The Celtics–Lakers rivalry is a National Basketball Association rivalry between the Boston Celtics and the Los Angeles Lakers. The Celtics and the Lakers are the two most storied franchises in the NBA, the rivalry has been called the best in the NBA; the Boston Celtics and the Los Angeles Lakers have met a record 12 times in the NBA Finals, starting with their first Finals meeting in 1959. They would both go on to dominate the league in the 1960s and 1980s, facing each other six times in the 1960s, three times in the 1980s, two times in 2008 and 2010; the two teams have won the two highest numbers of championships in the NBA: the Celtics have won 17, the Lakers have won 16. Together, they account for 33 of the 72 championships in NBA history; as of 2018, the Celtics and Lakers have a..596 all-time winning records respectively. As of the end of the 2017–18 season, Boston is the only team with a winning overall record against the Lakers; the rivalry had been less intense since the retirements of Larry Bird and Magic Johnson in the early 1990s, but in 2008 when the two teams met in the Finals for the first time since 1987, with the Celtics winning the series 4–2.
They met again in the 2010 NBA Finals. During the first decade of the NBA in the 1950s, the Minneapolis Lakers had the first NBA dynasty. Minneapolis would win the first Championship Series of the newly formed NBA in 1950. Under Hall of Fame head coach John Kundla, with the NBA's first superstar in George Mikan, they would win three more titles in 1952, 1953, 1954; the Celtics would emerge behind early NBA star Bob Cousy by winning the 1957 NBA Finals and losing in 1958. The first NBA Finals match-up between the two teams was in 1959 when on April 9, the Boston Celtics swept the Minneapolis Lakers 4-0 for the first sweep in the history of the NBA Finals; this would mark the first Finals loss for the dominant Lakers, the first of eight straight titles for Boston. The Lakers relocated to Los Angeles in 1960, it was after this move, during this decade, that the rivalry would escalate. The two teams emerged as the strongest in the NBA, featuring greats such as Bill Russell, Tom Heinsohn, John Havlicek, Sam Jones and head coach Red Auerbach for Boston and Elgin Baylor, Jerry West, Gail Goodrich, coach/GM Fred Schaus for Los Angeles.
However, it would prove to be the decade of the Celtics, who won the finals every year in the 1960s except for 1967. The Lakers would be the Celtics opponent in six of those series: 1962, 1963, 1965, 1966, 1968, 1969; the Celtics won all of those match-ups. Three of those series went seven games; the Celtics win over the Lakers in 1966 marked an unprecedented eight consecutive championships, the longest streak of any North American professional sports team. The Lakers acquired Wilt Chamberlain in 1968, which brought the personal rivalry between him and Bill Russell a feature of the Celtics-76ers rivalry, to Celtics-Lakers; the Lakers posted the best record in the West during the 1968–1969 season. By contrast, the aging Celtics struggled to obtain the fourth seed, with Russell and Jones playing in their final seasons. Despite this, the Celtics upset the Philadelphia 76ers and the New York Knicks and made it to the Finals; the Lakers had home court advantage for the first time and won the first two games, but the Celtics rebounded to force and win a dramatic Game 7 at the Los Angeles Forum, defying Laker's owner Jack Kent Cooke's infamous prediction of a Lakers celebration.
West was named Finals MVP despite being on the losing team, but it was small consolation in a decade where the Lakers went without a championship, every one of their Finals' losses in that decade coming at the hands of the Celtics. The 1969 Finals caused a deterioration in the relationship between Russell and Chamberlain, friends despite their rivalry, into one of intense loathing, when Chamberlain took himself out of the decisive Game 7 with six minutes left, Russell thereafter accused Chamberlain of being a malingerer and of "copping out" of the game when it seemed that the Lakers would lose. Chamberlain saw him as a backstabber; the two men did not talk to each other for over 20 years until Russell attempted to patch things up, although he never uttered a genuine apology. When Chamberlain died in 1999, Chamberlain's nephew stated that Russell was the second person he was ordered to break the news to; the Celtics and Lakers both found success in the 1970s, but there would be no rematch between the two teams.
