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68-471: Unlike other self-contained game systems that fit entirely in a controller, the Net Jet itself can not play games on its own. It does not contain any circuitry to display graphics, generate sound or a microprocessor to play games, instead it relies entirely on the host computer to act as the "game console". For this reason, its performance was potentially hindered by slower and less capable computers. Games consist of

136-458: A lookup table ), and the perspective correctness was about 16 times more expensive. The Doom engine restricted the world to vertical walls and horizontal floors/ceilings, with a camera that could only rotate about the vertical axis. This meant the walls would be a constant depth coordinate along a vertical line and the floors/ceilings would have a constant depth along a horizontal line. After performing one perspective correction calculation for

204-405: A Face index list. This is usually done only when the geometry changes. Winged-edge meshes are ideally suited for dynamic geometry, such as subdivision surfaces and interactive modeling, since changes to the mesh can occur locally. Traversal across the mesh, as might be needed for collision detection, can be accomplished efficiently. See Baumgart (1975) for more details. Winged-edge meshes are not

272-640: A data source, as the point of view nears the objects. As an optimization, it is possible to render detail from a complex, high-resolution model or expensive process (such as global illumination ) into a surface texture (possibly on a low-resolution model). Baking is also known as render mapping . This technique is most commonly used for light maps , but may also be used to generate normal maps and displacement maps . Some computer games (e.g. Messiah ) have used this technique. The original Quake software engine used on-the-fly baking to combine light maps and colour maps (" surface caching "). Baking can be used as

340-657: A form of level of detail generation, where a complex scene with many different elements and materials may be approximated by a single element with a single texture, which is then algorithmically reduced for lower rendering cost and fewer drawcalls . It is also used to take high-detail models from 3D sculpting software and point cloud scanning and approximate them with meshes more suitable for realtime rendering. Various techniques have evolved in software and hardware implementations. Each offers different trade-offs in precision, versatility and performance. Affine texture mapping linearly interpolates texture coordinates across

408-431: A four-sided box as represented by a VV mesh. Each vertex indexes its neighboring vertices. The last two vertices, 8 and 9 at the top and bottom center of the "box-cylinder", have four connected vertices rather than five. A general system must be able to handle an arbitrary number of vertices connected to any given vertex. For a complete description of VV meshes see Smith (2006). Face-vertex meshes represent an object as

476-429: A free demo key. The demo key allowed a user to try each of the Net Jet games for 30 minutes. Players would then have to purchase one of twelve different game keys in order to unlock each individual game already on the computer. Despite the actual games being stored on the computer's hard drive , without a key inserted the game system would not function. Later on, the demo key was replaced in branded bundled with some of

544-406: A game key, the user was also granted a free 30-minute demo, with the option to purchase the full game, of a variety of standard computer games, called "Deluxe Games." These titles were played with the keyboard and mouse, rather than the Net Jet controller itself. As of January 1, 2009, Hasbro was no longer legally obligated to maintain or continue its website operation. Support, however, was extended

612-525: A general rule, face-vertex meshes are used whenever an object must be rendered on graphics hardware that does not change geometry (connectivity), but may deform or morph shape (vertex positions) such as real-time rendering of static or morphing objects. Winged-edge or render dynamic meshes are used when the geometry changes, such as in interactive modeling packages or for computing subdivision surfaces. Vertex-vertex meshes are ideal for efficient, complex changes in geometry or topology so long as hardware rendering

680-450: A line of pixels to simplify the set-up (compared to 2d affine interpolation) and thus again the overhead (also affine texture-mapping does not fit into the low number of registers of the x86 CPU; the 68000 or any RISC is much more suited). A different approach was taken for Quake , which would calculate perspective correct coordinates only once every 16 pixels of a scanline and linearly interpolate between them, effectively running at

748-497: A mix of real-time 3D texture mapping , to simple 2D Java and flash-based games. The Net Jet makes itself plug n play by containing a small bit of firmware in the controller. When plugged in, a virtual CD-ROM image (400K in size) would automatically mount, then proceed to download and install its software via the Internet so to display a menu selection screen. Any games selected would then also need to be downloaded and stored on

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816-420: A modern evolution of tile map graphics ). Modern hardware often supports cube map textures with multiple faces for environment mapping. Texture maps may be acquired by scanning / digital photography , designed in image manipulation software such as GIMP , Photoshop , or painted onto 3D surfaces directly in a 3D paint tool such as Mudbox or ZBrush . This process is akin to applying patterned paper to

