{"history":[],"article":{"id":"eeef4ef7-0672-4d8f-b10d-c45b5b12db67","title":"Triangle mesh","slug":"triangle-mesh","content":":::summary\n:::\n\n# \nIn [computer graphics](https://bliptext.com/articles/computer-graphics), a **triangle mesh** is a [type](https://bliptext.com/articles/types-of-mesh) of [polygon mesh](https://bliptext.com/articles/polygon-mesh).\nIt comprises a set of [triangles](https://bliptext.com/articles/triangle) (typically in [three dimensions](https://bliptext.com/articles/three-dimensions)) that are connected by their common [edges](https://bliptext.com/articles/edge-geometry) or [vertices](https://bliptext.com/articles/vertex-geometry).\nMany [graphics software](https://bliptext.com/articles/graphics-software) packages and hardware devices can operate more efficiently on triangles that are grouped into meshes than on a similar number of triangles that are presented individually.\nThis is typically because computer graphics do operations on the vertices at the corners of triangles.\nWith individual triangles, the system has to operate on three vertices for every triangle.\nIn a large mesh, there could be eight or more triangles meeting at a single vertex - by processing those vertices just once, it is possible to do a fraction of the work and achieve an identical effect.\nIn many computer graphics applications it is necessary to manage a mesh of triangles.\nThe mesh components are vertices, edges, and triangles.\nAn application might require knowledge of the various connections between the mesh components.\nThese connections can be managed independently of the actual vertex positions.\nThis document describes a simple data structure that is convenient for managing the connections.\nThis is not the only possible data structure.\nMany other types exist and have support for various queries about meshes.\n# [Example of a triangle mesh representing a dolphin | upload.wikimedia.org/wikipedia/commons/f/fb/Dolphin///triangle///mesh.png]\n# [A triangle mesh created by contouring an implicit surface | upload.wikimedia.org/wikipedia/commons/3/35/Impl-flaeche-geschl2.svg]\n\n# Representation\nVarious methods of storing and working with a mesh in computer memory are possible.\nWith the [OpenGL](https://bliptext.com/articles/opengl) and [DirectX](https://bliptext.com/articles/directx) [APIs](https://bliptext.com/articles/api) there are two primary ways of passing a triangle mesh to the graphics hardware, [triangle strips](https://bliptext.com/articles/triangle-strip) and index arrays.\n\n## Triangle strip\nOne way of sharing vertex data between triangles is the triangle strip.\nWith [strips of triangles](https://bliptext.com/articles/triangle-strip) each triangle shares one complete edge with one neighbour and another with the next.\nAnother way is the triangle fan which is a set of connected triangles sharing one central vertex.\nWith these methods [vertices](https://bliptext.com/articles/vertex-computer-graphics) are dealt with efficiently resulting in the need to only process N+2 [vertices](https://bliptext.com/articles/vertex-computer-graphics) in order to draw N triangles.\nTriangle strips are efficient, however the drawback is that it may not be obvious how or convenient to translate an arbitrary triangle mesh into strips.\n\n## The Data Structure\nThe data structure representing the mesh provides support for two basic operations: inserting triangles and removing triangles.\nIt also supports an edge collapse operation that is useful in triangle decimation schemes.\nThe structure provides no support for the vertex positions, but it does assume that each vertex is assigned a unique integer identifier, typically the index of that vertex in an array of contiguous vertex positions.\nA mesh vertex is defined by a single integer and is denoted by hvi.\nA mesh edge is defined by a pair of integers hv0,v1i, each integer corresponding to an end point of the edge.\nTo support edge maps, the edges are stored so that v0 = min(v0,v1).\nA triangle component is defined by a triple of integers hv0,v1,v2i, each integer corresponding to a vertex of the triangle.\nTo support triangle maps, the triangles are stored so that v0 = min(v0,v1,v2).\nObserve that hv0,v1,v2i and hv0,v2,v1i are treated as different triangles.\nAn application requiring double–sided triangles must insert both triples into the data structure.\nFor the sake of avoiding constant reminders about order of indices, in the remainder of the document the pair/triple information does not imply the vertices are ordering in any way (although the implementation does handle the ordering).\nConnectivity between the components is completely determined by the set of triples representing the triangles.\nA triangle t = hv0,v1,v2i has vertices v0, v1, and v2.