The start of the decade saw the Lakers' woes in the NBA Finals continue, with a loss to the New York Knicks in 1970. However, the Lakers rebounded two years to win the 1972 NBA Finals and their first championship in Los Angeles against the Knicks; this would prove to be Laker great Jerry West's only NBA title. The following year, the Lakers again lost, they would not make it to the Finals again in this decade, but in 1975 they acquired Kareem Abdul-Jabbar. The Celtics won the NBA Finals in 1974, won again in 1976, under the leadership of coach Tom Heinsohn and players Dave Cowens, Paul Silas and Jo Jo White. Neither team won another championship until the 1980s. However, the foundation for the renewed Celtics–Lakers rivalry of the 1980s was laid down in college basketball of the late 1970s. During the 1978–79 NCAA season, Michigan State was led by Magic Joh
Video game genre
A video game genre is a classification assigned to a video game based on its gameplay interaction rather than visual or narrative differences. A video game genre is defined by a set of gameplay challenges and are classified independently of their setting or game-world content, unlike other works of fiction such as films or books. For example, a shooter game is still a shooter game, regardless of when it takes place; as with nearly all varieties of genre classification, the matter of any individual video game's specific genre is open to personal interpretation. Moreover, each individual game may belong to several genres at once; the first attempt to classify different genres of video games was made by Chris Crawford in his book The Art of Computer Game Design in 1984. In this book, Crawford focused on the player's experience and activities required for gameplay. Here, he stated that "the state of computer game design is changing quickly. We would therefore expect the taxonomy presented to become obsolete or inadequate in a short time."
Since among other genres, the platformer and 3D shooter genres, which hardly existed at the time, have gained a lot of popularity. As hardware capabilities have increased, new genres have become possible, with examples being increased memory, the move from 2D to 3D, new peripherals and location. Though genres were just interesting for game studies in the 1980s, the business of video games expanded in the 1990s and both smaller and independent publishers had little chance of surviving; because of this, games settled more into set genres that larger publishers and retailers could use for marketing. Due to "direct and active participation" of the player, video game genres differ from literary and film genres. Though one could state that Space Invaders is a science-fiction video game, such a classification "ignores the differences and similarities which are to be found in the player's experience of the game." In contrast to the visual aesthetics of games, which can vary it is argued that it is interactivity characteristics that are common to all games.
Descriptive names of genres take into account the goals of the game, the protagonist and the perspective offered to the player. For example, a first-person shooter is a game, played from a first-person perspective and involves the practice of shooting; the term "subgenre" may be used to refer to a category within a genre to further specify the genre of the game under discussion. Whereas "shooter game" is a genre name, "first-person shooter" and "third-person shooter" are common subgenres of the shooter genre. Other examples of such prefixes are real-time, turn based, side-scrolling; the target audience, underlying theme or purpose of a game are sometimes used as a genre identifier, such as with "games for girls," games for cats,"Christian game" and "Serious game" respectively. However, because these terms do not indicate anything about the gameplay of a video game, these are not considered genres. Video game genres vary in specificity, with popular video game reviews using genre names varying from "action" to "baseball."
In this practice, basic themes and more fundamental characteristics are used alongside each other. A game may combine aspects of multiple genres in such a way that it becomes hard to classify under existing genres. For example, because Grand Theft Auto III combined shooting and roleplaying in an unusual way, it was hard to classify using existing terms. Since the term Grand Theft Auto clone has been used to describe games mechanically similar to Grand Theft Auto III; the term roguelike has been developed for games that share similarities with Rogue. Elements of the role-playing genre, which focuses on storytelling and character growth, have been implemented in many different genres of video games; this is because the addition of a story and character enhancement to an action, strategy or puzzle video game does not take away from its core gameplay, but adds an incentive other than survival to the experience. According to some analysts, the count of each broad genre in the best selling physical games worldwide is broken down as follows.
The most popular genres are Shooter, Role-playing and Sports, with Platformer and Racing having both declined in the last decade. Puzzle games have declined when measured by sales, however, on mobile, where the majority of games are free-to-play, this genre remains the most popular worldwide. List of video game genres
Lakers versus Celtics and the NBA Playoffs
Lakers versus Celtics and the NBA Playoffs is a basketball video game developed and released by Electronic Arts. It was first released in 1989 for MS-DOS-compatible PCs and for the Mega Drive/Genesis in 1991; the game was successful. The game's title alludes to the 1980s rivalry between the Boston Celtics and Los Angeles Lakers, it is the first game in the NBA Playoffs series of games. The game would be the first team basketball video game to acquire full players names and rights without relying on the NBA Players Association. Established stars in the game include Larry Bird, Kareem Abdul-Jabbar, Magic Johnson, Michael Jordan, Patrick Ewing; the game introduced innovations in presentation. It was the first NBA game to feature TV style starting line up prior to the opening tip, it featured halftime shows and announcers to simulate an authentic feel. Games would go further by replacing music during gameplay with audience cheers; the game was conceived by Producer Don Traeger, principal work done by programmer Robert Weatherby and Art Director Michael Kosaka.
