leeds Metropolitan building was completed in 2009. Independent economic impact assessments conducted by Yorkshire Forward show that the new arena will bring over £25 million to the local economy every year and bring 300 full-time jobs. The money provided by Leeds City Council will come from part of

1566-933: The Four Seasons , Bette Midler , Neil Diamond , Simple Minds , Duran Duran , Bad Company , Paul Weller , alt-J , Natalie Imbruglia , Simply Red , Slipknot , Foals , Wet Wet Wet , Peter Andre , Ellie Goulding , Mariah Carey , Jeff Lynne's ELO , Noel Gallagher's High Flying Birds , Gladys Knight , Jean-Michel Jarre , Green Day , Drake , Pet Shop Boys , Kings of Leon , The Weeknd , Adnan Sami , Craig David , Richard Ashcroft , Shreya Ghoshal , Bruno Mars , Iron Maiden , The Cribs , Shawn Klush , Adam Ant , The Stone Roses , Celine Dion , The Killers , Steps , The Kooks , Jools Holland , Elbow , Black Sabbath , James Taylor , Alice Cooper , REO Speedwagon , Kylie Minogue , Nickelback , David Byrne , Gabrielle , Rick Astley , Tom Bailey , Belinda Carlisle , Culture Club , The Prodigy , Courteeners , Bring Me

1620-1621: The Horizon , Mumford & Sons , James , The Charlatans , Jane McDonald , Alison Moyet , Tears for Fears , Cher , Rita Ora , Bonnie Raitt , Frank Turner and The Sleeping Souls , The Wombats , George Ezra , Shawn Mendes , Westlife , A-ha , Little Mix , Blink-182 , Genesis , Orchestral Manoeuvres in the Dark , Deacon Blue , Liam Gallagher , Sam Fender , Dua Lipa , Blondie , Diana Ross , Diljit Dosanjh , Shania Twain , Ghost , Snoop Dogg , Tenacious D among others. Miranda Hart , Russell Howard , Micky Flanagan , Lee Evans , John Bishop , Mrs. Brown's Boys , Harry Enfield and Paul Whitehouse , Michael McIntyre , Vic Reeves and Bob Mortimer , Ricky Gervais , Chris Rock , Russell Peters , The League of Gentlemen , Whoopi Goldberg , Joanna Lumley , Jason Manford , Jack Whitehall . Cirque du Soleil , Jesus Christ Superstar , Dancing on Ice , Disney on Ice , The X Factor , Strictly Come Dancing , Diversity , BBC Radio 1 Xtra Live , The MOBO Awards , Dynamo , Bear Grylls , The ARIAS Awards , Leeds Gymfest, Harlem Globetrotters , Brian Cox , Nashville in Concert . The arena has hosted WWE Live in November 2013, May 2014, November 2014, April 2015, November 2015, April 2016, November 2016, November 2017 and November 2021. NXT visited in June 2017. It played host to

1674-492: The Leeds Arena would acquire funding from Yorkshire Forward . However, Yorkshire Forward stated that they fully support the development of an arena in Leeds and that an independent economic impact assessment also showed that a Leeds Arena would be of benefit to the entire region. Subsequently, Yorkshire Forward reaffirmed support to provide funding to the scheme. However, following a decision by BIS, central government initially did not authorise Yorkshire Forward to provide funding to

Leeds Arena - Misplaced Pages Continue

1728-483: The Voronoi cells may not be convex or even connected. In the usual Euclidean space, we can rewrite the formal definition in usual terms. Each Voronoi polygon R k {\textstyle R_{k}} is associated with a generator point P k {\textstyle P_{k}} . Let X {\textstyle X} be the set of all points in the Euclidean space. Let P 1 {\textstyle P_{1}} be

1782-501: The Voronoi diagram share a boundary, it is a line segment , ray , or line, consisting of all the points in the plane that are equidistant to their two nearest sites. The vertices of the diagram, where three or more of these boundaries meet, are the points that have three or more equally distant nearest sites. Let X {\textstyle X} be a metric space with distance function d {\textstyle d} . Let K {\textstyle K} be

1836-474: The Voronoi diagram. The Jump Flooding Algorithm can generate approximate Voronoi diagrams in constant time and is suited for use on commodity graphics hardware. Lloyd's algorithm and its generalization via the Linde–Buzo–Gray algorithm (aka k-means clustering ) use the construction of Voronoi diagrams as a subroutine. These methods alternate between steps in which one constructs the Voronoi diagram for

