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The Battle Cats

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The Battle Cats is a free-to-play tower defense video game developed and published by PONOS Corporation for iOS and Android , originally released in Japan under the name Nyanko Great War ( にゃんこ大戦争 , Nyanko Dai Sensō ) . The Battle Cats debuted on the Japanese iOS App Store in November 2012 under the name Battle cats , with Android support following in December of that same year, and became popular in both Japan and Korea. It later was deleted from both the US Google Play stores and App Store , with an improved version released on 17 September 2014. A Microsoft Windows port was released in August 2017, and was available for download until late 2018, when PONOS Corporation ended support for it. It was ported to the Nintendo 3DS in 2016 and Nintendo Switch in Japan in 2018 and Southeast Asia in 2021.

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68-578: The plot of The Battle Cats takes place across four main story sagas, three subchapter sagas in the Legends Stages, and various miscellaneous stages. Dialogue in the form of scrolling text before and after the completion of Chapters, unit and enemy descriptions, and battles during gameplay provide most of the game's lore and story. In total, there are four sagas in the main story, which each contain three chapters, except for "The Aku Realms", which has only one chapter. There are ten chapters in total, but with

136-482: A 2D battlefield to defend the Cat Base and defeat the enemy base. These enemies can have traits such as "Red", "Floating", "Black", "Metal", "Angel", "Alien", "Zombie", "Aku", "Relic", or "Traitless". To win the battle, the player must deploy different types of Cats and use their unique abilities to their advantage. In battle, each Cat and enemy unit has different stats and abilities, which gives units different roles on

204-598: A rigid motion : other rigid motions include rotations and reflections. A translation can also be interpreted as the addition of a constant vector to every point, or as shifting the origin of the coordinate system . A translation operator is an operator T δ {\displaystyle T_{\mathbf {\delta } }} such that T δ f ( v ) = f ( v + δ ) . {\displaystyle T_{\mathbf {\delta } }f(\mathbf {v} )=f(\mathbf {v} +\mathbf {\delta } ).} If v

272-562: A rotation in Euclidean space . rotates points in the xy - Cartesian plane counterclockwise through an angle θ about the origin of the Cartesian coordinate system . To perform the rotation using a rotation matrix R , the position of each point must be represented by a column vector v , containing the coordinates of the point. A rotated vector is obtained by using the matrix multiplication R v . Since matrix multiplication has no effect on

340-413: A scale factor that is the same in all directions. The result of uniform scaling is similar (in the geometric sense) to the original. A scale factor of 1 is normally allowed, so that congruent shapes are also classed as similar. (Some school text books specifically exclude this possibility, just as some exclude squares from being rectangles or circles from being ellipses.) More general is scaling with

408-614: A vector v = ( v x , v y , v z ), each homogeneous coordinate vector p = ( p x , p y , p z , 1) would need to be multiplied with this projective transformation matrix: As shown below, the multiplication will give the expected result: Since the last component of a homogeneous coordinate can be viewed as the denominator of the other three components, a uniform scaling by a common factor s (uniform scaling) can be accomplished by using this scaling matrix: For each vector p = ( p x , p y , p z , 1) we would have which would be homogenized to A convenient way to create

476-430: A complex image is to start with a blank "canvas" raster map (an array of pixels , also known as a bitmap ) filled with some uniform background color and then "draw", "paint" or "paste" simple patches of color onto it, in an appropriate order. In particular the canvas may be the frame buffer for a computer display . Some programs will set the pixel colors directly, but most will rely on some 2D graphics library or

544-438: A description of a document based on 2D computer graphics techniques can be much smaller than the corresponding digital image—often by a factor of 1/1000 or more. This representation is also more flexible since it can be rendered at different resolutions to suit different output devices . For these reasons, documents and illustrations are often stored or transmitted as 2D graphic files . 2D computer graphics started in

612-653: A game where the player gathers a crew of Cats to try and steal from other players' manors while evading various obstacles such as traps and guards. The player can also defend and fortify their own manor to ward off other crews and protect their treasure. On 13 November 2021, PONOS Corp released Battle Cats Quest . In this game, players roll a spherical Cat on land to knock off enemies from various stages and earn coins. Similar to Go! Go! Pogo Cat , players can complete in-game missions for rewards redeemable in The Battle Cats . 2D computer graphics 2D computer graphics

