G53GRA

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G53GRA Modelling
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Modelling Overview

• A model is a description of an object. Modeling
  is the process of describing the object
• Sometimes the description is an end product in
  itself, e.g., computer aided design
• More typically in graphics, the model is then
  used for rendering
• The model only exists to produce a picture
• It can be an approximation, as long as the visual
  result is ok
• The computer graphics motto If it looks right
  it is right, but it doesnt work for CAD

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Modelling Techniques

• There are many ways to represent an object. What
  should be considered when choosing a
  representation?
• Surface vs. Volume
• Sometimes we only care about the surface
• Rendering and geometric computations
• Sometimes we want to know about the volume
• Some representations are best thought of defining
  the space filled, rather than the surface around
  the space
• Medical data with information attached to the
  space
• Parametric vs. Implicit
• Parametric generates all the points on a surface
  (volume) by plugging in a parameter eg (sin?
  cos?, sin? sin?, cos?)
• Implicit models tell you if a point in on (in)
  the surface (volume) eg x2 y2 z2- 1 0

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Choosing a Representation

• How well does it represents the objects of
  interest?
• How easy is it to create?
• By hand, procedurally, by fitting to
  measurements,
• How easy is it to render (or convert to
  polygons)?
• How compact is it (how cheap to store and
  transmit)?
• How easy is it to interact with?
• Modifying it, animating it
• How easy is it to perform geometric computations?
• Distance, intersection, normal vectors,
  curvature,

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Level of Detail

• Represent objects in a scene at several
  resolutions, or decrease the complexity of
  object representation as it moves away from the
  viewer or according other metrics

• http//lodbook.com/
• The basic concept is to provide various models to
  represent the same object
• Must have a way to switch from one mesh to
  another
• Also called mesh decimation, multi-resolution
  modeling etc

http//www.cs.unc.edu/geom/SUCC_MAP/
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Polygonal Models

• Polygons are the most popular in modeling for
  real-time graphics
• Everything can be turned into polygons (almost
  everything)
• The concept is easy to understand.
• A common output format from CAD modelers
• Many operations are easy to do with polygons
• We know how to render polygons quickly
• Memory and disk space is cheap

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Polygonal Models

• A polygonal model has several components, or
  geometric entities
• Faces
• Edges (boundaries between faces)
• Vertices (boundaries between edges)
• Normals, texture coordinates, colors, shading
  coefficients, etc
• Some components are implicit, given the others
• For instance, given faces and vertices can
  determine edges
• Eulers formula Faces Vertices Edges 2,
  for any polyhedron that doesn't intersect itself
• A polygonal mesh is a set of polygons connected
  to form a mesh

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Polygonal Models

• There are reasons not to store the vertices
  explicitly at each polygon
• Wastes memory - each vertex repeated many times
• Very messy to find neighboring polygons
• Difficult to ensure that polygons meet correctly
• Solution Indirection
• Put all the vertices in a list
• Each face stores the list indices of its vertices
• Minimizes amount of data sent to the renderer
• Fewer function calls, faster!

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Polygonal Models

• Polygonal data structures are application
  dependent. Different applications require
  different operations to be fast
• Find the neighbours of a given face
• Find the faces that surround a vertex
• Intersect two polygon meshes
• You typically choose
• Which features to store explicitly (vertices,
  edge faces, normals)
• Which relationships you want to be explicit
  (vertices belonging to edges, edges belonging to
  faces, faces at a vertex, etc)

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Modelling a Cube

• A model can be obtained by slicing along certain
  edges of a solid and unfolding it to lie flat so
  that all polygons are seen from the outside