\nIt has edges e0 = hv0,v1i, e1 = hv1,v2i, and e2 = hv2,v0i.\nThe inverse connections are also known.\nVertex v0 is adjacent to edges e0 and e2 and to triangle t. Vertex v1 is adjacent to edges e0 and e1 and to triangle t. Vertex v2 is adjacent to edges e1 and e2 and to triangle t. All three edges e0, e1, and e2 are adjacent to t.\nHow much of this information a data structure stores is dependent on the needs of an application.\nMoreover, the application might want to have additional information stored at the components.\nThe information stored at a vertex, edge, or triangle is referred to as the vertex attribute, edge attribute, or triangle attribute.\nThe abstract representations of these for the simple data structure described here are\nVertex = ; // v Edge = ; // v0, v1 Triangle ; // v0, v1, v2 VData = ;\nEData = ;\nTData = ;\nVAttribute = ,set>; // data, eset, tset\nEAttribute = >; // data, tset\nTAttribute = ; // data\nVPair = pair;\nEPair = pair;\nTPair = pair;\nVMap = map;\nEMap = map;\nTMap = map;\nMesh = ; // vmap, emap, tmap\nThe maps support the standard insertion and removal functions for a hash table.\nInsertion occurs only if the item does not already exist.\nRemoval occurs only if the item does exist.\n\n## Edge collapse\nThis operation involves identifying an edge hvk, vti where vk is called the keep vertex and vt is called the throw vertex.\nThe triangles that share this edge are removed from the mesh.\nThe vertex vt is also removed from the mesh.\nAny triangles that shared vt have that vertex replaced by vk.\nFigure 1 shows a triangle mesh and a sequence of three edge collapses applied to the mesh.\n\n## Index array\nWith index arrays, a mesh is represented by two separate arrays, one array holding the vertices, and another holding sets of three indices into that array which define a triangle.\nThe graphics system processes the vertices first and renders the triangles afterwards, using the index sets working on the transformed data.\nIn OpenGL, this is supported by the **glDrawElements** primitive when using [Vertex Buffer Object](https://bliptext.com/articles/vertex-buffer-object) (VBO).\nWith this method, any arbitrary set of triangles sharing any arbitrary number of vertices can be stored, manipulated, and passed to the graphics API, without any intermediary processing.","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-26T07:57:55.109Z","updated_at":"2025-01-26T07:57:55.109Z","search_vector":"'mesh':2 'triangl':1","search_content":" \n \n\n# \nIn [computer graphics](https://bliptext.com/articles/computer-graphics), a **triangle mesh** is a [type](https://bliptext.com/articles/types-of-mesh) of [polygon mesh](https://bliptext.com/articles/polygon-mesh).\nIt comprises a set of [triangles](https://bliptext.com/articles/triangle) (typically in [three dimensions](https://bliptext.com/articles/three-dimensions)) that are connected by their common [edges](https://bliptext.com/articles/edge-geometry) or [vertices](https://bliptext.com/articles/vertex-geometry).\nMany [graphics software](https://bliptext.com/articles/graphics-software) packages and hardware devices can operate more efficiently on triangles that are grouped into meshes than on a similar number of triangles that are presented individually.\nThis is typically because computer graphics do operations on the vertices at the corners of triangles.\nWith individual triangles, the system has to operate on three vertices for every triangle.\nIn a large mesh, there could be eight or more triangles meeting at a single vertex - by processing those vertices just once, it is possible to do a fraction of the work and achieve an identical effect.\nIn many computer graphics applications it is necessary to manage a mesh of triangles.\nThe mesh components are vertices, edges, and triangles.\nAn application might require knowledge of the various connections between the mesh components.\nThese connections can be managed independently of the actual vertex positions.\nThis document describes a simple data structure that is convenient for managing the connections.\nThis is not the only possible data structure.\nMany other types exist and have support for various queries about meshes.\n# [Example of a triangle mesh representing a dolphin | upload.wikimedia.org/wikipedia/commons/f/fb/Dolphin///triangle///mesh.png]\n# [A triangle mesh created by contouring an implicit surface | upload.wikimedia.org/wikipedia/commons/3/35/Impl-flaeche-geschl2.svg]\n\n# Representation\nVarious methods of storing and working with a mesh in computer memory are possible.\nWith the [OpenGL](https://bliptext.com/articles/opengl) and [DirectX](https://bliptext.com/articles/directx) [APIs](https://bliptext.com/articles/api) there are two primary ways of passing a triangle mesh to the graphics hardware, [triangle strips](https://bliptext.com/articles/triangle-strip) and index arrays.