Representation Of Virtual Objects

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Representation Of Virtual Objects
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2D v 3D

• Computer screen 2D
• Virtual environment 3D
• Map 3D to 2D
• Movement of objects is in 3 dimensional space
• High processing power

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What Makes a Picture 3-D?

• Requires many more lines and angles to produce 3D
• Must be able to define the lines and calculate
  which are visible as viewpoint changes to show
  motion
• Colour, size and clarity of image indicate where
  shape is in relation to viewer

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Steps to create a 3D scene

• Creating a virtual 3-D world by defining a
  heirarchy of objects
• Determining what part of the world will be shown
  on the screen.
• Determining how every pixel on the screen will
  look so that the whole image appears as realistic
  as possible.

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View the scene from one point
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Another view
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Inducing reality

• Define the shapes so that they can be seen in
  motion 3D
• Apply a surface texture
• Show perspective
• Lighting
• Blurring the edges

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A simple shape
Defined as a set of points or Vertices With
surfaces or facets defined by spline boundaries
created by joining points with lines
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More complex shapes

• Shape defined as polygons(triangles)
• Rounded surfaces created by more polygons

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Defining the surface

• Colour
• variation over surface
• Texture
• rough, smooth, etc
• Lighting
• creates shadowing
• Reflectance
• dependant on texture and colour

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Realism added by surface mapping
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Lighting and reflectance

• Exhibits shadowing and shading
• Gouraud shading
• Ray-tracing used to calculate light paths based
  on reflectance values

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Perspective and Z-buffering

• Objects appear smaller further away
• Zero-point
• Uses Z co-ordinate to compute
• Relative position
• Occlusion

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Depth of field

• Further away objects become hazier
• Focus attention on nearer objects
• Occurs naturally but must be added to virtual
  environment

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Anti-aliasing

• Sharp contrast unreal
• Curved lines - stepped
• Edges blurred removing stepping
• More natural

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Basic Analysis

• 1 Define points in 3D space
• 2 Define lines and Facets which join points
  together
• 3 Apply texture to facets
• 4 Define reflectance properties and colour of
  surface
• 5 Define light sources to generate shadows and
  shading
• 6 Redraw image as viewpoint changes applying
  perspective and occlusion to induce Reality

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Definition of a vertex in 2D

• Distance Between Two Points
• The distance between two points ltAx,Aygt and
  ltBx,By gt can be found using the pythagorus
  theorem
• dx Ax-Bx
• dy Ay-By
• distance sqrt(dxdx dydy)

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A vector

• Vectors themselves can be added by adding each of
  their components
• or they can be multiplied (scaled) by multiplying
  each component by some constant k (where k ltgt 0).

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2D Vectors

• Vectors can be
• added
• multiplied
• scaled
• To calculate the length of a vector we simply
  calculate the distance between the origin and the
  point at ltx,ygt
• length ltx,ygt - lt0,0gt
• sqrt( (x-0)(x-0) (y-0)(y-0) )
• sqrt(xx yy)

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Definition of a 2D Line

• First we can imagine a line as having 2 endpoints
  in 2D space
• P1ltx1,y1gt and
• P2ltx2,y2gt.
• Alternatively we can imagine the line as having
  an origin (starting point) and a direction
  (vector)
• Origin ltXo, Yo gt
• Direction ltXd, Yd gt
• we can assume that the line's origin is at one
  endpoint and it's direction is the difference
  between both endpoints
• Origin P1 ltx1, y1 gt
• Direction P2-P1 ltx2-x1, y2-y1gt
• ltx,ygt ltxo, yogt k ltxd, ydgt

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Definition of a 3D Point

• A point is similar to it's 2D counterpart, we
  simply add an extra component, Z, for the 3rd
  axis
• Points are now represented with 3 numbers
  ltx,y,zgt.
• Left-hand rule

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Distance Between Two 3D Points

• The distance between two points ltAx,Ay,Azgt and
  ltBx,By,Bzgt can be found by again using the
  pythagorus theorem
• dx Ax-Bx
• dy Ay-By
• dz Az-Bz
• distance sqrt(dxdx dydy dzdz)

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Definition of a 3D Vector

• Either a point at ltx,y,zgt or a line going from
  the origin lt0,0,0gt to the point ltx,y,zgt.
• 3D Vector addition and subtraction is virtually
  identical to the 2D case. You can add a 3D vector
  ltvx,vy,vzgt to a 3D point ltx,y,zgt to get the new
  point ltx',y',z'gt like so
• x' x vx
• y' y vy
• z' z vz

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• Vectors themselves can be added by adding each of
  their components, or they can be multiplied
  (scaled) by multiplying each component by some
  constant k (where k ltgt 0)
• To calculate the length of a vector we simply
  calculate the distance between the origin and the
  point at ltx,y,zgt
• length ltx,y,zgt - lt0,0,0gt
• sqrt( (x-0)(x-0) (y-0)(y-0)
  (z-0)(z-0) )
• sqrt(xx yy zz)

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Definition of Object
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Translation
Three degrees of Freedom in x, y, and z plane
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Six degrees of freedom
3 degrees of freedom correspond to the
rotational movement about each of the axes Pitch
Roll Yaw
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Scaling
Multiply one or more components of the line
vectors
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MatricesA Point in Space

• x
• y
• z
• w lt- w1
• a point or vector can be represented in 3D space
  as ltx,y,zgt. When using matrix maths it helps to
  represent it as ltx,y,z,wgt.