680-536: A menacing message: "We'll surely meet again, Battle Cats. Just you wait..." Legend Stages are stages that supplement the main story of Battle Cats. It begins with the Cats using the time machine they obtained at the end of Empire of Cats to travel twenty years into the past, prior to the Cat God's birth/creation. These stages are split into two main groups: Event Stages and Sub-Chapter sagas. While all Sub-Chapter sagas are canon to

748-400: A new world. Zero Legends (sometimes shortened to ZL) is the third and final Sub-Chapter saga and features bosses previously exclusive to Advent Stages. After travelling through a spacetime distortion to the "other world", the Cats find themselves in an alternate universe at its Big Bang and face Metafilibuster, a corrupted and altered version of Filibuster Obstructa. After emerging victorious,

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816-509: A second Talent Orb after reaching level 60. Cats have level caps , or maximum levels, which differ across Cat units, ranging from as high as 60 to as low as 20. Some Cats can evolve into a True Form after being leveled up to their evolved form, which requires Catfruit or Catfruit Seeds; these are obtained from special stages or from merging Seeds into Fruit. Some Cats require Behemoth Stones or Behemoth Gems to reach their True Form, which are obtained from Behemoth Culling stages. The 13.0 update adds

884-406: A separate scale factor for each axis direction. Non-uniform scaling ( anisotropic scaling , inhomogeneous dilation ) is obtained when at least one of the scaling factors is different from the others; a special case is directional scaling or stretching (in one direction). Non-uniform scaling changes the shape of the object; e.g. a square may change into a rectangle, or into a parallelogram if

952-474: A sound model for certain techniques such as mitered joints and the even–odd rule . Layered models are also used to allow the user to suppress unwanted information when viewing or printing a document, e.g. roads or railways from a map , certain process layers from an integrated circuit diagram, or hand annotations from a business letter. In a layer-based model, the target image is produced by "painting" or "pasting" each layer, in order of decreasing depth, on

1020-546: A specific order. The ordering is usually defined by a single number (the layer's depth , or distance from the viewer). Layered models are sometimes called "2 1 ⁄ 2 -D computer graphics". They make it possible to mimic traditional drafting and printing techniques based on film and paper, such as cutting and pasting; and allow the user to edit any layer without affecting the others. For these reasons, they are used in most graphics editors . Layered models also allow better spatial anti-aliasing of complex drawings and provide

1088-406: A translation matrix can be obtained by reversing the direction of the vector: Similarly, the product of translation matrices is given by adding the vectors: Because addition of vectors is commutative , multiplication of translation matrices is therefore also commutative (unlike multiplication of arbitrary matrices). In linear algebra , a rotation matrix is a matrix that is used to perform

1156-405: Is a fixed vector, then the translation T v will work as T v ( p ) = p + v . If T is a translation, then the image of a subset A under the function T is the translation of A by T . The translation of A by T v is often written A + v . In a Euclidean space , any translation is an isometry . The set of all translations forms the translation group T , which

1224-615: Is an area where the player can spend Cat Tickets, Rare Cat Tickets, Lucky Tickets, and Cat Food to obtain Cat Units and Ability Capsules. The rewards earned are then either used to add a "plus-level" to a Cat, exchanged for NP/XP, saved in the Cat Storage to be used later, or exchanged for a Rare Ticket. While many cats can be obtained outside of gacha, some Rare and Super Rare Cats, as well as Uber Super Rare and Legend Rare Cats, are exclusive to it. The Battle Cats POP! (often abbreviated to BCP)

1292-407: Is isomorphic to the space itself, and a normal subgroup of Euclidean group E ( n ). The quotient group of E ( n ) by T is isomorphic to the orthogonal group O ( n ): Since a translation is an affine transformation but not a linear transformation , homogeneous coordinates are normally used to represent the translation operator by a matrix and thus to make it linear. Thus we write

1360-610: Is known as inverting color or color inversion , and is often used in graphical user interfaces for highlighting, rubber-band drawing, and other volatile painting—since re-painting the same shapes with the same color will restore the original pixel values. The models used in 2D computer graphics usually do not provide for three-dimensional shapes, or three-dimensional optical phenomena such as lighting, shadows , reflection , refraction , etc. However, they usually can model multiple layers (conceptually of ink, paper, or film; opaque, translucent , or transparent —stacked in

1428-415: Is not commonly used in dimensions higher than 3; there is a notion of a rotational displacement , which can be represented by a matrix, but no associated single axis or angle. Rotation matrices are square matrices , with real entries. More specifically they can be characterized as orthogonal matrices with determinant  1: The set of all such matrices of size n forms a group , known as