1890-470: The Voronoi polygon are closer to the generator point of that polygon than any other generator point in the Voronoi diagram in Euclidean plane". As a simple illustration, consider a group of shops in a city. Suppose we want to estimate the number of customers of a given shop. With all else being equal (price, products, quality of service, etc.), it is reasonable to assume that customers choose their preferred shop simply by distance considerations: they will go to

1944-722: The case of Voronoi cells defined using a distance which is a metric , in this case some of the Voronoi cells may be empty. A power diagram is a type of Voronoi diagram defined from a set of circles using the power distance ; it can also be thought of as a weighted Voronoi diagram in which a weight defined from the radius of each circle is added to the squared Euclidean distance from the circle's center. The Voronoi diagram of n {\displaystyle n} points in d {\displaystyle d} -dimensional space can have O ( n ⌈ d / 2 ⌉ ) {\textstyle O(n^{\lceil d/2\rceil })} vertices, requiring

1998-489: The developer competition and chosen Claypit Lane as the site for the new arena and would proceed as the developer. This resulted in a legal challenge from Montpellier Estates. Montpellier's claim for more than £43.5 million was rejected by Mr Justice Supperstone in a Judgement handed down on 6 February 2013. The arena's location, on Claypit Lane is situated at the Northern Quarter of Leeds city centre, behind

2052-433: The diagram itself) or indirectly by starting with a Delaunay triangulation and then obtaining its dual. Direct algorithms include Fortune's algorithm , an O ( n log( n )) algorithm for generating a Voronoi diagram from a set of points in a plane. Bowyer–Watson algorithm , an O ( n log( n )) to O ( n ) algorithm for generating a Delaunay triangulation in any number of dimensions, can be used in an indirect algorithm for

2106-654: The east and north boundary, with Woodhouse Lane acting as the district's western boundary. The main two facilities located in Arena Quarter are the Merrion Centre and Leeds Arena . The Yorkshire Bank HQ is also located in Arena Quarter. The vibrant and upcoming part of Leeds has seen major developments around the Arena Quarter. Most of the development has come in the way of student housing, with many tall buildings constructed or under construction. This included two of Leeds' tallest buildings, including Sky Plaza and Opal 3 . In October 2018, plans were announced for

2160-502: The familiar Euclidean distance : or the Manhattan distance : The corresponding Voronoi diagrams look different for different distance metrics. Informal use of Voronoi diagrams can be traced back to Descartes in 1644. Peter Gustav Lejeune Dirichlet used two-dimensional and three-dimensional Voronoi diagrams in his study of quadratic forms in 1850. British physician John Snow used a Voronoi-like diagram in 1854 to illustrate how

2214-537: The first concert on 24 July 2013, with an audience of 13,000. The arena's opening season in 2013 later included acts including Kaiser Chiefs , Rod Stewart , Status Quo and Depeche Mode . The building was named the "best new venue in the world" in 2014 by the Stadium Business Awards. Leeds had been the largest city in the United Kingdom without a major venue to hold music or indoor sporting events. As of February 2010, Bristol and Leeds were

Leeds Arena - Misplaced Pages Continue

2268-590: The future operators of the Leeds arena, following worldwide interest from potential operators in the scheme. SMG also operate arenas in Newcastle, Belfast and Manchester and over 200 other venues worldwide. The addition of Leeds strengthens their position in the UK Arena market. SMG's European managing director has stated that Leeds will be its "flagship venue" in Europe and that they expect First Direct Arena "to be in

2322-985: The majority of people who died in the Broad Street cholera outbreak lived closer to the infected Broad Street pump than to any other water pump. Voronoi diagrams are named after Georgy Feodosievych Voronoy who defined and studied the general n -dimensional case in 1908. Voronoi diagrams that are used in geophysics and meteorology to analyse spatially distributed data are called Thiessen polygons after American meteorologist Alfred H. Thiessen , who used them to estimate rainfall from scattered measurements in 1911. Other equivalent names for this concept (or particular important cases of it): Voronoi polyhedra, Voronoi polygons, domain(s) of influence, Voronoi decomposition, Voronoi tessellation(s), Dirichlet tessellation(s). Voronoi tessellations of regular lattices of points in two or three dimensions give rise to many familiar tessellations. Certain body-centered tetragonal lattices give

2376-519: The only major cities without an arena style venue. Previously, the Queen's Hall was used for popular music concerts but this was demolished in 1989. Since 1989, there have been a number of failed attempts at building a major venue in Leeds. An arena has been long requested by residents in Leeds. The consultation on the Vision for Leeds 2004 to 2020 showed a strong demand from the public for a new arena, and

