Modeling is the process of describing an object or scene so that we can construct an image of it.

Points & polygons

• Points - three-dimensional locations (or coordinate triples)

• Vectors - have direction and magnitude; can also be thought of as displacement

• Polygons - sequences of ``correctly'' co-planar points; or an initial point and a sequence of vectors
  
  

• Geometry definition = position and topology (connectivity)
  
  

A cube described in .obj file format

g

v -2.500000 2.500000 2.500000
v -2.500000 -2.500000 2.500000
v 2.500000 -2.500000 2.500000
v 2.500000 2.500000 2.500000
v -2.500000 2.500000 -2.500000
v -2.500000 -2.500000 -2.500000
v 2.500000 -2.500000 -2.500000
v 2.500000 2.500000 -2.500000
# 8 vertices

g cube

f 1 2 3 4
f 8 7 6 5
f 4 3 7 8
f 5 1 4 8
f 5 6 2 1
f 2 6 7 3
# 6 elements

Primitives

Primitives are the fundamental geometric entities within a given data structure.

• We have already touched on point, vector and polygon primitives

• Regular Polygon Primitives - square, triangle, circle, n-gon, etc.
  
  

• Polygon strips or meshes
  
  
• Meshes provide a more economical description than multiple individual polygons
  
  For example, 100 individual triangles, each requiring 3 vertices, would require
  100 x 3 or 300 vertex definitions to be stored in the 3-D database.
  
  By contrast, triangle strips require n + 2 vertex definitions for any n number or triangles
  in the strip. Hence, a 100 triangle strip requires only 102 unique vertex definitions.
  
  
• Meshes also provide continuity across surfaces which is important for shading calculations
  
  

• 3D primitives in a polygonal database
  
  3D shapes are represented by polygonal meshes that define or approximate geometric surfaces.
  
  

• With curved surfaces, the accuracy of the approximation is directly proportional to the number of polygons used in the representation.
  
  
• More polygons (when well used) yields a better approximation.
  
  
• But more polygons also exact greater computational overhead, thereby degrading interactive performance, increasing render times, etc.
  
  

Rendering - The process of computing a two dimensional image using a combination of a three-dimensional database, scene characteristics, and viewing transformations. Various algorithms can be employed for rendering, depending on the needs of the application.

Tessellation - The subdivision of an entity or surface into one or more non-overlapping primitives. Typically, renderers decompose surfaces into triangles as part of the rendering process.

Sampling - The process of selecting a representative but finite number of values along a continuous function sufficient to render a reasonable approximation of the function for the task at hand.

Level of Detail (LOD) - To improve rendering efficiency when dynamically viewing a scene, more or less detailed versions of a model may be swapped in and out of the scene database depending on the importance (usually determined by image size) of the object in the current view.

Polygons and rendering

• Clockwise versus counterclockwise
  
  

Surface normal - a vector that is perpendicular to a surface and ``outward'' facing

• Surface normals are used to determine visibility and in the calculation of shading values (among other things)
  
  

• Convex versus concave

  A shape is convex if any two points within the shape can be connected with a straight line that never goes out of the shape. If not, the shape is concave.
  

• Concave polygons can cause problems during rendering (e.g. tears, etc., in apparent surface).
  
  

• Polygon meshes and shared vertices
  
  

• Polygons consisting of non-co-planar vertices can cause problems when rendering (e.g. visible tearing of the surface, etc.)
  
  
• With quad meshes, for example, vertices within polygons can be inadvertently transformed into non-co-planer positions during modeling or animation transformations.
  
  
• With triangle meshes, all polygons are triangles and therefore all vertices within any given polygon will be coplanar.
  
  

With polygonal databases:

• Explicit, low-level descriptions of geometry tend to be employed
  
  
• Object database files can become very large relative to more economical, higher order descriptions.
  
  
• Organic forms or free-form surfaces can be difficult to model.
  
  

A cylinder described in VRML

#VRML V2.0 utf8 CosmoWorlds V1.0

Group {
children Transform {
children Shape {
appearance Appearance {
material Material {
}

}

geometry Cylinder {
}

}

translation 0.0 0.0 0.0

}

}

Surface models

• Surfaces can be constructed from mathematical descriptions
  
  
• Resolution independent - surfaces can be tesselated at rendering with an appropriate level of approximation for current display devices and/or viewing parameters
  
  
• Tessellation can be adaptive to the local degree of curvature of a surface.
  
  

• Primitives
  
  

• Free-form surfaces can be built from curves
  
  

• Construction history, while also used in polygonal modeling, can be particularly useful with curve and surface modeling techniques.
  
  

• Parameterization
  
  

• Curve direction and surface construction
  
  

• Surface parameterization (u, v, w)
  
  
  • For placing texture maps, etc.
    
    
  • For locating trimming curves, etc.
    
    

Constructive solid geometry and boolean operations

• Algebraic combination (union, intersection, or difference) of two geometric solids
  
  
• Used in both polygonal and surface modelers
  
  
• Volumes usually must be closed
  
  
• Trimmed portions may continue to exist even though invisible
  
  

Metaballs (blobby surfaces)

• Potential functions (usually radially symmetric Gaussian functions) are used to define surfaces surrounding points
  

  
  

  • Potential function is proportional to distance from a point
    
    
  • Potential = 1 at point location; potential falls off to 0 at some distance from location
    
    
  • An implicit surface could be drawn arbitrarily at any potential value (e.g. potential = .5)
    
    
  • For a single point, the potential function results in a sphere
    
    
  • For clusters of points that are sufficiently close for the radii of their potential functions to overlap, the surfaces derived from uniform potential values may be ``blobby'' like coalescing droplets of water, for example.

Isosurfaces

• Boundary layers in a three-dimensional data cloud
  
  
• The 3-D equivalent of contour lines
  
  

Binary Space Partitioning - for modeling/rendering volumetric data

• 
  
  
  
• Data volume is repeatedly subdivided
  
• Data extremes within each new subdivision or cell are tested to see if they exceed a pre-determined threshold
  
• If threshold is exceeded, current cell is further subdivided
  
• If threshold is not exceeded, current cell is no longer subdivided
  

Grouping and hierarchies

• Allows selective control of the frame of reference to which modeling operations and/or transformations are applied.

Back to course outline

This file was last modified on January 22, 2002.