884-500: A perspective correct texture mapping. To do this, we first calculate the reciprocals at each vertex of our geometry (3 points for a triangle). For vertex n {\displaystyle n} we have u n z n , v n z n , 1 z n {\displaystyle {\frac {u_{n}}{z_{n}}},{\frac {v_{n}}{z_{n}}},{\frac {1}{z_{n}}}} . Then, we linearly interpolate these reciprocals between

952-502: A place on the screen, a forward texture mapping renderer iterates through each texel on the texture, splatting each one onto a pixel of the frame buffer . This was used by some hardware, such as the 3DO , the Sega Saturn and the NV1 . The primary advantage is that the texture will be accessed in a simple linear order, allowing very efficient caching of the texture data. However, this benefit

1020-536: A plain white box. Every vertex in a polygon is assigned a texture coordinate (which in the 2d case is also known as UV coordinates ). This may be done through explicit assignment of vertex attributes , manually edited in a 3D modelling package through UV unwrapping tools . It is also possible to associate a procedural transformation from 3D space to texture space with the material . This might be accomplished via planar projection or, alternatively, cylindrical or spherical mapping. More complex mappings may consider

1088-405: A quad looks less incorrect than the same quad split into two triangles (see affine texture mapping above). The NV1 hardware also allowed a quadratic interpolation mode to provide an even better approximation of perspective correctness. Polygon mesh In 3D computer graphics and solid modeling , a polygon mesh is a collection of vertices , edge s and face s that defines

1156-401: A set of faces and a set of vertices. This is the most widely used mesh representation, being the input typically accepted by modern graphics hardware. Face-vertex meshes improve on VV-mesh for modeling in that they allow explicit lookup of the vertices of a face, and the faces surrounding a vertex. The above figure shows the "box-cylinder" example as an FV mesh. Vertex v5 is highlighted to show

1224-533: A surface, and so is the fastest form of texture mapping. Some software and hardware (such as the original PlayStation ) project vertices in 3D space onto the screen during rendering and linearly interpolate the texture coordinates in screen space between them. This may be done by incrementing fixed point UV coordinates , or by an incremental error algorithm akin to Bresenham's line algorithm . In contrast to perpendicular polygons, this leads to noticeable distortion with perspective transformations (see figure:

1292-430: A texture to the screen: Inverse texture mapping is the method which has become standard in modern hardware. With this method, a pixel on the screen is mapped to a point on the texture. Each vertex of a rendering primitive is projected to a point on the screen, and each of these points is mapped to a u,v texel coordinate on the texture. A rasterizer will interpolate between these points to fill in each pixel covered by

1360-510: A time on a polygon. For instance, a light map texture may be used to light a surface as an alternative to recalculating that lighting every time the surface is rendered. Microtextures or detail textures are used to add higher frequency details, and dirt maps may add weathering and variation; this can greatly reduce the apparent periodicity of repeating textures. Modern graphics may use more than 10 layers, which are combined using shaders , for greater fidelity. Another multitexture technique

1428-745: A year past said date, as at start up the software displays a pop-up disclaimer stating the product has been discontinued and support ended as of December 31, 2009. All online service has since been shut down, and as a result, the product no longer functions. The actual message from Net Jet software when controller is plugged-in: " PLEASE NOTE: We hope you have enjoyed your NET JET Online Gaming System. Unfortunately, NET JET products have been discontinued. This site will no longer be available on or about December 31, 2009. We appreciate your interest in Hasbro products. If you have any questions, please contact Hasbro Customer Service. " Texture mapping Texture mapping

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1496-502: Is bump mapping , which allows a texture to directly control the facing direction of a surface for the purposes of its lighting calculations; it can give a very good appearance of a complex surface (such as tree bark or rough concrete) that takes on lighting detail in addition to the usual detailed coloring. Bump mapping has become popular in recent video games, as graphics hardware has become powerful enough to accommodate it in real-time. The way that samples (e.g. when viewed as pixels on

1564-451: Is a means of using data streams for textures, where each texture is available in two or more different resolutions, as to determine which texture should be loaded into memory and used based on draw distance from the viewer and how much memory is available for textures. Texture streaming allows a rendering engine to use low resolution textures for objects far away from the viewer's camera, and resolve those into more detailed textures, read from

