3D graphics eBook - Course Materials Repository
3D graphics eBook - Course Materials Repository 3D graphics eBook - Course Materials Repository
Euler operator 43 Euler operator In mathematics, Euler operators are a small set of functions to create polygon meshes. They are closed and sufficient on the set of meshes, and they are invertible. Purpose A "polygon mesh" can be thought of as a graph, with vertices, and with edges that connect these vertices. In addition to a graph, a mesh has also faces: Let the graph be drawn ("embedded") in a two-dimensional plane, in such a way that the edges do not cross (which is possible only if the graph is a planar graph). Then the contiguous 2D regions on either side of each edge are the faces of the mesh. The Euler operators are functions to manipulate meshes. They are very straightforward: Create a new vertex (in some face), connect vertices, split a face by inserting a diagonal, subdivide an edge by inserting a vertex. It is immediately clear that these operations are invertible. Further Euler operators exist to create higher-genus shapes, for instance to connect the ends of a bent tube to create a torus. Properties Euler operators are topological operators: They modify only the incidence relationship, i.e., which face is bounded by which face, which vertex is connected to which other vertex, and so on. They are not concerned with the geometric properties: The length of an edge, the position of a vertex, and whether a face is curved or planar, are just geometric "attributes". Note: In topology, objects can arbitrarily deform. So a valid mesh can, e.g., collapse to a single point if all of its vertices happen to be at the same position in space. References • Sven Havemann, Generative Mesh Modeling [1] , PhD thesis, Braunschweig University, Germany, 2005. • Martti Mäntylä, An Introduction to Solid Modeling, Computer Science Press, Rockville MD, 1988. ISBN 0-88175-108-1. References [1] http:/ / www. eg. org/ EG/ DL/ dissonline/ doc/ havemann. pdf
False radiosity 44 False radiosity False Radiosity is a 3D computer graphics technique used to create texture mapping for objects that emulates patch interaction algorithms in radiosity rendering. Though practiced in some form since the late 1990s, the term was coined around 2002 by architect Andrew Hartness, then head of 3D and real-time design at Ateliers Jean Nouvel. During the period of nascent commercial enthusiasm for radiosity-enhanced imagery, but prior to the democratization of powerful computational hardware, architects and graphic artists experimented with time-saving 3D rendering techniques. By darkening areas of texture maps corresponding to corners, joints and recesses, and applying maps via self-illumination or diffuse mapping in a 3D program, a radiosity-like effect of patch interaction could be created with a standard scan-line renderer. Successful emulation of radiosity required a theoretical understanding and graphic application of patch view factors, path tracing and global illumination algorithms. Texture maps were usually produced with image editing software, such as Adobe Photoshop. The advantage of this method is decreased rendering time and easily modifiable overall lighting strategies. Another common approach similar to false radiosity is the manual placement of standard omni-type lights with limited attenuation in places in the 3D scene where the artist would expect radiosity reflections to occur. This method uses many lights and can require an advanced light-grouping system, depending on what assigned materials/objects are illuminated, how many surfaces require false radiosity treatment, and to what extent it is anticipated that lighting strategies be set up for frequent changes. References • Autodesk interview with Hartness about False Radiosity and real-time design [1] References [1] http:/ / usa. autodesk. com/ adsk/ servlet/ item?siteID=123112& id=5549510& linkID=10371177
- Page 1 and 2: 3D Rendering PDF generated using th
- Page 3 and 4: Image-based lighting 64 Image plane
- Page 5 and 6: References Article Sources and Cont
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- Page 37 and 38: Conversion between quaternions and
- Page 39 and 40: Cube mapping 34 Advantages Cube map
- Page 41 and 42: Cube mapping 36 Related A large set
- Page 43 and 44: Diffuse reflection 38 2), or, of co
- Page 45 and 46: Displacement mapping 40 Meaning of
- Page 47: DooSabin subdivision surface 42 Ext
- Page 51 and 52: Geometry pipelines 46 Geometry pipe
- Page 53 and 54: Global illumination 48 Rendering wi
- Page 55 and 56: Gouraud shading 50 Gouraud shading
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- Page 71 and 72: Irregular Z-buffer 66 Applications
- Page 73 and 74: Lambert's cosine law 68 than would
- Page 75 and 76: Lambertian reflectance 70 Lambertia
- Page 77 and 78: Level of detail 72 Well known appro
- Page 79 and 80: Level of detail 74 Hierarchical LOD
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Euler operator 43<br />
Euler operator<br />
In mathematics, Euler operators are a small set of functions to create polygon meshes. They are closed and<br />
sufficient on the set of meshes, and they are invertible.<br />
Purpose<br />
A "polygon mesh" can be thought of as a graph, with vertices, and with edges that connect these vertices. In addition<br />
to a graph, a mesh has also faces: Let the graph be drawn ("embedded") in a two-dimensional plane, in such a way<br />
that the edges do not cross (which is possible only if the graph is a planar graph). Then the contiguous 2D regions on<br />
either side of each edge are the faces of the mesh.<br />
The Euler operators are functions to manipulate meshes. They are very straightforward: Create a new vertex (in some<br />
face), connect vertices, split a face by inserting a diagonal, subdivide an edge by inserting a vertex. It is immediately<br />
clear that these operations are invertible.<br />
Further Euler operators exist to create higher-genus shapes, for instance to connect the ends of a bent tube to create a<br />
torus.<br />
Properties<br />
Euler operators are topological operators: They modify only the incidence relationship, i.e., which face is bounded<br />
by which face, which vertex is connected to which other vertex, and so on. They are not concerned with the<br />
geometric properties: The length of an edge, the position of a vertex, and whether a face is curved or planar, are just<br />
geometric "attributes".<br />
Note: In topology, objects can arbitrarily deform. So a valid mesh can, e.g., collapse to a single point if all of its<br />
vertices happen to be at the same position in space.<br />
References<br />
• Sven Havemann, Generative Mesh Modeling [1] , PhD thesis, Braunschweig University, Germany, 2005.<br />
• Martti Mäntylä, An Introduction to Solid Modeling, Computer Science Press, Rockville MD, 1988. ISBN<br />
0-88175-108-1.<br />
References<br />
[1] http:/ / www. eg. org/ EG/ DL/ dissonline/ doc/ havemann. pdf
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