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A subdivision surface, in the field of 3D computer graphics, is a method of representing a smooth surface via the specification of a coarser piecewise linear polygon mesh. The smooth surface can be calculated from the coarse mesh as the limit of an iterative process of subdividing each polygonal face into smaller faces that better approximate the smooth surface. A 3D rendering with raytracing and ambient occlusion using Blender and Yafray 3D computer graphics are works of graphic art that were created with the aid of digital computers and specialized 3D software. ...
An open surface with X-, Y-, and Z-contours shown. ...
In mathematics, a piecewise linear function , where V is a vector space and is a subset of a vector space, is any function with the property that can be decomposed into finitely many convex polytopes, such that f is equal to a linear function on each of these polytopes. ...
A mesh is a collection of vertices and polygons that define the shape of an object in 3D computer graphics. ...
In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements, as their index increases indefinitely. ...
In geometry, a face of a polyhedron is any of the polygons that make up its boundaries. ...
Subdivision surfaces have, in the past 5 years, replaced NURBS surfaces as the preferred method of modeling character and scene geometry in the film and television visual effects industries. NURBS, short for nonuniform rational B-spline, is a computer graphics technique for drawing curves. ...
Visual effects (vfx) is the term given to a sub-category of special effects in which images or film frames are created or manipulated for film and video. ...
 First three steps of Catmull-Clark subdivision of a cube with subdivision surface below
Overview
The subdivision surfaces algorithm is recursive in nature. The process starts with a given polygonal mesh. A Refinement Scheme is then applied to this mesh. This process takes that mesh and subdivides it, creating new vertices and new faces. The position of the new vertices in the mesh are computed based on the positions of nearby old vertices. In some refinement schemes, the position of old vertices is also altered based on the positions of new vertices. See: Recursion Recursively enumerable language Recursively enumerable set Recursive filter Recursive function Recursive set Primitive recursive function This is a disambiguation page — a list of pages that otherwise might share the same title. ...
This process produces a new mesh containing many more polygonal faces than the original one did. The resulting mesh can be passed through the same refinement scheme again, this allowing for greater refinement.
The ultimate goal of iteratively applying a refinement scheme is to take the a given mesh and make it smoother.
Refinement Schemes Subdivision surface refinement schemes can be broadly classified into two categories: interpolating and approximating. Interpolating schemes are required to match the original position of vertices in the original mesh. Approximating schemes are not; they can and will adjust these positions as needed. In general, approximating schemes have greater smoothness, but the user has less overall control of the outcome. This is analogous to spline surfaces and curves, where Bézier splines are required to interpolate certain control points, while B-Splines are not. One type of spline, a bézier curve In the mathematical subfield of numerical analysis, a spline is a special function defined piecewise by polynomials. ...
In the mathematical subfield of numerical analysis a B-spline is a spline function which has minimal support with respect to a given degree, smoothness, and domain partition. ...
There is another division in subdivision surface schemes as well: the type of polygon that they operate on. Some function for quadrilaterals (quads), while others operate on triangles.
Approximating - Catmull-Clark, Quads - generalization of bi-cubic uniform B-splines
- Loop, Triangles - generalization of quartic triangular box splines
First three steps of Catmull-Clark subdivision of a cube with subdivision surface below In short: An algorithm used in CGI Subdivision Surfaces to create smooth surfaces. ...
In the mathematical subfield of numerical analysis a B-spline is a spline function which has minimal support with respect to a given degree, smoothness, and domain partition. ...
Interpolating - Doo-Sabin, Quads - generalization of bi-quadratic uniform B-splines
- Butterfly, Triangles - named after the scheme's shape
- Midedge, Quads
- Kobbelt, Quads - a variational subdivision method that tries to overcome uniform subdivision drawbacks
In the mathematical subfield of numerical analysis a B-spline is a spline function which has minimal support with respect to a given degree, smoothness, and domain partition. ...
Benefits over NURBS surfaces The main benefits of subdivision surfaces over NURBS subdivision surfaces from an artist's perspective are: - They support complex topologies
- The artist has local control over surface smoothness
- The artist can specify the degree of surface refinement
- They can be rendered directly (there is no need to tessellate a polygonal surface as a precursor to rendering.)
A tessellated plane A tessellation of the plane is a collection of plane figures that fill the plane with no overlaps and no gaps. ...
Disadvantages over NURBS surfaces - Flat sections in the original mesh are prone to having subtle waves, particularly after many iterations.
- Some approximating algorithms can over-smooth a surface, removing specific details after several iterations.
Key developments - 1978: Subdivision surfaces were discovered simultaneously by Edwin Catmull and Jim Clark (see Catmull-Clark subdivision surface), and by Daniel Doo and Malcom Sabin (see Doo-Sabin subdivision surfaces.)
- 1995: Ulrich Reif solved subdivision surface behaviour near extraordinary vertices [1].
- 1998: Jos Stam contributed a method for exact evaluation for Catmull-Clark subdivision surfaces under arbitrary parameter values [2].
Edwin Catmull after receiving a medal at SIGGRAPH 2001. ...
First three steps of Catmull-Clark subdivision of a cube with subdivision surface below In short: An algorithm used in CGI Subdivision Surfaces to create smooth surfaces. ...
External links - Resources about subdvisions
- Geri's Game : Oscar winning animation by Pixar completed in 1997 that introduced subdivision surfaces (along with cloth simulation)
- Subdivision for Modeling and Animation tutorial, SIGGRAPH 2000 course notes
- A unified approach to subdivision algorithms near extraordinary vertices, Ulrich Reif (Computer Aided Geometric Design 12(2):153-174 March 1995)
Pixar Animation Studios is an award-winning American computer animation studio based in Emeryville, California (USA). ...
SIGGRAPH 2005 official logo SIGGRAPH (short for Special Interest Group in Graphics) is the name of the annual conference on computer graphics convened by the ACM SIGGRAPH organization. ...
References - ^ Ulrich Reif. 1995. A unified approach to subdivision algorithms near extraordinary vertices. Computer Aided Geometric Design. 12(2)153-174
- ^ Jos Stam, "Exact Evaluation of Catmull-Clark Subdivision Surfaces at Arbitrary Parameter Values", Proceedings of SIGGRAPH'98. In Computer Graphics Proceedings, ACM SIGGRAPH, 1998, 395-404
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