It was the first game to feature the EASN logo, a stand-alone sports brand that Traeger had conceived with marketing partner Don Transeth and Artist Michael Kosaka. This brand evolved into EA Sports; the game can be played in various ways: players could play against each other, or against the computer. Games against the computer were divided into two modes, "Exhibition" or "Playoffs". There are three difficulty levels: Pre-season, Regular season and Showtime. Games could be configured for 5, 8 or 12 minute quarters. Signature moves for individual star players have their own were introduced, such as Charles Barkley's gorilla dunk. Players can pick from one of eight teams who were among the 16 that had competed in the playoffs the year before the game was released; the MS-DOS version contained teams from the 1989 NBA Playoffs, while the Genesis version contained the playoff teams from the 1990 NBA Playoffs. MS-DOS version: Genesis version: Both conferences' All-Star Teams were available as well; this was the only way to use players from teams that were not represented in the game - Hakeem Olajuwon, Dominique Wilkins, Chris Mullin and Reggie Miller being examples.
Computer Gaming World praised Lakers versus Celtics' graphics, but stated that it was too easy to play for action gamers and that scoring was too high for statistics-oriented players, unfavorably noted the lack of league play or custom lineups. The game scored 88 % in Video Games. Mega placed the game at #24 in their Top Mega Drive Games of All Time. Lakers versus Celtics and the NBA Playoffs at GameFAQs Lakers versus Celtics and the NBA Playoffs at GameFAQs Lakers versus Celtics and the NBA Playoffs at MobyGames
Michael Jeffrey Jordan known by his initials, MJ, is an American former professional basketball player, the principal owner and chairman of the Charlotte Hornets of the National Basketball Association. He played 15 seasons in the NBA for the Chicago Bulls and Washington Wizards, his biography on the official NBA website states: "By acclamation, Michael Jordan is the greatest basketball player of all time." He was one of the most marketed athletes of his generation and was considered instrumental in popularizing the NBA around the world in the 1980s and 1990s. Jordan played three seasons for coach Dean Smith at the University of North Carolina; as a freshman, he was a member of the Tar Heels' national championship team in 1982. Jordan joined the Bulls in 1984 as the third overall draft pick, he emerged as a league star and entertained crowds with his prolific scoring. His leaping ability, demonstrated by performing slam dunks from the free throw line in Slam Dunk Contests, earned him the nicknames Air Jordan and His Airness.
He gained a reputation for being one of the best defensive players in basketball. In 1991, he won his first NBA championship with the Bulls, followed that achievement with titles in 1992 and 1993, securing a "three-peat". Although Jordan abruptly retired from basketball before the beginning of the 1993–94 NBA season, started a new career in Minor League Baseball, he returned to the Bulls in March 1995 and led them to three additional championships in 1996, 1997, 1998, as well as a then-record 72 regular-season wins in the 1995–96 NBA season. Jordan retired for a second time in January 1999, but returned for two more NBA seasons from 2001 to 2003 as a member of the Wizards. Jordan's individual accolades and accomplishments include six NBA Finals Most Valuable Player Awards, ten scoring titles, five MVP Awards, ten All-NBA First Team designations, nine All-Defensive First Team honors, fourteen NBA All-Star Game selections, three All-Star Game MVP Awards, three steals titles, the 1988 NBA Defensive Player of the Year Award.
He holds the NBA records for highest career regular season scoring average and highest career playoff scoring average. In 1999, he was named the greatest North American athlete of the 20th century by ESPN, was second to Babe Ruth on the Associated Press' list of athletes of the century. Jordan is a two-time inductee into the Naismith Memorial Basketball Hall of Fame, having been enshrined in 2009 for his individual career, again in 2010 as part of the group induction of the 1992 United States men's Olympic basketball team, he became a member of the FIBA Hall of Fame in 2015. Jordan is known for his product endorsements, he fueled the success of Nike's Air Jordan sneakers, which were introduced in 1984 and remain popular today. Jordan starred as himself in the 1996 film Space Jam. In 2006, he became part-owner and head of basketball operations for the Charlotte Bobcats, bought a controlling interest in 2010. In 2014, Jordan became the first billionaire player in NBA history, he is the third-richest African-American, behind Robert F. Oprah Winfrey.