1632-465: Is a method for mapping a texture on a computer-generated graphic . "Texture" in this context can be high frequency detail , surface texture , or color . The original technique was pioneered by Edwin Catmull in 1974 as part of his doctoral thesis. Texture mapping originally referred to diffuse mapping , a method that simply mapped pixels from a texture to a 3D surface ("wrapping" the image around

1700-409: Is also its disadvantage: as a primitive gets smaller on screen, it still has to iterate over every texel in the texture, causing many pixels to be overdrawn redundantly. This method is also well suited for rendering quad primitives rather than reducing them to triangles, which provided an advantage when perspective correct texturing was not available in hardware. This is because the affine distortion of

1768-418: Is constant time. However, the edges are implicit, so a search is still needed to find all the faces surrounding a given face. Other dynamic operations, such as splitting or merging a face, are also difficult with face-vertex meshes. Introduced by Baumgart in 1975, winged-edge meshes explicitly represent the vertices, faces, and edges of a mesh. This representation is widely used in modeling programs to provide

1836-460: Is possible to use the alpha channel (which may be convenient to store in formats parsed by hardware) for other uses such as specularity . Multiple texture maps (or channels ) may be combined for control over specularity , normals , displacement , or subsurface scattering e.g. for skin rendering. Multiple texture images may be combined in texture atlases or array textures to reduce state changes for modern hardware. (They may be considered

1904-582: Is ubiquitous as most SOCs contain a suitable GPU. Some hardware combines texture mapping with hidden-surface determination in tile based deferred rendering or scanline rendering ; such systems only fetch the visible texels at the expense of using greater workspace for transformed vertices. Most systems have settled on the Z-buffering approach, which can still reduce the texture mapping workload with front-to-back sorting . Among earlier graphics hardware, there were two competing paradigms of how to deliver

1972-457: Is used on them. The reason this technique works is that the distortion of affine mapping becomes much less noticeable on smaller polygons. The Sony PlayStation made extensive use of this because it only supported affine mapping in hardware but had a relatively high triangle throughput compared to its peers. Software renderers generally preferred screen subdivision because it has less overhead. Additionally, they try to do linear interpolation along

2040-747: The n {\displaystyle n} vertices (e.g., using barycentric coordinates ), resulting in interpolated values across the surface. At a given point, this yields the interpolated u i , v i {\displaystyle u_{i},v_{i}} , and z R e c i p r o c a l i = 1 z i {\displaystyle zReciprocal_{i}={\frac {1}{z_{i}}}} . Note that this u i , v i {\displaystyle u_{i},v_{i}} cannot be yet used as our texture coordinates as our division by z {\displaystyle z} altered their coordinate system. To correct back to

2108-989: The u , v {\displaystyle u,v} space we first calculate the corrected z {\displaystyle z} by again taking the reciprocal z c o r r e c t = 1 z R e c i p r o c a l i = 1 1 z i {\displaystyle z_{correct}={\frac {1}{zReciprocal_{i}}}={\frac {1}{\frac {1}{z_{i}}}}} . Then we use this to correct our u i , v i {\displaystyle u_{i},v_{i}} : u c o r r e c t = u i ⋅ z i {\displaystyle u_{correct}=u_{i}\cdot z_{i}} and v c o r r e c t = v i ⋅ z i {\displaystyle v_{correct}=v_{i}\cdot z_{i}} . This correction makes it so that in parts of

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2176-491: The forward texture mapping used by the Nvidia NV1 , was able to offer efficient quad primitives. With perspective correction (see below) triangles become equivalent and this advantage disappears. For rectangular objects that are at right angles to the viewer, like floors and walls, the perspective only needs to be corrected in one direction across the screen, rather than both. The correct perspective mapping can be calculated at

2244-449: The Net Jet's marquee titles, such as Transformers Battle Universe or Bubble Bonanza , for example. Game keys primarily function as jumper blocks (an activation switch) to unlock games on the computer. Actual games are downloaded and stored, along with save game settings, onto the computer through an Internet connection. Games or save-status are not stored on the keys. The keys however permanently "remember" which three choice games

2312-409: The above table, explicit indicates that the operation can be performed in constant time, as the data is directly stored; list compare indicates that a list comparison between two lists must be performed to accomplish the operation; and pair search indicates a search must be done on two indices. The notation avg(V,V) means the average number of vertices connected to a given vertex; avg(E,V) means

2380-445: The appearance of a texture mapped landscape without the use of traditional geometric primitives. Every triangle can be further subdivided into groups of about 16 pixels in order to achieve two goals. First, keeping the arithmetic mill busy at all times. Second, producing faster arithmetic results. For perspective texture mapping without hardware support, a triangle is broken down into smaller triangles for rendering and affine mapping

