WEEK 2: 3D Computer Graphics

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WEEK 2 3D Computer Graphics
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Introduction

• Computer graphics is a very broad subject area
  and supports the disciplines of image processing,
  visualization, virtual reality, computer-aided
  design (CAD) and computer animation
• Here, you will be introduced about the basic
  concepts of computer graphics so that in the
  following chapter on computer animation
  techniques, there will be no need to continuosly
  pause and define individual computer graphics
  terms

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Principles of computer graphics

• The internal world of a computer is organized as
  binary code, therefore we must create some more
  convenient world if we are to make any headway in
  creating 3D images
• Here, we use various branches of mathematics that
  have been around for thousands of years
• In particular, the work of Euclid and Pythagoras
  is very relevant, as they laid the foundations
  for the geometric problems encountered in
  computer graphics

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Cartesian coordinates

• One of the central principles of computer
  graphics is the use of Cartesian coordinate
• Figure 2.1 shows a set of 3D Cartesian axes(90o
  to each other) labeled X, Y, and Z, intersecting
  at the origin
• The position of the point A is uniquely
  identified by the three measurements x, y and z,
  which are called the coordinates of A

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• These coordinates are enclosed in brackets
  (x,y,z), and are always defined in this
  alphabetic sequence ? this triplet of numbers is
  a unique definition of the point P
• The point P, or any other point, can be used to
  locate the position of a camera, light source or
  a specific point on an object

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• What is important to realize is that inside a
  computer, three numbers represent a point in
  space once we have access to some numeric
  quantity, it can be altered within a computer
  program
• The coordinate system showed in Fig 2.1 is a
  right-handed coordinate system. This means that
  when using your right hand, you can align your
  thumb with the X-axis, your index finger with the
  Y-axis, and your middle finger with the Z-axis

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• If you attempt to use your left hand, you will
  discover that the Z-axis points in the opposite
  direction commercial animation systems use both
  conventions not only that, some choose the
  Z-axis as the vertical axis ? so you should pay
  special attention to these conventions when
  working with animation system

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Modeling simple objects

• Many objects in the real world such as tables,
  cupboards, desks, boxes, etc are constructed from
  flat surfaces where each surface is defined by a
  series of edges, and each edge is associated with
  two corners
• In world of computer graphics, geometric shapes
  called polygons, with 3,4,5 or more edges
  represent such surfaces BUT instead of corners,
  we use the word vertices

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• Fig 2.2 shows a triangular polygon with vertices
  A1, A2, and A3 . Each vertex has three
  coordinates, nine number represent it For
  example, if coordinates for the vertices are P1
  (1,5,10), P2 (10,1,9) and P3 (12,10,1), the
  triangle could be stored inside a computer as
  three groups of coordinates, as shown in Table 3.1

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• Table 2.1 A triangle is stored inside a computer
  as a table of coordinates

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Y

• X
• Fig 2.2 A triangular polygon

A1
A3
A2
Z
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• Even though a polygon has two sides, some
  computer animation systems may only recognize one
  of them as visible For example the triangle in
  Fig 2.2 could be visible from above, but
  invisible when looking from below
• Another convention applies to the direction of
  the vertices. For instance, in Fig 2.2, the
  vertex sequence A1, A2, A3 is anti-clockwise when
  viewed from above, but clockwise when viewed from
  below.When polygon is defined it is conventional
  to be consistent about vertex sequences

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• The simplest 3D solid object than can be
  constructed from triangles is the tetrahedron,
  which has four triangular sides, as shown in Fig
  2.3. A cube (6 squares), a tetrahedron (4
  triangles), an octahedron (8 triangles), a
  dodecahedron (12 hexagons), and an icosahedron(20
  triangles)

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• Fig 2.3 A tetrahedron constructed from 4 triangles

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• The mathematician Euler (1707 1783), discovered
  the following relationship between the edges,
  faces and vertices of any polyhedron
• Faces Vertices Edges 2
• For example, a cube has 6 faces, 8 vertices and
  12 edges and an octahedron has 8 faces, 6
  vertices and 12 edges