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5
6
2
1
6
6
5
3
2
4
1
7
3
0
4
7
5
7
4
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Modelling a Cube
y
Vertex array Vi012 V1(-1,-1,1) V2(-1,1,1
) V3(1,1,1) V4(1,-1,1) V5(1,-1,-1) V6(1,1,-1)
V7(-1,1,-1) V8(-1,-1,-1)
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-1,1,-1
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1,1,-1
1,2,3,4, // front face 5,6,7,8 // back face
3,2,7,6 // top face 1,3,5,4 // bottom face
3,6,5,4 // right face 1,2,7,8 // left face
2
-1,1,1
o
3
8
1,1,1
-1,-1,-1
5
1
x
1,-1,-1
-1,-1,1
z
4
1,-1,1
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Normal Vectors

• Normal vectors give information about the true
  surface shape
• Per-Face normals
• One normal vector for each face, stored as part
  of face
• Flat shading
• Per-Vertex normals
• A normal specified for every vertex (smooth
  shading)
• Can keep an array of normals analogous to array
  of vertices
• Faces store vertex indices and normal indices
  separately
• Allows for normal sharing independent of vertex
  sharing

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A Cube
Vertices (1,1,1) (-1,1,1) (-1,-1,1) (1,-1,1) (1,1
,-1) (-1,1,-1) (-1,-1,-1) (1,-1,-1)
Normals (1,0,0) (-1,0,0) (0,1,0) (0,-1,0) (0,0,1)
(0,0,-1)
Faces ((vert,norm), ) ((0,4),(1,4),(2,4),(3,4))
((0,0),(3,0),(7,0),(4,0)) ((0,2),(4,2),(5,2),(1,2)
) ((2,1),(1,1),(5,1),(6,1)) ((3,3),(2,3),(6,3),(7,
3)) ((7,5),(6,5),(5,5),(4,5))
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Modelling Tetrahedron
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Modelling Prism
y
x
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Whats Bad About Polygons?

• They are an approximation to curved surfaces.
  Most real-world surfaces are curved, particularly
  natural surfaces, but approximation can be as
  good as you want
• Quite difficult to create a complex model
• They are unstructured
• They are hard to globally parameterize
• It is difficult to perform many geometric
  operations

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Meshes from Scanning

• Laser scanners sample 3D positions
• One method uses laser triangulation a thin
  stripe of laser light is projected across the
  object surface and is viewed by a camera.
  Stereoscopic method uses the positions of the
  camera and laser emitter to compute the 3D
  positions of points on the laser stripe
• Another uses time of flight shoots a laser pulse
  at the object and measures the time taken for the
  pulse to return to the scanner, distancespeed of
  light time
• Software then takes thousands of points and
  builds a polygon mesh out of them
• Research topics
• Reduce the number of points in the mesh
• Reconstruction and re-sampling!

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Scanning in Action

• http//www-graphics.stanford.edu/projects/mich/

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Alternative Object Representations

• Sweep Objects
• Hierarchical modeling
• Constructive Solid Geometry
• Octrees
• Metaballs/Blobs
• Production rules
• Procedural methods

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Sweep Objects

• Define a polygon by its edges
• Sweep it along a path
• The path taken by the edges form a surface - the
  sweep surface
• Special cases
• Surface of revolution Rotate edges about an axis
• Extrusion Sweep along a straight line

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Hierarchical Modelling

• A hierarchical model unites several instances
  into one object, for example a desk is made up
  of many blocks
• Generally represented as a tree, with
  transformations and instances at each node
• Rendered by traversing the tree, applying the
  transformations, and rendering the instances
• Particularly useful for animation
• Human is a hierarchy of body, head, upper arm,
  lower arm, etc
• Animate by changing the transformations at the
  nodes

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Hierarchical Model Example

• Vitally Important Point
• Every node has its own local coordinate system.
• This makes specifying transformations much easier.

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OpenGL Support

• OpenGL defines glPushMatrix() and glPopMatrix()
• Takes the current matrix and pushes it onto a
  stack, or pops the matrix off the top of the
  stack and makes it the current matrix
• Note Pushing does not change the current matrix
• Rendering a hierarchy (recursive)

RenderNode(tree) glPushMatrix() Apply node
transformation Draw node contents RenderNode(c
hildren) glPopMatrix()
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Recursive Hierarchical Modeling

• By applying small transforms to every level of
  the scene graph, a recursive model can quickly
  generate some amazing images.