2448-410: The average number of edges connected to a given vertex, and avg(F,V) is the average number of faces connected to a given vertex. The notation "V → f1, f2, f3, ... → v1, v2, v3, ..." describes that a traversal across multiple elements is required to perform the operation. For example, to get "all vertices around a given vertex V" using the face-vertex mesh, it is necessary to first find the faces around

2516-448: The benefit that changes in shape, but not geometry, can be dynamically updated by simply resending the vertex data without updating the face connectivity. Modeling requires easy traversal of all structures. With face-vertex meshes it is easy to find the vertices of a face. Also, the vertex list contains a list of faces connected to each vertex. Unlike VV meshes, both faces and vertices are explicit, so locating neighboring faces and vertices

2584-448: The checker box texture appears bent), especially as primitives near the camera . Such distortion may be reduced with the subdivision of the polygon into smaller ones. For the case of rectangular objects, using quad primitives can look less incorrect than the same rectangle split into triangles, but because interpolating 4 points adds complexity to the rasterization, most early implementations preferred triangles only. Some hardware, such as

2652-413: The computer. The Net Jet is a proprietary controller, it can only be used for its own games and not as a standard PC gamepad controller (it was detected as a USB hub, as opposed to a game controller by the computer). The controller itself had 7 buttons (two shoulder, four front and one 'start'), d-pad and analog thumb stick. The Net Jet system is played by inserting a key into the controller. It came with

2720-596: The corner-table (triangle fan) is commonly incorporated into low-level rendering APIs such as DirectX and OpenGL . Vertex-vertex meshes represent an object as a set of vertices connected to other vertices. This is the simplest representation, but not widely used since the face and edge information is implicit. Thus, it is necessary to traverse the data in order to generate a list of faces for rendering. In addition, operations on edges and faces are not easily accomplished. However, VV meshes benefit from small storage space and efficient morphing of shape. The above figure shows

2788-427: The data, and the operations to be performed. For example, it is easier to deal with triangles than general polygons, especially in computational geometry . For certain operations it is necessary to have a fast access to topological information such as edges or neighboring faces; this requires more complex structures such as the winged-edge representation. For hardware rendering, compact, simple structures are needed; thus

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2856-429: The depth, the rest of the line could use fast affine mapping. Some later renderers of this era simulated a small amount of camera pitch with shearing which allowed the appearance of greater freedom whilst using the same rendering technique. Some engines were able to render texture mapped Heightmaps (e.g. Nova Logic 's Voxel Space , and the engine for Outcast ) via Bresenham -like incremental algorithms, producing

2924-458: The distance along a surface to minimize distortion. These coordinates are interpolated across the faces of polygons to sample the texture map during rendering. Textures may be repeated or mirrored to extend a finite rectangular bitmap over a larger area, or they may have a one-to-one unique " injective " mapping from every piece of a surface (which is important for render mapping and light mapping , also known as baking ). Texture mapping maps

2992-461: The face list contains an index of vertices. In addition, traversal from vertex to face is explicit (constant time), as is from face to vertex. RD meshes do not require the four outgoing edges since these can be found by traversing from edge to face, then face to neighboring edge. RD meshes benefit from the features of winged-edge meshes by allowing for geometry to be dynamically updated. See Tobler & Maierhofer ( WSCG 2006) for more details. In

3060-447: The faces that surround it. Notice that, in this example, every face is required to have exactly 3 vertices. However, this does not mean every vertex has the same number of surrounding faces. For rendering, the face list is usually transmitted to the GPU as a set of indices to vertices, and the vertices are sent as position/color/normal structures (in the figure, only position is given). This has

3128-473: The fly, making it unnecessary to store a mesh in a triangulated form. Polygon meshes may be represented in a variety of ways, using different methods to store the vertex, edge and face data. These include: Each of the representations above have particular advantages and drawbacks, further discussed in Smith (2006). The choice of the data structure is governed by the application, the performance required, size of

3196-445: The given vertex V using the vertex list. Then, from those faces, use the face list to find the vertices around them. Winged-edge meshes explicitly store nearly all information, and other operations always traverse to the edge first to get additional info. Vertex-vertex meshes are the only representation that explicitly stores the neighboring vertices of a given vertex. As the mesh representations become more complex (from left to right in