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• To model really smooth objects such as a sphere
  or a torus, three techniques are available the
  first is to use many small polygons the second
  is to use surface patches and the third is to
  use mathematical equations
• Fig 2.6 shows a sphere and a torus constructed
  from polygons. The faceted surfaces are very
  obvious however we could improve matters by
  doubling the number of polygons

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• This would improve the appearance and at the same
  time, would double the amount of information
  stored inside the computer, and also increase the
  time needed to render the image
• In some circumstances, this does not matter, but
  in computer games, VR and flight simulation, it
  is essential to keep the number of polygons to a
  minimum

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• Surface patches employ mathematical techniques
  for representing a small portion of a smooth
  surface, which can be joined together to form
  very complex surfaces such as a face or a car
  body
• The ideas behind Bezier and NURBS patches are
  described later on.

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• Certain object such as a sphere, a cone, a torus,
  a cylinder, and an ellipsoid, can be represented
  by mathematical equations
• These can be used to form more complex structures
  and are used by a computer-aided design strategy
  called constructive solid geometry(CSG). Although
  this is a very important area of computer
  graphics, it is not widely used within the
  computer animation sector

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Modeling assemblies

• More complex objects can be built from assemblies
  of geometric primitives. For example, a table may
  be modeled from a rectangular top and four
  identical legs, and a chest of drawers would only
  require one drawer copied several times, with the
  associated cabinet
• However, there would be no need to model the
  individual drawers if they were not to be
  animated. It would be sufficient to model the
  surface detail to give the impression that the
  drawers actually existed

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3D libraries

• Animation systems generally provide the user with
  a library of 3D objects, but they may only
  contain simple geometric primitives. If you
  require something more esoteric such as human
  heart, an elephant, a submarine, or a tree, then
  its highly likely that Viewpoint DataLabs can
  help
• They provide a valuable service to the computer
  animation industry by maintaining and selling a
  large database of 3D models that can be purchased
  individually or as a collection

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• For example, Fig 2.9 shows a human heart
  revealing its complex interior. The model is
  extremely accurate as it is based upon the
  geometry of a real heart
• Fig 2.10 shows the geometry of an elephant, which
  has probably been digitized from a physical model

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• Fig 2.11 shows a 3D polygonal model of a tree.
  Such a model could have been digitized by hand,
  or even grown using a program
• The latter technique would have involved
  providing a program with a set of rules
  describing the characteristics of the tree
• The rules would describe how branches would be
  formed how they would be distributed about the
  trunk the angle between a major branch a minor
  branch how the branches reduce in size and how
  leaves are distributed

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• If stochastic (random) procedures are introduces
  into a program, in the form of random numbers,
  the tree acquires characteristics that give it a
  life-like quality, rather than looking as though
  it has been grown to a formula

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Internal representation of geometry

• It is obvious that faceted objects can be created
  from a collection of polygons defined by edges
  and vertices
• However, we still require a way to store these
  numbers inside a computer such that the integrity
  of the geometry is maintained no matter how its
  modified
• A data structure is used to hold data in
  particular order within a computer system, and
  one such technique is called a winged-edge data
  structure.

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• The name comes from the basic atom of geometry
  stored within the data structure, and illustrated
  in Fig 2.12
• The figure shows a rectangular box constructed
  from 6 sides, 12 edges, 8 vertices. Because each
  pair of adjacent sides share a common edge, and
  each edge is defined by two vertices, which
  belong to other edges dividing other adjacent
  sides, a simple data structure can be used to
  link everything together

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• The data structure is shown in Fig 2.12 as a
  vertical line forming an edge, and wing-like
  elements on the top and bottom, identify the
  vertices and their associated edges
• Whenever a vertex is moved to a new position, the
  integrity of the edge geometry remains sound, so
  too, is the boundary of the polygon so no matter
  how vertices, edges and polygons are moved
  around, the winged-edge data structure will not
  introduce any holes of gaps

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• Remember that the geometry of a polygonal object
  is stored in the form of vertices represented as
  three coordinates, and a data structure is used
  to relate the vertices to edges, and edges to
  polygons. The winged-edge data structure may not
  be found in every CAS, but something similar is
  required