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1496-542: Is the computer-based generation of digital images —mostly from two-dimensional models (such as 2D geometric models , text, and digital images) and by techniques specific to them. It may refer to the branch of computer science that comprises such techniques or to the models themselves. 2D computer graphics are mainly used in applications that were originally developed upon traditional printing and drawing technologies, such as typography , cartography , technical drawing , advertising , etc. In those applications,

1564-406: Is used, with the x axis to the right and the y axis up, the rotation R( θ ) is counterclockwise. If a left-handed Cartesian coordinate system is used, with x directed to the right but y directed down, R( θ ) is clockwise. Such non-standard orientations are rarely used in mathematics but are common in 2D computer graphics, which often have the origin in the top left corner and the y -axis down

1632-446: The special orthogonal group SO( n ) . In two dimensions every rotation matrix has the following form: This rotates column vectors by means of the following matrix multiplication : So the coordinates (x',y') of the point (x,y) after rotation are: The direction of vector rotation is counterclockwise if θ is positive (e.g. 90°), and clockwise if θ is negative (e.g. -90°). If a standard right-handed Cartesian coordinate system

1700-693: The 1950s, based on vector graphics devices . These were largely supplanted by raster-based devices in the following decades. The PostScript language and the X Window System protocol were landmark developments in the field. 2D graphics models may combine geometric models (also called vector graphics ), digital images (also called raster graphics ), text to be typeset (defined by content, font style and size, color, position, and orientation), mathematical functions and equations , and more. These components can be modified and manipulated by two-dimensional geometric transformations such as translation , rotation , and scaling . In object-oriented graphics ,

1768-402: The 3-dimensional vector w = ( w x , w y , w z ) using 4 homogeneous coordinates as w = ( w x , w y , w z , 1). To translate an object by a vector v , each homogeneous vector p (written in homogeneous coordinates) would need to be multiplied by this translation matrix : As shown below, the multiplication will give the expected result: The inverse of

1836-506: The 3DS version." The Battle Cats Unite! was released for the Nintendo Switch by Bandai Namco Entertainment in Japan and Southeast Asia on 9 December 2021. The game is a revamped port of The Battle Cats , with a new two-player co-op mode and many minigames exclusive to this version. Battle Cats Rangers , an idle/clicker game published by MEMORY Inc and developed with YD Online Corp,

1904-426: The 4 entries of a rotation matrix with 2 rows and 2 columns. In 3-dimensional space, every rotation can be interpreted as a rotation by a given angle about a single fixed axis of rotation (see Euler's rotation theorem ), and hence it can be simply described by an angle and a vector with 3 entries. However, it can also be represented by the 9 entries of a rotation matrix with 3 rows and 3 columns. The notion of rotation

1972-540: The Aku Altars are universally deactivated, allowing them to venture into areas they had been previously obstructed from. With his plans foiled and his army in shambles, Mamon loses motivation to continue his plans. However, this allows him to refocus his mind on his family, whom he reunites with for the first time in nearly a decade. He thanks the Battle Cats for giving him an opportunity to change his ways, but leaves them with

2040-482: The Aku Realms to destroy the altars and regain their strength. Upon reaching Norway and destroying half of the altars, the Cats arrive at "Mount Aku", where Dark Priest Mamon attacks them and they face him with half their strength. The Cats prevail and force Mamon to retreat, but he flees with the crystal of awakening; the last component needed to awaken Jagando. The Cats finish destroying the altars on Earth and travel to

2108-564: The Aku's power possesses her and she becomes the Possessed Empress. Upon entering the Aku Realms, the Cats find more altars that weaken them, which the dark priest Mamon built to prevent others from interfering with his plan. He seeks to unleash the god of destruction, Jagando, the Lord of Ruin, by undoing the seals of power the heroes of legend placed on him. With their power limited, the Cats fight across

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2176-429: The Aku, but it is unknown what the latter refers to. Near the end of the saga, the Cats begin a project to "Catify" all of humanity and come across the ancient Sacred Tree, whose guardian is Zero Luza, the ancestor of humanity. After defeating him, a God emerges from the tree and tells the Cats to find a sage in another world while drawing a can of Cat Food in the sky. The Cats, with a new journey ahead of them, venture into

2244-491: The Cat army travels the world to conquer its nations. Their inferred creator/leader is the fictional deity, The Cat God, who can be unlocked early on as a power-up and as a unit upon further progression. Throughout the saga, the Cat army recruits special Legend units, who are similar to the mythical "The Heroes of Legend"; ancient warriors who wielded immense power. After conquering Earth, the Cats end resistance to their rule by conquering