3264-490: The greatest flexibility in dynamically changing the mesh geometry, because split and merge operations can be done quickly. Their primary drawback is large storage requirements and increased complexity due to maintaining many indices. A good discussion of implementation issues of Winged-edge meshes may be found in the book Graphics Gems II . Winged-edge meshes address the issue of traversing from edge to edge, and providing an ordered set of faces around an edge. For any given edge,

3332-468: The left and right edges of the floor, and then an affine linear interpolation across that horizontal span will look correct, because every pixel along that line is the same distance from the viewer. Perspective correct texturing accounts for the vertices' positions in 3D space, rather than simply interpolating coordinates in 2D screen space. This achieves the correct visual effect but it is more expensive to calculate. To perform perspective correction of

3400-625: The line of constant distance for arbitrary polygons and rendering along it. Texture mapping hardware was originally developed for simulation (e.g. as implemented in the Evans and Sutherland ESIG and Singer-Link Digital Image Generators DIG), and professional graphics workstations such as Silicon Graphics , broadcast digital video effects machines such as the Ampex ADO and later appeared in Arcade cabinets , consumer video game consoles , and PC video cards in

3468-883: The mesh's edges are rendered instead of the faces, then the model becomes a wireframe model . Several methods exist for mesh generation , including the marching cubes algorithm. Volumetric meshes are distinct from polygon meshes in that they explicitly represent both the surface and interior region of a structure, while polygon meshes only explicitly represent the surface (the volume is implicit). [REDACTED] Objects created with polygon meshes must store different types of elements. These include vertices, edges, faces, polygons and surfaces. In many applications, only vertices, edges and either faces or polygons are stored. A renderer may support only 3-sided faces, so polygons must be constructed of many of these, as shown above. However, many renderers either support quads and higher-sided polygons, or are able to convert polygons to triangles on

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3536-718: The mid-1990s. In flight simulation , texture mapping provided important motion and altitude cues necessary for pilot training not available on untextured surfaces. It was also in flight simulation applications, that texture mapping was implemented for real-time processing with prefiltered texture patterns stored in memory for real-time access by the video processor. Modern graphics processing units (GPUs) provide specialised fixed function units called texture samplers , or texture mapping units , to perform texture mapping, usually with trilinear filtering or better multi-tap anisotropic filtering and hardware for decoding specific formats such as DXTn . As of 2016, texture mapping hardware

3604-430: The model surface (or screen space during rasterization) into texture space ; in this space, the texture map is visible in its undistorted form. UV unwrapping tools typically provide a view in texture space for manual editing of texture coordinates. Some rendering techniques such as subsurface scattering may be performed approximately by texture-space operations. Multitexturing is the use of more than one texture at

3672-1154: The number of polygons and lighting calculations needed to construct a realistic and functional 3D scene. A texture map is an image applied (mapped) to the surface of a shape or polygon . This may be a bitmap image or a procedural texture . They may be stored in common image file formats , referenced by 3D model formats or material definitions , and assembled into resource bundles . They may have one to three dimensions, although two dimensions are most common for visible surfaces. For use with modern hardware, texture map data may be stored in swizzled or tiled orderings to improve cache coherency . Rendering APIs typically manage texture map resources (which may be located in device memory ) as buffers or surfaces, and may allow ' render to texture ' for additional effects such as post processing or environment mapping . They usually contain RGB color data (either stored as direct color , compressed formats , or indexed color ), and sometimes an additional channel for alpha blending ( RGBA ) especially for billboards and decal overlay textures. It

3740-559: The number of outgoing edges may be arbitrary. To simplify this, winged-edge meshes provide only four, the nearest clockwise and counter-clockwise edges at each end. The other edges may be traversed incrementally. The information for each edge therefore resembles a butterfly, hence "winged-edge" meshes. The above figure shows the "box-cylinder" as a winged-edge mesh. The total data for an edge consists of 2 vertices (endpoints), 2 faces (on each side), and 4 edges (winged-edge). Rendering of winged-edge meshes for graphics hardware requires generating

3808-437: The object). In recent decades, the advent of multi-pass rendering, multitexturing , mipmaps , and more complex mappings such as height mapping , bump mapping , normal mapping , displacement mapping , reflection mapping , specular mapping , occlusion mapping , and many other variations on the technique (controlled by a materials system ) have made it possible to simulate near- photorealism in real time by vastly reducing