\n\n## Triangle strip\nOne way of sharing vertex data between triangles is the triangle strip.\nWith [strips of triangles](https://bliptext.com/articles/triangle-strip) each triangle shares one complete edge with one neighbour and another with the next.\nAnother way is the triangle fan which is a set of connected triangles sharing one central vertex.\nWith these methods [vertices](https://bliptext.com/articles/vertex-computer-graphics) are dealt with efficiently resulting in the need to only process N+2 [vertices](https://bliptext.com/articles/vertex-computer-graphics) in order to draw N triangles.\nTriangle strips are efficient, however the drawback is that it may not be obvious how or convenient to translate an arbitrary triangle mesh into strips.\n\n## The Data Structure\nThe data structure representing the mesh provides support for two basic operations: inserting triangles and removing triangles.\nIt also supports an edge collapse operation that is useful in triangle decimation schemes.\nThe structure provides no support for the vertex positions, but it does assume that each vertex is assigned a unique integer identifier, typically the index of that vertex in an array of contiguous vertex positions.\nA mesh vertex is defined by a single integer and is denoted by hvi.\nA mesh edge is defined by a pair of integers hv0,v1i, each integer corresponding to an end point of the edge.\nTo support edge maps, the edges are stored so that v0 = min(v0,v1).\nA triangle component is defined by a triple of integers hv0,v1,v2i, each integer corresponding to a vertex of the triangle.\nTo support triangle maps, the triangles are stored so that v0 = min(v0,v1,v2).\nObserve that hv0,v1,v2i and hv0,v2,v1i are treated as different triangles.\nAn application requiring double–sided triangles must insert both triples into the data structure.\nFor the sake of avoiding constant reminders about order of indices, in the remainder of the document the pair/triple information does not imply the vertices are ordering in any way (although the implementation does handle the ordering).\nConnectivity between the components is completely determined by the set of triples representing the triangles.\nA triangle t = hv0,v1,v2i has vertices v0, v1, and v2.\nIt has edges e0 = hv0,v1i, e1 = hv1,v2i, and e2 = hv2,v0i.\nThe inverse connections are also known.\nVertex v0 is adjacent to edges e0 and e2 and to triangle t. Vertex v1 is adjacent to edges e0 and e1 and to triangle t. Vertex v2 is adjacent to edges e1 and e2 and to triangle t. All three edges e0, e1, and e2 are adjacent to t.\nHow much of this information a data structure stores is dependent on the needs of an application.\nMoreover, the application might want to have additional information stored at the components.\nThe information stored at a vertex, edge, or triangle is referred to as the vertex attribute, edge attribute, or triangle attribute.\nThe abstract representations of these for the simple data structure described here are\nVertex = ; // v Edge = ; // v0, v1 Triangle ; // v0, v1, v2 VData = ;\nEData = ;\nTData = ;\nVAttribute = ,set >; // data, eset, tset\nEAttribute = >; // data, tset\nTAttribute = ; // data\nVPair = pair ;\nEPair = pair ;\nTPair = pair ;\nVMap = map ;\nEMap = map ;\nTMap = map ;\nMesh = ; // vmap, emap, tmap\nThe maps support the standard insertion and removal functions for a hash table.\nInsertion occurs only if the item does not already exist.\nRemoval occurs only if the item does exist.\n\n## Edge collapse\nThis operation involves identifying an edge hvk, vti where vk is called the keep vertex and vt is called the throw vertex.\nThe triangles that share this edge are removed from the mesh.\nThe vertex vt is also removed from the mesh.\nAny triangles that shared vt have that vertex replaced by vk.\nFigure 1 shows a triangle mesh and a sequence of three edge collapses applied to the mesh.\n\n## Index array\nWith index arrays, a mesh is represented by two separate arrays, one array holding the vertices, and another holding sets of three indices into that array which define a triangle.\nThe graphics system processes the vertices first and renders the triangles afterwards, using the index sets working on the transformed data.\nIn OpenGL, this is supported by the **glDrawElements** primitive when using [Vertex Buffer Object](https://bliptext.com/articles/vertex-buffer-object) (VBO).\nWith this method, any arbitrary set of triangles sharing any arbitrary number of vertices can be stored, manipulated, and passed to the graphics API, without any intermediary processing.","revisionCount":0},"hasMore":false,"nextCursor":null}

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