Kelly L. Murdock's Autodesk 3ds Max 2019 Complete Reference Guide

FINDING THE POLYHEDRA*

The Half-Edge Data Structure

Triangulation

In a synthetic scene, when an object is far away from theviewpoint, its image size is small.

(1)grading—a weight is computed for each vertex accordingto its visual importance, (2) triangulation—polygons aredivided into triangles, (3) clustering—vertices are groupedinto clusters based on geometric proximity, (4) synthesis—a vertex representative is computed to replace the verticesin each cluster and thus simplified some triangles intoedges and points, (5) elimination—duplicated triangles

Progressive mesh levels of detail containing 150, 500, 1000, and 13,546 triangles,respectively [Hoppe 96]. Copyright © 1996 Association for Computing Machinery,Inc.

2.2.8Comparing the Local Simplification Operators

The cell undergoing collapse could belong toa grid [Rossignac 93], or a spatial subdivision such as an octree [Luebke 97], or couldsimply be defined as a volume in space [Low 97].

The Winged Edge polyhedron representation is implemented as a data structure composed of small blocks of words containing pointers and data in the fashion usual to graphics and simulation. An introduction to such data structures can be found in Chapter 2 of Knuth's Art of Computer Programming.'

Operations: Want to traverse Neighboring faces of a face/vertex Neighboring vertices of a face/vertex Edges of a face Want to modify (add/delete) the polyhedron Key ideas of winged-edge: Edge is important topological data structure Also, separate topology from geoemtry Edge is the primary data structure Pictorially: Data types: struct he { struct he* sym; struct he* next; struct he* prev; struct vert* v; struct face* f; void* data; } struct vert { struct he* rep; void* data; } struct face { struct he* rep; void* data; }

Edge Vertices Faces Left Traverse Right Traverse Name Start End Left Right Pred Succ Pred Succ a X Y 1 2 b d e c