2430-423: The particular case where the space is a finite-dimensional Euclidean space , each site is a point, there are finitely many points and all of them are different, then the Voronoi cells are convex polytopes and they can be represented in a combinatorial way using their vertices, sides, two-dimensional faces, etc. Sometimes the induced combinatorial structure is referred to as the Voronoi diagram. In general however,

2484-489: The proceeds from their sale of Leeds Bradford International Airport in October 2007, Yorkshire Forward grant support and revenues from rent and commercial activities. The total scheme costs are estimated to be in the region of £80 million including land costs. Construction work on the Leeds Arena commenced in February 2011. Construction was completed by May 2013, with a public open day scheduled for 25 May. The arena

2538-481: The project became one of the city's 12 priorities. The Leeds Initiative formed a Cultural facilities task group to consider options. It appointed PMP consultants whose report outlined the viability of a Leeds Arena, and other potential projects such as a concert hall. The task group recommended that the Council proceed with the development of a 12,500-seat arena. Whilst this study was taking place, campaigners including

2592-513: The same bound for the amount of memory needed to store an explicit description of it. Therefore, Voronoi diagrams are often not feasible for moderate or high dimensions. A more space-efficient alternative is to use approximate Voronoi diagrams . Voronoi diagrams are also related to other geometric structures such as the medial axis (which has found applications in image segmentation, optical character recognition , and other computational applications), straight skeleton , and zone diagrams . It

2646-1189: The scheme. Soon after, it was announced that £9.9 million funding was authorised by central government. This was met with fierce criticism from Sheffield MPs, but received positive responses in the Leeds City Region. The first event in the arena was a concert by Bruce Springsteen on 24 July 2013 although the arena officially opened in September 2013. A wide variety of events have already taken place or been scheduled, including: Bruce Springsteen , Elton John , Leonard Cohen , Kaiser Chiefs , Stereophonics , Depeche Mode , Rod Stewart , Andrea Bocelli , James Blunt , Bryan Adams , Slash , Boyzone , JLS , 5 Seconds of Summer , Miley Cyrus , Prince , Dolly Parton , Eagles , The 1975 , Status Quo , Pearl Jam , Biffy Clyro , Manic Street Preachers , Catatonia , Eric Clapton , Il Divo , The Who , Alfie Boe , Michael Bublé , The Script , Madness , Morrissey , Lionel Richie , Paloma Faith , Queen + Adam Lambert , Olly Murs , Nicki Minaj , Pharrell Williams , Fleetwood Mac , John Legend , Frankie Valli and

2700-413: The shop located nearest to them. In this case the Voronoi cell R k {\displaystyle R_{k}} of a given shop P k {\displaystyle P_{k}} can be used for giving a rough estimate on the number of potential customers going to this shop (which is modeled by a point in our city). For most cities, the distance between points can be measured using

2754-675: The sponsor of the arena, with the name officially becoming the First Direct Arena. The naming rights were set to expire in 2018. However, it was announced in March 2017 that First Direct had extended its naming sponsorship for a further 5 years, continuing to be called the First Direct Arena until at least 2023. The naming rights have now been extended until 2031. Yorkshire Forward's support and intended £18 million funding has gained much publicity and has been subject to fierce political fighting. Some critics outlined that public funding

SECTION 50

#1732786756429

2808-495: The success of hosting week 7 of the 2014 Premier League Darts. The first boxing event took place on 21 December 2013, which saw Stuart Hall win the vacant IBF championship. Leeds-based boxer Josh Warrington first fought at the arena 21 May 2014. Warrington made the arena his home, headlining the next seven events. The arena hosted the count of various Leeds seats in the 2015 general election including Morley and Outwood where Shadow Chancellor Ed Balls lost his seat. The arena

2862-520: The top 10 in the world within two to three years of opening". Two main bidders were involved in the competition to develop the arena. These were the Council owned land at Elland Road as one location which would have been developed by GMI Construction and land owned by Montpellier Estates in Sweet Street, Holbeck with Montpellier Estates being developer of the latter site. However, in November 2008, Leeds City Council announced they had terminated

2916-463: Was announced in August 2010. The external design has a honeycomb design. This design uses – a voronoi diagram – to create cellular shapes that are visible on key approaches to the arena. On an evening, the front of the building has the ability to change colour or pattern dependent on the show or mood of the arena at the time. On 2 May 2013, telephone and internet bank First Direct were announced as

#428571