Jordan was born in Brooklyn, New York, to Deloris, who worked in banking, James R. Jordan Sr. an equipment supervisor. His family moved to Wilmington, North Carolina. Jordan is the fourth of five children, he has two older brothers, Larry Jordan and James R. Jordan, Jr. one older sister and one younger sister, Roslyn. Jordan's brother James retired in 2006 as the Command Sergeant Major of the 35th Signal Brigade of the XVIII Airborne Corps in the U. S. Army. Jordan attended Emsley A. Laney High School in Wilmington, where he highlighted his athletic career by playing basketball and football, he tried out for the varsity basketball team during his sophomore year, but at 5'11", he was deemed too short to play at that level. His taller friend, Harvest Leroy Smith, was the only sophomore to make the team. Motivated to prove his worth, Jordan became the star of Laney's junior varsity team, tallied several 40-point games; the following summer, he trained rigorously. Upon earning a spot on the varsity roster, Jordan averaged more than 25 points per game over his final two seasons of high school play.
As a senior, he was selected to play in the 1981 McDonald's All-American Game and scored 30 points, after averaging 27 points, 12 rebounds and 6 assists per game for the season. Jordan was recruited by numerous college basketball programs, including Duke, North Carolina, South Carolina and Virginia. In 1981, Jordan accepted a basketball scholarship to North Carolina, where he majored in cultural geography; as a freshman in coach Dean Smith's team-oriented system, he was named ACC Freshman of the Year after he averaged 13.4 ppg on 53.4% shooting. He made the game-winning jump shot in the 1982 NCAA Championship game against Georgetown, led by future NBA rival Patrick Ewing. Jordan described this shot as the major turning point in his basketball career. During his three seasons at North Carolina, he averaged 17.7 ppg on 54.0% shooting, added 5.0 rpg. He was selected by consensus to the NCAA All-American First Team in both his sophomore and junior seasons. After winning the Naismith and the Wooden College Player of the Year awards in 1984, Jordan left North Carolina one year before his scheduled graduation to enter the 1984 NBA draft.
The Chicago Bulls selected Jordan after Hakeem Olajuwon and Sam Bowie. One of the primary reasons why Jordan was not drafted
The Sega Genesis, known as the Mega Drive in regions outside of North America, is a 16-bit home video game console developed and sold by Sega. The Genesis was the successor to the Master System. Sega released it as the Mega Drive in Japan in 1988, followed by North America as the Genesis in 1989. In 1990, it was distributed as the Mega Drive by Virgin Mastertronic in Europe, Ozisoft in Australasia, Tec Toy in Brazil. In South Korea, it was distributed by Samsung as the Super Gam*Boy and the Super Aladdin Boy. Designed by an R&D team supervised by Hideki Sato and Masami Ishikawa, the Genesis was adapted from Sega's System 16 arcade board, centered on a Motorola 68000 processor as the CPU, a Zilog Z80 as a sound controller, a video system supporting hardware sprites and scrolling, it plays a library of more than 900 games created by Sega and a wide array of third-party publishers and delivered on ROM-based cartridges. Several add-ons were released, including a Power Base Converter to play Master System games.
It was released in several different versions, some created by third parties. Sega created two network services to support the Genesis: Sega Channel. In Japan, the Mega Drive fared poorly against its two main competitors, Nintendo's Super Famicom and NEC's PC Engine, but it achieved considerable success in North America and Europe. Contributing to its success were its library of arcade game ports, the popularity of Sega's Sonic the Hedgehog series, several popular sports franchises, aggressive youth marketing that positioned the system as the cool console for adolescents; the release of the Super Nintendo Entertainment System two years after the Genesis resulted in a fierce battle for market share in the United States and Europe, termed as a "console war" by journalists and historians. As this contest drew increasing attention to the video game industry among the general public, the Genesis and several of its highest-profile games attracted significant legal scrutiny on matters involving reverse engineering and video game violence.
Controversy surrounding violent games such as Night Trap and Mortal Kombat led Sega to create the Videogame Rating Council, a predecessor to the Entertainment Software Rating Board. 30.75 million first-party Genesis units were sold worldwide. In addition, Tec Toy sold an estimated three million licensed variants in Brazil, Majesco projected it would sell 1.5 million licensed variants of the system in the United States, much smaller numbers were sold by Samsung in South Korea. By the mid-2010s, licensed third-party Genesis rereleases were still being sold by AtGames in North America and Europe. Many games have been rereleased in compilations or on online services such as the Nintendo Virtual Console, Xbox Live Arcade, PlayStation Network, Steam; the Genesis was succeeded in 1994 by the Sega Saturn. In the early 1980s, Sega Enterprises, Inc. a subsidiary of Gulf & Western, was one of the top five arcade game manufacturers active in the United States, as company revenues surpassed $200 million between July 1981 and June 1982.