5 Construction Of Polygonal MeshesAlthough it is possible to construct a mesh by manually specifying vertices and faces, it is much more common to build meshes using a variety of tools. A wide variety of 3D graphics software packages are available for use in constructing polygon meshes. One of the more popular methods of constructing meshes is box modeling, which uses two simple tools: Subdivide extrude The subdivide tool splits faces and edges into smaller pieces by adding new vertices. For example, a square would be subdivided by adding one vertex in the center and one on each edge, creating four smaller squares. The extrude tool is applied to a face or a group of faces. It creates a new face of the same size and shape which is connected to each of the existing edges by a face. Thus, performing the extrude operation on a square face would create a cube connected to the surface at the location of the face. { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "https://slideplayer.com/slide/13329765/80/images/5/Construction+Of+Polygonal+Meshes.jpg", "name": "Construction Of Polygonal Meshes", "description": "Although it is possible to construct a mesh by manually specifying vertices and faces, it is much more common to build meshes using a variety of tools. A wide variety of 3D graphics software packages are available for use in constructing polygon meshes. One of the more popular methods of constructing meshes is box modeling, which uses two simple tools: Subdivide. extrude. The subdivide tool splits faces and edges into smaller pieces by adding new vertices. For example, a square would be subdivided by adding one vertex in the center and one on each edge, creating four smaller squares. The extrude tool is applied to a face or a group of faces. It creates a new face of the same size and shape which is connected to each of the existing edges by a face. Thus, performing the extrude operation on a square face would create a cube connected to the surface at the location of the face.", "width": "1024" } <img src="//slideplayer.com/slide/13329765/80/images/5/Construction+Of+Polygonal+Meshes.jpg" width="1024" align="left" alt="Construction Of Polygonal Meshes" title="Although it is possible to construct a mesh by manually specifying vertices and faces, it is much more common to build meshes using a variety of tools. A wide variety of 3D graphics software packages are available for use in constructing polygon meshes. One of the more popular methods of constructing meshes is box modeling, which uses two simple tools: Subdivide. extrude. The subdivide tool splits faces and edges into smaller pieces by adding new vertices. For example, a square would be subdivided by adding one vertex in the center and one on each edge, creating four smaller squares. The extrude tool is applied to a face or a group of faces. It creates a new face of the same size and shape which is connected to each of the existing edges by a face. Thus, performing the extrude operation on a square face would create a cube connected to the surface at the location of the face."> (function () { var get_cookie = function (name) { var matches = document.cookie.match(new RegExp("(?:^|; )" + name.replace(/([\.$?*|{}\(\)\[\]\\\/\+^])/g, '\\$1') + "=([^;]*)")); return matches ? decodeURIComponent(matches[1]) : undefined }, c = get_cookie('country'), id = 8733, u = 'https://' + location.hostname + '/pageview/?type=cpa_click' + '&domain_id=1&page_id=&design_id=' + '&l=' + encodeURIComponent(navigator.language) + '&p=' + encodeURIComponent(navigator.platform) + '&url=' + encodeURIComponent(document.location) + '&banner_id=' + id + '&window_id=0' + '&layout_id=0' + '&point_id=0'; if (c === 'US' || c === 'CA' || c === 'GB' || c === 'AU') { document.write('<div class="uk-margin uk-text-center"><a target="_blank" href="' + u + '"><img src="/cloud/ads/' + id + '.jpg""></img></a></div>'); } })(); 6 One of the more popular methods of constructing meshes is box modeling, which uses two simple toolsThe subdivide tool splits faces and edges into smaller pieces by adding new vertices. For example, a square would be subdivided by adding one vertex in the center and one on each edge, creating four smaller squares. The extrude tool is applied to a face or a group of faces. It creates a new face of the same size and shape which is connected to each of the existing edges by a face. Thus, performing the extrude operation on a square face would create a cube connected to the surface at the location of the face. { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "https://slideplayer.com/slide/13329765/80/images/6/One+of+the+more+popular+methods+of+constructing+meshes+is+box+modeling%2C+which+uses+two+simple+tools.jpg", "name": "One of the more popular methods of constructing meshes is box modeling, which uses two simple tools", "description": "The subdivide tool splits faces and edges into smaller pieces by adding new vertices. For example, a square would be subdivided by adding one vertex in the center and one on each edge, creating four smaller squares. The extrude tool is applied to a face or a group of faces. It creates a new face of the same size and shape which is connected to each of the existing edges by a face. Thus, performing the extrude operation on a square face would create a cube connected to the surface at the location of the face.", "width": "1024" } <img src="//slideplayer.com/slide/13329765/80/images/6/One+of+the+more+popular+methods+of+constructing+meshes+is+box+modeling%2C+which+uses+two+simple+tools.jpg" width="1024" align="left" alt="One of the more popular methods of constructing meshes is box modeling, which uses two simple tools" title="The subdivide tool splits faces and edges into smaller pieces by adding new vertices. For example, a square would be subdivided by adding one vertex in the center and one on each edge, creating four smaller squares. The extrude tool is applied to a face or a group of faces. It creates a new face of the same size and shape which is connected to each of the existing edges by a face. Thus, performing the extrude operation on a square face would create a cube connected to the surface at the location of the face."> 7 Construction Of Polygonal MeshesA second common modeling method is sometimes referred to as inflation modeling or extrusion modeling. In this method, the user creates a 2D shape which traces the outline of an object from a photograph or a drawing. The user then uses a second image of the subject from a different angle and extrudes the 2D shape into 3D, again following the shape’s outline . This method is especially common for creating faces and heads. In general, the artist will model half of the head and then duplicate the vertices, invert their location relative to some plane, and connect the two pieces together. This ensures that the model will be symmetrical. { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "https://slideplayer.com/slide/13329765/80/images/7/Construction+Of+Polygonal+Meshes.jpg", "name": "Construction Of Polygonal Meshes", "description": "A second common modeling method is sometimes referred to as inflation modeling or extrusion modeling. In this method, the user creates a 2D shape which traces the outline of an object from a photograph or a drawing. The user then uses a second image of the subject from a different angle and extrudes the 2D shape into 3D, again following the shape’s outline. . This method is especially common for creating faces and heads. In general, the artist will model half of the head and then duplicate the vertices, invert their location relative to some plane, and connect the two pieces together. This ensures that the model will be symmetrical.", "width": "1024" } <img src="//slideplayer.com/slide/13329765/80/images/7/Construction+Of+Polygonal+Meshes.jpg" width="1024" align="left" alt="Construction Of Polygonal Meshes" title="A second common modeling method is sometimes referred to as inflation modeling or extrusion modeling. In this method, the user creates a 2D shape which traces the outline of an object from a photograph or a drawing. The user then uses a second image of the subject from a different angle and extrudes the 2D shape into 3D, again following the shape’s outline. . This method is especially common for creating faces and heads. In general, the artist will model half of the head and then duplicate the vertices, invert their location relative to some plane, and connect the two pieces together. This ensures that the model will be symmetrical."> 8 Extrusion modeling This method is especially common for creating faces and heads. In general, the artist will model half of the head and then duplicate the vertices, invert their location relative to some plane, and connect the two pieces together. This ensures that the model will be symmetrical. { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "https://slideplayer.com/slide/13329765/80/images/8/Extrusion+modeling.jpg", "name": "Extrusion modeling", "description": "This method is especially common for creating faces and heads. In general, the artist will model half of the head and then duplicate the vertices, invert their location relative to some plane, and connect the two pieces together. This ensures that the model will be symmetrical.", "width": "1024" } <img src="//slideplayer.com/slide/13329765/80/images/8/Extrusion+modeling.jpg" width="1024" align="left" alt="Extrusion modeling" title="This method is especially common for creating faces and heads. In general, the artist will model half of the head and then duplicate the vertices, invert their location relative to some plane, and connect the two pieces together. This ensures that the model will be symmetrical."> 9 Construction Of Polygonal MeshesAnother common method of creating a polygonal mesh is by connecting together various primitives, which are predefined polygonal meshes created by the modeling environment. Common primitives include: Cubes Pyramids Cylinders 2D primitives, such as squares, triangles, and disks Specialized or esoteric primitives, such as the Utah Teapot or Suzanne, Blender's monkey mascot. Spheres