2312-435: The Cats soon find that the laws of this universe are drastically different from their own, with natural phenomena such as snow in volcanoes and forests filled with poisonous gas. As well, enemies that are individuals in their world are commonplace here, and a new type of enemies, the mysterious Sage enemies, appear. The Battle Cats is a tower defense game where the player selects a team of unlockable "Cats" and sends them onto

2380-674: The Cats' time machine brings them to an ancient world, where Relic enemies, who have the ability to Curse their opponents, have not yet gone extinct. The Cats soon realize that some of the Legend units recruited during Stories of Legend are descendants of the true Heroes of Legend. Like before, the Cat Army defeats and recruits these ancient Legend units into their army. During their journey, the Cats also find "a dark guild researching forbidden knowledge" and "the REAL ultimate superweapon". The former likely refers to

2448-858: The Future (ItF), and Cats of the Cosmos (CotC), with each story containing three chapters, and an extra story, the Aku Realms. In addition, there are the Legend Stages, which are extra and less linear stages that supplement the main sagas. There are also two additional areas known as the Catclaw Dojo and the Underground Labyrinth, which are bonus areas designed to challenge the player in unique ways. The Legend Stages are further divided into three main categories: Stories of Legend (SoL), Uncanny Legends (UL), and Zero Legends (ZL), with each being unlocked after completing

2516-517: The Heroes of Legend. After the Cats accomplish their goals, they travel back to the present; however, upon emerging from the time machine, they find themselves in an unfamiliar place. Uncanny Legends (often shortened to UL) is the second of the Sub-Chapter Sagas and takes place after Stories of Legend and Cats of the Cosmos. After defeating and recruiting Mecha-Bun, the final boss of Stories of Legend,

2584-518: The Moon and obtaining a time machine. The second main story saga, Into the Future (shortened to ItF), begins after the Cats use the time machine they obtained at the end of Empire of Cats, transporting them to the year 29XX, where Aliens have conquered Earth. The Cats fight across new locations, including the Floating Continent and Great Abyss, and battle new enemies to reconquer the world. After driving

2652-405: The Moon to destroy the last altar and confront High Priest Mamon. With Mamon's defeat, the last altar is seized and the Cats' power is almost fully restored. However, Mamon succeeds in reawakening Jagando, and with him holding the last of the Cats' power, they go back to Mount Aku to face him. They defeat and seal away Jagando once more and take his egg to raise as their own. With the Cats' triumph,

2720-517: The ability for some Uber Rare Cats to evolve into Ultra Forms after reaching level 60, after spending Behemoth Stones or Gems, Catfruit, and XP. Duplicates obtained in the game's Gacha systems can be used to add additional "plus-levels" to a Cat unit. The Cats are divided into six rarities, which are in increasing rarity: Normal, Special, Rare, Super Rare, Uber Super Rare (commonly shortened to Uber Rare or just Uber), and Legend Rare. The Battle Cats contains three main stories: Empire of Cats (EoC), Into

2788-427: The aliens off the planet, the Cats arrive on the moon, as they had during Empire of Cats. However, this time they face corrupted versions of the Legend units (except for the first chapter) they recruited during Empire of Cats. After defeating them, the Legend units unlock their true forms, gaining more power and abilities. The saga ends with the Cat army deciding to build a rocket to venture further into space and continue

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2856-417: The battlefield, such as being a meat shield , crowd controller , or sniper. Along with base stats, a unit can also have abilities, which can improve their performance against a specific type of enemy or gives them a unique ability that can aid them in battle. For example, the Cat "Axe Cat" is strong against the "Red" enemy type, meaning it deals more and takes less damage against "Red" enemies. The Cats are

2924-475: The case of one or more negative scale factors. The latter corresponds to a combination of scaling proper and a kind of reflection: along lines in a particular direction we take the reflection in the point of intersection with a plane that need not be perpendicular; therefore it is more general than ordinary reflection in the plane. In projective geometry , often used in computer graphics , points are represented using homogeneous coordinates . To scale an object by

2992-468: The case where v x = v y = v z = k , the scaling is also called an enlargement or dilation by a factor k, increasing the area by a factor of k and the volume by a factor of k . Scaling in the most general sense is any affine transformation with a diagonalizable matrix . It includes the case that the three directions of scaling are not perpendicular. It includes also the case that one or more scale factors are equal to zero ( projection ), and