A downturn in the arcade business starting in 1982 hurt the company, leading Gulf & Western to sell its North American arcade manufacturing organization and the licensing rights for its arcade games to Bally Manufacturing. The company retained Sega's North American R&D operation, as well as its Japanese subsidiary, Sega Enterprises, Ltd. With its arcade business in decline, Sega Enterprises, Ltd. president Hayao Nakayama advocated that the company leverage its hardware expertise to move into the home console market in Japan, in its infancy at the time. Nakayama received permission to proceed with this project, leading to the release of Sega's first home video game system, the SG-1000, in July 1983; the SG-1000 was not successful. Sega estimated; the SG-1000 was replaced by the Sega Mark III within two years. In the meantime, Gulf & Western began to divest itself of its non-core businesses after the death of company founder Charles Bluhdorn, so Nakayama and former Sega CEO David Rosen arranged a management buyout of the Japanese subsidiary in 1984 with financial backing from CSK Corporation, a prominent Japanese software company.
Nakayama was installed as CEO of the new Sega Enterprises, Ltd. In 1986, Sega redesigned the Mark III for release in North America as the Sega Master System; this was followed by a European release the next year. Although the Master System was a success in Europe, in Brazil, it failed to ignite significant interest in the Japanese or North American markets, which, by the mid-to-late 1980s, were both dominated by Nintendo. With Sega continuing to have difficulty penetrating the home market, Sega's console R&D team, led by Masami Ishikawa and supervised by Hideki Sato, began work on a successor to the Master System immediately after that console launched. In 1987, Sega faced another threat to its console business when Japanese computer giant NEC released the PC Engine amid great publicity. To remain competitive against the two more established consumer electronics companies and his team decided they needed to incorporate a 16-bit microprocessor into their new system to make an impact in the marketplace and once again turned to Sega's strengths in the arcade industry to adapt the successful Sega System 16 arcade board into architecture for a home console.
The decision to use a Motorola 68000 as the system's main CPU was made late in development, while a Zilog Z80 was used as a secondary CPU to handle the sound due to f
3D computer graphics
3D computer graphics or three-dimensional computer graphics, are graphics that use a three-dimensional representation of geometric data, stored in the computer for the purposes of performing calculations and rendering 2D images. Such images may be stored for viewing or displayed in real-time. 3D computer graphics rely on many of the same algorithms as 2D computer vector graphics in the wire-frame model and 2D computer raster graphics in the final rendered display. In computer graphics software, 2D applications may use 3D techniques to achieve effects such as lighting, 3D may use 2D rendering techniques. 3D computer graphics are referred to as 3D models. Apart from the rendered graphic, the model is contained within the graphical data file. However, there are differences: a 3D model is the mathematical representation of any three-dimensional object. A model is not technically a graphic. A model can be displayed visually as a two-dimensional image through a process called 3D rendering or used in non-graphical computer simulations and calculations.
With 3D printing, 3D models are rendered into a 3D physical representation of the model, with limitations to how accurate the rendering can match the virtual model. William Fetter was credited with coining the term computer graphics in 1961 to describe his work at Boeing. One of the first displays of computer animation was Futureworld, which included an animation of a human face and a hand that had appeared in the 1972 experimental short A Computer Animated Hand, created by University of Utah students Edwin Catmull and Fred Parke.3D computer graphic s software began appearing for home computers in the late 1970s. The earliest known example is 3D Art Graphics, a set of 3D computer graphics effects, written by Kazumasa Mitazawa and released in June 1978 for the Apple II. 3D computer graphics creation falls into three basic phases: 3D modeling – the process of forming a computer model of an object's shape Layout and animation – the placement and movement of objects within a scene 3D rendering – the computer calculations that, based on light placement, surface types, other qualities, generate the image The model describes the process of forming the shape of an object.