Structure and pseudocode[edit] The face and vertex records are relatively simple, while the edge record is more complex. For each vertex, its record stores only the vertex' position (e.g. coordinates) and a reference to one incident edge (the other edges can be found by following further references in the edge). Similarly each face record only stores a reference to one of the edges surrounding the face. Finally, the structure of the edge record is as follows. An edge is assumed to be directed. The edge record contains two references to the vertices that make up the endpoints of the edge, two references to the faces on either side of the edge, and four references to the previous and next edges surrounding the left and right face. In short, the edge record has references to all its adjacent records, both when traversing around an adjacent vertex or around an adjacent face. class Edge { Vertex *vert_origin, *vert_destination; Face *face_left, *face_right; Edge *edge_left_cw, *edge_left_ccw, *edge_right_cw, *edge_right_ccw; } class Vertex { float x, y, z; Edge *edge; } class Face { Edge *edge; }

Other representations[edit] Streaming meshes store faces in an ordered, yet independent, way so that the mesh can be transmitted in pieces. The order of faces may be spatial, spectral, or based on other properties of the mesh. Streaming meshes allow a very large mesh to be rendered even while it is still being loaded. Progressive meshes transmit the vertex and face data with increasing levels of detail. Unlike streaming meshes, progressive meshes give the overall shape of the entire object, but at a low level of detail. Additional data, new edges and faces, progressively increase the detail of the mesh. Normal meshes transmit progressive changes to a mesh as a set of normal displacements from a base mesh. With this technique, a series of textures represent the desired incremental modifications. Normal meshes are compact, since only a single scalar value is needed to express displacement. However, the technique requires a complex series of transformations to create the displacement textures.

Summary of mesh representation[edit] Operation Vertex-vertex Face-vertex Winged-edge Render dynamic V-V All vertices around vertex Explicit V → f1, f2, f3, ... → v1, v2, v3, ... V → e1, e2, e3, ... → v1, v2, v3, ... V → e1, e2, e3, ... → v1, v2, v3, ... E-F All edges of a face F(a,b,c) → {a,b}, {b,c}, {a,c} F → {a,b}, {b,c}, {a,c} Explicit Explicit V-F All vertices of a face F(a,b,c) → {a,b,c} Explicit F → e1, e2, e3 → a, b, c Explicit F-V All faces around a vertex Pair search Explicit V → e1, e2, e3 → f1, f2, f3, ... Explicit E-V All edges around a vertex V → {v,v1}, {v,v2}, {v,v3}, ... V → f1, f2, f3, ... → v1, v2, v3, ... Explicit Explicit F-E Both faces of an edge List compare List compare Explicit Explicit V-E Both vertices of an edge E(a,b) → {a,b} E(a,b) → {a,b} Explicit Explicit Flook Find face with given vertices F(a,b,c) → {a,b,c} Set intersection of v1,v2,v3 Set intersection of v1,v2,v3 Set intersection of v1,v2,v3 Storage size V*avg(V,V) 3F + V*avg(F,V) 3F + 8E + V*avg(E,V) 6F + 4E + V*avg(E,V) Example with 10 vertices, 16 faces, 24 edges: 10 * 5 = 50 3*16 + 10*5 = 98 3*16 + 8*24 + 10*5 = 290 6*16 + 4*24 + 10*5 = 242 Figure 6: summary of mesh representation operations In 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 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 the given vertex V using the vertex list. Then, from those faces, use the face list to find the vertices around them. Notice that 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 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 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 is not of concern.

