Modelling

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Modelling

• The creation and manipulation of a system
  representation is termed modelling
• Any single representation is called a model of
  the system.
• There are mainly two kinds (conventional)
• Descriptive (e.g. a set of equations or rules to
  define parameter relationships)
• Graphical/volumetric (e.g. architectural and
  engineering system).

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How to represent these objects in computer
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Example Methods
Polygonal
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Polygonal Models

• The most common type of model used in 3D - store
  faces of the object as planar polygons
• Each polygonal face may be represented by its
  vertices or edges
• Physical properties of the object may also be
  held as part of the representation, for example
  colour, light, texture
• Representation methods such as this are called
  Boundary Representations or B-reps.
• This type of representation may also be used to
  represent an approximation to a curved surface,
  where the curved surface is approximated by
  planar polygonal patches

5
3D Modeling with Polygons

• We construct 3D models using groups of polygons.
• Each polygon is planar ? we need a large number
  of small polygons to give the impression of
  curved surfaces

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Polygon Mesh
A polygon mesh approximates the surface shape of
an object by specifying a set of points in space
these points representing vertices of various
polygonal faces.
we have 8 vertices and 6 (polygons) faces. A
list of vertices for a polygon is created by
looking at the polygon from the outside, listing
the vertices in a counter-clockwise direction
until a complete circle is made. Poly1 v1,
v2, v3, v4 (front face) Poly2 v4, v3, v6, v5
(right face)
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Polygon Mesh
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Example Methods
Parametric surface
9
Example Methods
Parametric surface
mapping from R2(u,v) to R3(x,y,z) Parametrized
by u and v.
xf(u,v)yg(u,v)zh(u,v)


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Example Methods
Allows easy enumeration of points. Just plug in
values for u and v. Differentiable Can have more
than one z value for each (x,y)


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Example Methods
Implict surface F(x, y, z) Constant
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Example Methods
Voxels Uniform grid of volumetric samples
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Space Subdivision

• This splits space into some arrangement of cells
  that cover all of the space.
• With each cell a note is kept as to whether that
  cell is occupied by the model or not.
• Obviously this method has limited accuracy.
• Typically methods are based on quadtrees and
  octrees.

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Sweep Objects

• Define a polygon by its edges then sweep it along
  a path
• Special cases
• Surface of revolution rotate edges about an axis
• Extrusion Sweep along a straight line

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Geometric Models

• Geometric models are collections of components
  with well defined geometry and often
  interconnections between components, including
  engineering and architectural structures, and
  other chemical structures
• Geometric models often have a hierarchical
  structure included by a bottom up construction
  process. Components are used as building blocks
  to create higher levels entities, which in turn
  serve as building blocks for yet higher level
  entities, and so on
• Generally represented as a tree, with
  transformations and instances at any node

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Geometric Models
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Geometric Modeling

• Each node may have its own local coordinate
  system
• Most useful for animating polygonal meshes
• Rendered by traversing the tree, applying
  transformations, and rendering the instances
• Consider a walking robot
• Does the entire robot move in the same way?
• Does the position of one part of the robot depend
  on other parts?

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Constructive Solid Geometry

• Constructive solid geometry (CSG) is a technique
  used in solid modeling. CSG is often, but not
  always, a procedural modeling technique used in
  3D computer graphics and CAD.
• Constructive solid geometry allows a modeler to
  create a complex surface or object by using
  Boolean operators to combine objects.
• Often CSG presents a model or surface that
  appears visually complex, but is actually little
  more than cleverly combined or decombined
  objects.
• In some cases, constructive solid geometry is
  performed on polygonal meshes, and may or may not
  be procedural and/or parametric.

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Constructive Solid Geometry

• The simplest solid objects used for the
  representation are called primitives.
• Typically they are the objects of simple shape
  cuboids, cylinders, prisms, pyramids, spheres,
  cones.
• The set of allowable primitives is limited by
  each software package.
• Some software packages allow CSG on curved
  objects while other packages do not.
• It is said that an object is constructed from
  primitives by means of allowable operations,
  which are typically Boolean operations on sets
  union, intersection and difference.

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Constructive Solid Geometry
Union
Difference
Intersection
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Constructive Solid Geometry
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Constructive Solid Geometry

• This object could be produced by gluing two
  rectangular blocks together and then drilling the
  hole
• in CSG terms the the union of two blocks is taken
  and then the difference of the resultant solid
  and a cylinder is carried out
• The basic primitive objects, the blocks and the
  cylinder, may have to be scaled to the correct
  size, possibly oriented and then placed in the
  correct positions relative to each other before
  the logical operations

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Procedural Methods

• The pattern is produced by the code 
•         polyline(ns1,verts)         for (i0
  i lt nr i)                     
  transverts(verts,ns,fc,fs)           
  polyline(ns1,verts)          
• where ns is the number of sides in the polygon,
  nr is the number of repetitions of the polygon
  and the array verts holds the vertices of the
  current polygon. The function transverts applies
  a scaling and a rotation to the vertices at each
  step

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Smoke Particle System

• Constantly create particles
• Particles move upwards, with turbulence added
• Draw them as partially transparent circles that
  fade over time

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Particle Systems

• A particle has
• A position in the world
• Rules for how it moves over time
• Rules for how it is drawn
• A particle system
• Controls when particles are created and destroyed
• Makes sure that all the particles are updated

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Basic Ocean

• Ocean created with Computational Fluid Dynamics

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Procedural Approach
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Procedural Approach

• Whilst being far from a solved problem,
  techniques have been developed for
  algorithmically generating convincing computer
  graphics models of landscapes, sea scapes,
  cloudscapes, forests, urban environment

Substrate
http//www.bartlett.ucl.ac.uk/ve/weblog/2004/10/pr
ocedural-urban-modelling.html
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Modeling in OpenGL

• OpenGL provides a set of routines (API) for 3D
  graphics
• derived from Silicon Graphics GL
• draws simple primitives (points, lines, polygons)
• provides control over transformations, lighting,
  texture etc.
• In OpenGL, all geometry is specified by stating
  the type of object and then giving the vertices
  of the object
• glBegin() glEnd()
• glVertex34fdv
• Three or four components (regular or homogeneous)
• Float, double or vector (eg float3)
• Primitives are defined by vertices, for example
  to draw a triangle
• glBegin (GL_POLYGON)
• glVertex3v (0, 0, 0)
• glVertex3v (0, 1, 0)
• glVertex3v (1, 0, 1)
• glEnd()

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OpenGL Command Formats
glVertex3fv( v )
Number of components
Data Type
Vector
b - byte ub - unsigned byte s - short us -
unsigned short i - int ui - unsigned int f -
float d - double
omit v for scalar form glVertex2f( x, y )
2 - (x,y) 3 - (x,y,z) 4 - (x,y,z,w)
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Geometric Primitives

• All geometric objects in OpenGL are created from
  a set of basic primitives
• Certain primitives are provided to allow
  optimisation of geometry for improved rendering
  speed
• Line based primitives

GL_LINE_STRIP
GL_LINE_LOOP
GL_LINES
GL_POINTS
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Geometric Primitives

• Polygon primitives

v4
v4
v4
GL_POLYGON
GL_QUADS
GL_TRIANGLES
v4
v3
v8
v2
v4
v5
v6
v3
v2
v5
v2
v1
v4
v1
v6
v3
v5
v7
v1
GL_QUAD_STRIP
GL_TRIANGLE_STRIP
GL_TRIANGLE_FAN