The two most common sources of 3D models are those that an artist or engineer originates on the computer with some kind of 3D modeling tool, models scanned into a computer from real-world objects. Models can be produced procedurally or via physical simulation. A 3D model is formed from points called vertices that define the shape and form polygons. A polygon is an area formed from at least three vertexes. A polygon of n points is an n-gon; the overall integrity of the model and its suitability to use in animation depend on the structure of the polygons. Materials and textures are properties that the render engine uses to render the model, in an unbiased render engine like blender cycles, one can give the model materials to tell the engine how to treat light when it hits the surface. Textures are used to give the material color using a color or albedo map, or give the surface features using a bump or normal map, it can be used to deform the model itself using a displacement map. Before rendering into an image, objects must be laid out in a scene.
This defines spatial relationships including location and size. Animation refers to the temporal description of an object; these techniques are used in combination. As with animation, physical simulation specifies motion. Rendering converts a model into an image either by simulating light transport to get photo-realistic images, or by applying an art style as in non-photorealistic rendering; the two basic operations in realistic rendering are scattering. This step is performed using 3D computer graphics software or a 3D graphics API. Altering the scene into a suitable form for rendering involves 3D projection, which displays a three-dimensional image in two dimensions. Although 3D modeling and CAD software may perform 3D rendering as well, exclusive 3D rendering software exists. 3D computer graphics software produces computer-generated imagery through 3D modeling and 3D rendering or produces 3D models for analytic and industrial purposes. 3D modeling software is a class of 3D computer graphics. Individual programs of this class are called modeling modelers.
3D modelers allow users to alter models via their 3D mesh. Users can add, subtract and otherwise change the mesh to their desire. Models can be viewed from a variety of angles simultaneously. Models can be rotated and the view can be zoomed in and out. 3D modelers can export their models to files, which can be imported into other applications as long as the metadata are compatible. Many modelers allow importers and exporters to be plugged-in, so they can read and write data in the native formats of other applications. Most 3D modelers contain a number of related features, such as ray tracers and other rendering alternatives and texture mapping facilities; some contain features that support or allow animation of models. Some may be able to generate full-motion video of a series of rendered scenes. Computer aided design software may employ the same fundamental 3D modeling techniques that 3D modeling software use but their goal differs, they are used in computer-aided engineering, computer-aided man
Isometric projection is a method for visually representing three-dimensional objects in two dimensions in technical and engineering drawings. It is an axonometric projection in which the three coordinate axes appear foreshortened and the angle between any two of them is 120 degrees; the term "isometric" comes from the Greek for "equal measure", reflecting that the scale along each axis of the projection is the same. An isometric view of an object can be obtained by choosing the viewing direction such that the angles between the projections of the x, y, z axes are all the same, or 120°. For example, with a cube, this is done by first looking straight towards one face. Next, the cube is rotated ±45° about the vertical axis, followed by a rotation of 35.264° about the horizontal axis. Note that with the cube the perimeter of the resulting 2D drawing is a perfect regular hexagon: all the black lines have equal length and all the cube's faces are the same area. Isometric graph paper can be placed under a normal piece of drawing paper to help achieve the effect without calculation.
In a similar way, an isometric view can be obtained in a 3D scene. Starting with the camera aligned parallel to the floor and aligned to the coordinate axes, it is first rotated vertically by about 35.264° as above ±45° around the vertical axis. Another way isometric projection can be visualized is by considering a view within a cubical room starting in an upper corner and looking towards the opposite, lower corner; the x-axis extends diagonally down and right, the y-axis extends diagonally down and left, the z-axis is straight up. Depth is shown by height on the image. Lines drawn along the axes are at 120° to one another; the term "isometric" is mistakenly used to refer to axonometric projections, generally. There are, however three types of axonometric projections: isometric and trimetric. From the two angles needed for an isometric projection, the value of the second may seem counterintuitive and deserves some further explanation. Let’s first imagine a cube with sides of length 2, its center positioned at the axis origin.
We can calculate the length of the line from its center to the middle of any edge as √2 using Pythagoras' theorem. By rotating the cube by 45° on the x-axis, the point will therefore become as depicted in the diagram; the second rotation aims to bring the same point on the positive z-axis and so needs to perform a rotation of value equal to the arctangent of 1⁄√2, 35.264°. There are eight different orientations to obtain an isometric view, depending into which octant the viewer looks; the isometric transform from a point ax,y,z in 3D space to a point bx,y in 2D space looking into the first octant can be written mathematically with rotation matrices as: = = 1 6 where α = arcsin ≈ 35.264° and β = 45°. As explained above, this is a rotation around the vertical axis by β, followed by a rotation around the horizontal axis by α; this is followed by an orthographic projection to the xy-plane: =