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Constructive Solid Geometry

• Based on a tree structure, like hierarchical
  modeling, but now
• The internal nodes are set operations union,
  intersection or difference (sometimes complement)
• The edges of the tree have transformations
  associated with them
• The leaves contain only geometry
• Allows complex shapes with only a few primitives
• Common primitives are cylinders, cubes, etc, or
  quadric surfaces
• Motivated by computer aided design and
  manufacture
• Difference is like drilling or milling
• A common format in CAD products

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CSG Example
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CSG Summary

• Advantages
• Good for describing many things, particularly
  machined objects
• Better if the primitive set is rich
• Moderately intuitive and easy to understand
• Disadvantages
• Not a good match for polygon renderers
• Some objects may be very hard to describe, if at
  all
• Geometric computations are sometimes easy,
  sometimes hard

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Spatial Data Structures

• Basic idea describe object by the space it
  occupies
• For example, break a volume of interest into lots
  of tiny cubes, and say which cubes are inside the
  object
• Works well for things like medical data
• The process itself, like MRI or CAT scans,
  enumerates the volume
• Data is associated with each voxel (volume
  element)
• Problem to overcome
• For anything other than small volumes or low
  resolutions, the number of voxels explodes
• Note that the number of voxels grows with the
  cube of linear dimension

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Spatial Data Structures

• Octrees are an example of a spatial data
  structure
• A data structure specifically designed for
  storing information of a spatial nature
• In graphics, octrees are frequently used to store
  information about where polygons, or other
  primitives, are located in a scene
• Speeds up many computations by making it fast to
  determine when something is relevant or not
• Other spatial data structures include BSP (binary
  space partitioning) trees, KD-Trees, Interval
  trees,

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Quadtrees (and Octrees)

• Build a tree where successive levels represent
  better resolution (smaller voxels)
• Large uniform spaces result in shallow trees
• Quadtree is for 2D (four children for each node)
• Octree is for 3D (eight children for each node)

top left
top right
bot left
bot right
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Rendering Octrees

• Volume rendering renders octrees and associated
  data directly
• A special area of graphics, visualization, not
  covered in this class, will be covered in G54CGA
• Can convert to polygons by a few methods
• Just take faces of voxels that are on the
  boundary
• Find isosurfaces within the volume and render
  those
• Typically do some interpolation (smoothing) to
  get rid of the artifacts from the voxelization

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Implicit Functions

• Some objects are easy represent this way
• Spheres, ellipses, and similar
• More generally, quadratic surfaces
• Shapes depends on all the parameters
  a,b,c,d,e,f,g
• Some surfaces can be represented as the vanishing
  points of functions in 3D space, e.g., points
  where a function f(x,y,z)0
• Rendering implicit surfaces
• Raytracing - find ray and surface intersections
• For polygonal renderer, must convert to polygons

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Metaballs/Blobs

• Implicit modeling technique
• We can think of a metaball as a particle
  surrounded by a density field, where the density
  (particle influence) decreases with distance from
  the particle location
• A surface is implied by taking an isosurface
  through this density field - the higher the
  isosurface value, the nearer it will be to the
  particle

• An isosurface is a surface in 3D space whose
  points render a function constant think about a
  special instance of implicit function f(x,y,z)
  c, similar to the contours on a topographic map

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Metaballs

• The key to using metaballs is the definition of
  the equation for specifying the influence of a
  particle on an arbitrary point
• Blinn used exponentially decaying fields for each
  particle, with a Gaussian function with height b
  and standard deviation a - If r is the distance
  from the particle to a location in the field, the
  influence of the particle on the location is
  bexp(-ar). The resulting density of an arbitrary
  location in the field is simply the summation of
  the contributions from all the particles.