3060-412: The energy meter, it can be addictive like Pokémon. Too much of its free-to-play roots are still present, but even with these elements, it's still a worthwhile time killer." Matt S. of Digitally Downloaded gave 3.5 stars out of 5 and noted "what sets Battle Cats apart from the many similar games of similar depth and strategy is that hugely creative edge. This game is the distinctive and memorable example of

3128-439: The expected result: Such a scaling changes the diameter of an object by a factor between the scale factors, the area by a factor between the smallest and the largest product of two scale factors, and the volume by the product of all three. The scaling is uniform if and only if the scaling factors are equal ( v x = v y = v z ). If all except one of the scale factors are equal to 1, we have directional scaling. In

3196-406: The fabled checker paint which used to be available only in cartoons ). Painting a pixel with a given color usually replaces its previous color. However, many systems support painting with transparent and translucent colors, which only modify the previous pixel values. The two colors may also be combined in more complex ways, e.g. by computing their bitwise exclusive or . This technique

3264-489: The fight against the aliens. The third main saga, Cats of the Cosmos (often shortened to CotC), begins with the Cats invading the cosmos, presumably using the rocket they built at the end of Into the Future. However, the Aliens out in space are unlike those on Earth, as their power is enhanced with stars and they can warp Cat units away and nullify damage with barriers. The Cats battle across the solar system and venture further into

3332-431: The first three sagas, each chapter is essentially a reskin of the previous one. They contain the same levels, theme, and treasures across all three chapters, with the only difference being the amount, difficulty, and selection of enemies that appear in levels. Additionally, the opening and ending dialogue to each saga is reused across them. The story begins in the first saga, Empire of Cats (commonly shortened to EoC), where

3400-510: The galaxy, eventually arriving at "The Big Bang", where they fight the Cat God. After defeating him, he reveals he was the narrator throughout the saga and that, despite being an ancient deity, he is actually around twenty years old. After the cat and Alien forces come to a pseudo-ceasefire, the Cat God declares the cats should move onward to the Realm of Legends. However, an entity of chaos and destruction known as Filibuster Obstructa arrives from outside

3468-473: The galaxy, serving as the Cat Army's final test in their conquest of the Cosmos. The last main saga, The Aku Realms, takes place some time after Cats of the Cosmos. Its story begins with a Cat leader, Empress Cat, conducting research on the mysterious Aku, who are connected to altars that weaken the Cat Army's power. With help from the Cat Aku Researcher, she manages to open a portal to the Aku Realms, but

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3536-682: The game. Over time, players can add more Cats to their team and encounter stronger enemies they have to defeat. After two years, the game shut down its online servers on 29 March 2019 and could no longer be played. Five years after the release of The Battle Cats , PONOS Corp released Go! Go! Pogo Cat , a simple infinite platformer similar to Flappy Bird . The player directs a Cat riding a pogo stick to gain as much distance as possible, collecting items like gems. Players that have The Battle Cats downloaded can complete certain missions in-game for rewards redeemable in The Battle Cats . On 23 February 2021, PONOS Corp announced and released The Burgle Cats ,

3604-423: The genre, because the artists had the sense to do something that makes no sense, but we end up looking forward to each new level just to see the insanity that it brings." However, the stamina system was criticized by reviewers, as Louisa Bhairam with Nintendo Life stated "It's a bit of an odd system for a game which requires no micro-transactions   ... It's a feature which should have been removed entirely for

3672-442: The image is described indirectly by an object endowed with a self- rendering method —a procedure that assigns colors to the image pixels by an arbitrary algorithm. Complex models can be built by combining simpler objects, in the paradigms of object-oriented programming . In Euclidean geometry , a translation (geometry) moves every point a constant distance in a specified direction. A translation can be described as

3740-436: The machine's graphics card , which usually implement the following operations: Text, shapes and lines are rendered with a client-specified color. Many libraries and cards provide color gradients , which are handy for the generation of smoothly-varying backgrounds, shadow effects, etc. (See also Gouraud shading ). The pixel colors can also be taken from a texture, e.g. a digital image (thus emulating rub-on screentones and

3808-402: The player competes against other players to defeat the most enemies as quickly as possible. At the end of the event, a reward is given out to each participant based on their performance. The Underground Labyrinth is a collection of 100 stages with an unknown schedule that reward the player with rare items like Dark Catseyes. Despite the early stages being relatively easy, the challenge comes from

3876-551: The previous. There is also an additional category for event and Collaboration stages. The Catclaw Dojo contains a stage to test how many enemies the player can defeat within a given period of time. This challenge is amplified by the money being scarce, as the enemies don't drop money when killed, forcing the player to send out units strategically and save their money. While the Dojo usually offers no reward, special stages, ranked events, and collaboration events occur rather frequently, in which