3876-478: The only representation which allows for dynamic changes to geometry. A new representation which combines winged-edge meshes and face-vertex meshes is the render dynamic mesh , which explicitly stores both, the vertices of a face and faces of a vertex (like FV meshes), and the faces and vertices of an edge (like winged-edge). Render dynamic meshes require slightly less storage space than standard winged-edge meshes, and can be directly rendered by graphics hardware since

3944-489: The perspective with a faster calculation, such as a polynomial. Still another technique uses 1/z value of the last two drawn pixels to linearly extrapolate the next value. The division is then done starting from those values so that only a small remainder has to be divided but the amount of bookkeeping makes this method too slow on most systems. Finally, the Build engine extended the constant distance trick used for Doom by finding

4012-403: The player selected via a 1K embedded EEPROM IC. Physical releases sold at retail in the form of USB game keys. Purchasing a game key also enabled a user to unlock three Internet Java/Flash-based games playable with the controller, called "Choice Games". Many of these titles were also available on other web gaming portals but had been retooled to support Net Jet specific button inputs. With

4080-616: The polygon that are closer to the viewer the difference from pixel to pixel between texture coordinates is smaller (stretching the texture wider) and in parts that are farther away this difference is larger (compressing the texture). 3D graphics hardware typically supports perspective correct texturing. Various techniques have evolved for rendering texture mapped geometry into images with different quality/precision tradeoffs, which can be applied to both software and hardware. Classic software texture mappers generally did only simple mapping with at most one lighting effect (typically applied through

4148-414: The primitive. The primary advantage is that each pixel covered by a primitive will be traversed exactly once. Once a primitive's vertices are transformed, the amount of remaining work scales directly with how many pixels it covers on the screen. The main disadvantage versus forward texture mapping is that the memory access pattern in the texture space will not be linear if the texture is at an angle to

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4216-551: The screen) are calculated from the texels (texture pixels) is governed by texture filtering . The cheapest method is to use the nearest-neighbour interpolation , but bilinear interpolation or trilinear interpolation between mipmaps are two commonly used alternatives which reduce aliasing or jaggies . In the event of a texture coordinate being outside the texture, it is either clamped or wrapped . Anisotropic filtering better eliminates directional artefacts when viewing textures from oblique viewing angles. Texture streaming

4284-415: The screen. This disadvantage is often addressed by texture caching techniques, such as the swizzled texture memory arrangement. The linear interpolation can be used directly for simple and efficient affine texture mapping, but can also be adapted for perspective correctness . Forward texture mapping maps each texel of the texture to a pixel on the screen. After transforming a rectangular primitive to

4352-842: The shape of a polyhedral object's surface . It simplifies rendering , as in a wire-frame model . The faces usually consist of triangles ( triangle mesh ), quadrilaterals (quads), or other simple convex polygons ( n-gons ). A polygonal mesh may also be more generally composed of concave polygons , or even polygons with holes . The study of polygon meshes is a large sub-field of computer graphics (specifically 3D computer graphics) and geometric modeling . Different representations of polygon meshes are used for different applications and goals. The variety of operations performed on meshes may include: Boolean logic ( Constructive solid geometry ), smoothing , simplification , and many others. Algorithms also exist for ray tracing , collision detection , and rigid-body dynamics with polygon meshes. If

4420-402: The speed of linear interpolation because the perspective correct calculation runs in parallel on the co-processor. The polygons are rendered independently, hence it may be possible to switch between spans and columns or diagonal directions depending on the orientation of the polygon normal to achieve a more constant z but the effort seems not to be worth it. Another technique was approximating

4488-498: The summary), the amount of information explicitly stored increases. This gives more direct, constant time, access to traversal and topology of various elements but at the cost of increased overhead and space in maintaining indices properly. Figure 7 shows the connectivity information for each of the four technique described in this article. Other representations also exist, such as half-edge and corner tables. These are all variants of how vertices, faces and edges index one another. As

4556-400: The surface being textured. In contrast, the original z {\displaystyle z} , u {\displaystyle u} and v {\displaystyle v} , before the division, are not linear across the surface in screen space. We can therefore linearly interpolate these reciprocals across the surface, computing corrected values at each pixel, to result in

4624-536: The texture coordinates u {\displaystyle u} and v {\displaystyle v} , with z {\displaystyle z} being the depth component from the viewer's point of view, we can take advantage of the fact that the values 1 z {\displaystyle {\frac {1}{z}}} , u z {\displaystyle {\frac {u}{z}}} , and v z {\displaystyle {\frac {v}{z}}} are linear in screen space across

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