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Metaballs

• Can combine (add together) multiple primitives
  into a complex object - by summing the influences
  of each metaball on a given point, we can get a
  smooth blending of the influence fields
• This results in automatic blending of shapes

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Metaballs

• Metaballs are a type of implicit modeling
  technique.
• Think of a metaball as a particle surrounded by
  a density field, where the density decreases with
  distance from the particle location.
• A surface is implied by taking an isosurface
  through this density field - the higher the
  isosurface value, the nearer it will be to the
  particle.
• Metaballs can be combined. By simply summing the
  influences of the metaballs on a given point, we
  can get a very smooth blending of the spherical
  influence fields

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Rewriting Systems

• Rewriting is a technique for defining complex
  objects by successively replacing parts of a
  simple initial object using a set of rewriting
  rules or productions
• Works best for things like plants
• Start with a stem
• Replace it with stem branches
• Replace some part with more stem branches, and
  so on
• Best understood rewriting systems operate on
  character strings, and the methods for
  generating, recognizing and transforming them
• Essentially, generate a string that describes the
  object, replace sub-strings with some new strings
• Can generate a garden or forest

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Rewriting Systems
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A Simple L-System

• Biologist, Aristid Lindenmayer, introduced a
  string-rewriting mechanism in 1968 , termed
  L-systems. An example
• Production rules below are applied
• b ? a.
• a ? ab.
• We now have word ab.
• In the next derivation step, a is replaced by ab,
  b is replaced by a, so we have word aba
• Similar way, the string aba yields abaab which in
  turn yields abaababa, then abaababaabaab, and so
  on

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Graphical Representation of L-System

• Letters of the L-system alphabet are represented
  graphically as shorter or longer rectangles with
  rounded corners. The generated structures are
  one-dimensional chains of rectangles, reflecting
  the sequence of symbols in the corresponding
  strings.
• In order to model higher plants, a more
  sophisticated graphical interpretation of
  L-systems is needed.
• The geometric aspects, such as the lengths of
  line segments and the angle values, were added in
  a post-processing phase.

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Graphical Representation of L-System

• F Move forward a step of length d. The state of
  the turtlechanges to (x, y, a), where x x d
  cos a and y y d sin a. A line segment between
  points (x, y) and (x, y) is drawn.
• f Move forward a step of length d without
  drawing a line.
• Turn left by angle d. The next state (of the
  turtle) is
• (x, y, ad). The positive orientation of angles
  is counterclockwise.
• - Turn right by angle d. The next state (of the
  turtle) is
• (x, y, a - d).

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Representation of L-System

• Interpreting strings generated by the following
  L-system
• a
• F - F - F - F
• b
• F ? F - F F FF - F - F F

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Representation of L-System

• Axial trees (at each node, at most one straight
  segment, zero or more side segments) can be
  represented using strings with brackets
• Push the current state of the turtle onto a
  pushdown operations stack. The information saved
  on the stack contains the turtles position and
  orientation, and possibly other attributes such
  as the color and width of lines being drawn
• Pop a state from the stack and make it the
  current state of the turtle.

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L-Systems Generator
Prusinkiewicz, Hammel, Mech http//www.cpsc.ucalga
ry.ca/projects/bmv/vmm/title.html
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L-Systems
Prusinkiewicz, Hammel, Mech http//www.cpsc.ucalga
ry.ca/projects/bmv/vmm/title.html
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Procedural Methods Swarm Intelligence
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Procedural Methods - Particle Systems

• Particles have positions and velocity in the
  scene and these are tracked and updated
• To create fire
• Constantly create particles
• Particles move upwards, with turbulence added
• Draw them as partially transparent circles that
  fade over time

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Procedural Methods - Physical Systems

• Describe in mathematical terms how objects behave
  when forces are applied to them