3944-600: The protagonists that the player deploys throughout the game. Many Cats do not resemble typical cats, and are sometimes equipped with external objects such as weapons or machines, or are characters earned through collaboration events. A player can spend XP and "Catseyes", resources obtained by completing levels, to upgrade a Cat, as well as NP (Nyanko Points) to obtain and upgrade Talents, which give additional abilities. Version 12.1 added Ultra Talents, allowing certain Uber Rare Cats to unlock one to three additional Talents and equip

4012-516: The restrictions placed on the unit selection. In the Labyrinth, if a unit is used, it cannot be removed from the Equip menu. The only way to remove a unit from the player's lineup is for the unit to become "trapped". After defeating or losing a level, units can become trapped and are unusable, but the player can be given the choice to "rescue" a unit from being trapped, allowing them to still be usable. However,

4080-408: The screen or page. See below for other alternative conventions which may change the sense of the rotation produced by a rotation matrix . Particularly useful are the matrices for 90° and 180° rotations: In Euclidean geometry , uniform scaling ( isotropic scaling , homogeneous dilation , homothety ) is a linear transformation that enlarges (increases) or shrinks (diminishes) objects by

4148-410: The sides of the square are not parallel to the scaling axes (the angles between lines parallel to the axes are preserved, but not all angles). A scaling can be represented by a scaling matrix. To scale an object by a vector v = ( v x , v y , v z ), each point p = ( p x , p y , p z ) would need to be multiplied with this scaling matrix : As shown below, the multiplication will give

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4216-539: The story of Battle Cats, some Event stages, such as collaboration stages, are not. While all Legend stages take place after the first chapter of Empire of Cats, where they fit within the greater story and timeline is unknown. Stories of Legend (often shortened to SoL), is the first of the Sub-Chapter sagas and has 49 sub-chapters, with 1-8 stages in each one. They fight new and old enemies as they aim to collect Cat Food and recruit Legend units to join their army, who resemble

4284-476: The traps are unpredictable, since they vary in how many units they trap and sometimes do not allow the player to rescue units. The Labyrinth forces players to ration their units carefully and use different units, adding a unique challenge seen nowhere else in the game. It is inconclusive whether or not the events that take place in the Catclaw Dojo and the Underground Labyrinth are canon to the story of The Battle Cats . Cat Capsules, often referred to as gacha ,

4352-409: The two-dimensional image is not just a representation of a real-world object, but an independent artifact with added semantic value; two-dimensional models are therefore preferred, because they give more direct control of the image than 3D computer graphics (whose approach is more akin to photography than to typography). In many domains, such as desktop publishing , engineering , and business ,

4420-743: The virtual canvas. Conceptually, each layer is first rendered on its own, yielding a digital image with the desired resolution which is then painted over the canvas, pixel by pixel. Fully transparent parts of a layer need not be rendered, of course. The rendering and painting may be done in parallel, i.e., each layer pixel may be painted on the canvas as soon as it is produced by the rendering procedure. Layers that consist of complex geometric objects (such as text or polylines ) may be broken down into simpler elements ( characters or line segments , respectively), which are then painted as separate layers, in some order. However, this solution may create undesirable aliasing artifacts wherever two elements overlap

4488-452: The zero vector (i.e., on the coordinates of the origin), rotation matrices can only be used to describe rotations about the origin of the coordinate system. Rotation matrices provide a simple algebraic description of such rotations, and are used extensively for computations in geometry , physics , and computer graphics . In 2-dimensional space, a rotation can be simply described by an angle θ of rotation , but it can be also represented by

4556-449: Was released for Android and iOS devices on 27 April 2017. In Battle Cats Rangers , players tap the screen as quickly as possible to attack incoming enemies and bosses that swarm their team of Cats. Two weeks prior to release, a pre-registration campaign was announced which encouraged users to sign up for in-game currency. By using coins and cat food, the in-game currency, players can level up their Cats to deal more damage and progress through

4624-485: Was released for the Nintendo 3DS worldwide on 27 June 2016. The game features mechanics similar to The Battle Cats , but also introduces a two-player VS mode and made use of the 3DS' 3D capabilities. The Battle Cats POP! received average reviews. CJ Andriessen of Destructoid gave the game a 7 out of 10 and wrote that "The Battle Cats POP! is a game that's as fun as it is weird. When it's not putting you in time out with

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