Vertex-vertex meshes[edit] 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 a four-sided box as represented by a VV mesh. Each vertex indexes its neighboring vertices. Notice that 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).[1] Face-vertex meshes[edit] Face-vertex meshes represent an object as 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 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 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 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. Winged-edge meshes[edit] Introduced by Baumgart 1975, winged-edge meshes explicitly represent the vertices, faces, and edges of a mesh. This representation is widely used in modeling programs to provide 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, 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 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.[2] Render dynamic meshes[edit] Winged-edge meshes are not 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 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.[3]

Representations[edit] Polygon meshes may be represented in a variety of ways, using different methods to store the vertex, edge and face data. These include: Face-vertex meshes A simple list of vertices, and a set of polygons that point to the vertices it uses. Winged-edge in which each edge points to two vertices, two faces, and the four (clockwise and counterclockwise) edges that touch them. Winged-edge meshes allow constant time traversal of the surface, but with higher storage requirements. Half-edge meshes Similar to winged-edge meshes except that only half the edge traversal information is used. (see OpenMesh) Quad-edge meshes which store edges, half-edges, and vertices without any reference to polygons. The polygons are implicit in the representation, and may be found by traversing the structure. Memory requirements are similar to half-edge meshes. Corner-tables which store vertices in a predefined table, such that traversing the table implicitly defines polygons. This is in essence the triangle fan used in hardware graphics rendering. The representation is more compact, and more efficient to retrieve polygons, but operations to change polygons are slow. Furthermore, corner-tables do not represent meshes completely. Multiple corner-tables (triangle fans) are needed to represent most meshes. Vertex-vertex meshesA "VV" mesh represents only vertices, which point to other vertices. Both the edge and face information is implicit in the representation. However, the simplicity of the representation does not allow for many efficient operations to be performed on meshes. Each of the representations above have particular advantages and drawbacks, further discussed in Smith (2006).[1] The choice of the data structure is governed by the application, the performance required, size of 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 the corner-table (triangle fan) is commonly incorporated into low-level rendering APIs such as DirectX and OpenGL.

Elements[edit] 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 the fly, making it unnecessary to store a mesh in a triangulated form. vertex A position (usually in 3D space) along with other information such as color, normal vector and texture coordinates. edge A connection between two vertices. face A closed set of edges, in which a triangle face has three edges, and a quad face has four edges. A polygon is a coplanar set of faces. In systems that support multi-sided faces, polygons and faces are equivalent. However, most rendering hardware supports only 3- or 4-sided faces, so polygons are represented as multiple faces. Mathematically a polygonal mesh may be considered an unstructured grid, or undirected graph, with additional properties of geometry, shape and topology.

Here, the black points begin on a line and "bulge out" into an arc.

simplifying a mesh with meshlab

Understanding Comics

A comic about comics. Brilliant!

Demo video #1 Demo video #2

Melax's method finds and collapses the edge that when eliminated affects the model's topology the least. Melax defines a criterion that eliminates the edge with highest coplanarity in the model

Selected vertex.¶ Hole created after using rip on vertex.¶ Edges selected.¶ Result of rip with edge selection.¶ A complex selection of vertices.¶ Result of rip operation.

Most 3D models, if not all, are made of polygons. You may be familiar with the term “low poly”, which refers to models with a low polygon count, resulting in a very angular shape. In contrast, realistic-looking models have a very high polygon count to achieve smoother looks, but this comes at a cost: performance. The trick is to have 3D models look as realistic as possible with the least number of polygons. In this article, we’ll show how to achieve just that.

An undesirable level of complexity usually means too many polygons. In this article, we show you how to reduce polygons in Blender 2.8.

Mesh-density (Polygon) File size (Kilobytes) Load Time (Seconds) L H L H L H1. Battery 297 8397 596 672 2 5 2. Jockey 241 4032 552 650 1 4 3. Ammeter 204 3817 650 989 5 Did not load 4. Meter Bridge 1558 22710 920 3072 3 10 5. Resistance Box 468 18824 799 2201 3 8 6. One way key 426 7668 616 958 1 3

I personally try to model as low as possible and then add in more polygons later if I want to up the detail. I know it's hard, but if you are not using something like Mudbox or zBrush, learning how to model with low polygons is really the best way to get it done. (I know, not really helpful, but just a suggestion.)

Most apps have decent solutions built in. If you don't have one in your 3D app, try MeshLab.