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Modifying the Position of a Point

• Modifying the Position of a Point
• Let's say we want to take any point ltx,y,z,wgt
  given and do something to it to get a new point.
  A row vector can be used to represent how we want
  to change a point
• These values                      x
• contain the                       y       
  This is our
• info on how we ----gt A B C D  . z   lt---
  point in 3D
• want to change                    w       
  space
• the point
• To figure out how the row vector A B C D
  changes a point, we can visualize lying the
  point's column vector on it's side on top of it
  like this
• x  y  z  w
• A  B  C  D      (x A) (y B) (z C)
  (w D)

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• Let's say we want to be able to take any point
  ltx,y,z,wgt and figure out the coordinates for the
  point ltx',y',z',1gt which is exactly 4 units to
  the "right" of it, ie the point which is 4 units
  further along the x axis. We can do this by using
  4 row vectors. The first one will calculate the
  new x point (x'), the next one the new y point
  (y') and so on.
• First let's look at calculating x'.
• We know that the new x point can be calculated
  like this x' x 4, so a row vector to
  calculate this would be
•                      1  0  0  4
• i.e. when we multiply this out by a point
  ltx,y,z,wgt we'll get
• (x 1) (y 0) (z 0) (w 4) x 4

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• We also know that y' y, z' z and w 1, so we
  can calculate the row vectors for each of the
  other values, and stack them all on top of each
  other
• x row vector ----gt    1  0  0  4     x'1x
  0y 0z 4 x 4
• y row vector ----gt    0  1  0  0     x'0x
  1y 0z 0 y
• z row vector ----gt    0  0  1  0     x'0x
  0y 1z 0 z
• 1 row vector ----gt    0  0  0  1     x'0x
  0y 0z 1 1

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• To take a point ltx,y,z,wgt and calculate the new
  point we just mutiply the point by each of the
  row vectors.
• Here's a more generic representation of what we
  are doing
• x'         M11  M12  M13 M14     x
• y'        M21  M22  M23  M24     y
• z'         M31  M32  M33  M34     z
• w'         M41  M42  M43  M44     w
• In this case ltx,y,z,wgt is the point we are
  popping in, ltx',y',z',w'gt is the point we'll be
  getting out, and each number in the matrix is
  represented by Mij, where i row number and j
  column number.

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• x' (x M11) (y M12) (z M13) M14
• y' (x M21) (y M22) (z M23) M24
• z' (x M31) (y M32) (z M33) M34
• w' (x M41) (y M42) (z M43) M44

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•   1    0    0    0             x' x
•   0    1    0    0             y' y
•   0    0    1    0             z' z
•   0    0    0    1

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• Translation
• A translation is simply adding (or subtracting) a
  point to any given point. Let's say you want to
  add TX to x, TY to y and TZ to z. The matrix to
  do this is
•   1    0    0    tx             x' x tx
•   0    1    0    ty             y' y ty
•   0    0    1    tz             z' z tz
•   0    0    0    1 
• Scaling
• Sometimes we may need to scale a point, ie
  mutiply each axis by a given number. This is
  handy for things like zoom effects.
•   sx   0    0    0             x' sx x
•   0    sy   0    0             y' sy y
•   0    0    sz   0             z' sz tz
•   0    0    0    1
• Of course, if you only want to scale along the X
  axis (say) then you simply set SX to the scale
  value and set SY and SZ to 1.

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• Basic Rotations
• Things start getting a wee bit tricky here. For
  starters we can rotate things around the x axis,
  the y axis, or the z axis. Notice how each axis
  forms a line in 3D space? Well we can also rotate
  things around any arbitrary line. This is handy
  if we want to do effects like objects rotating
  about their own axis.
• The matrix for rotating around the x axis by
  angle é is
•   1      0      0     0             x'
  x
•   0    cos é  -sin é  0             y' (cos
  é) y - (sin é) z
•   0    sin é   cos é  0             z' (sin
  é) y (cos é) z
•   0      0      0     1
• The matrix for rotating around the y axis by
  angle é is
•   cos é   0    sin é   0            x' (cos
  é) x (sin é) z
•     0     1     0      0             y'
  y
•   -sin é  0    cos é   0             z' -(sin
  é) x (cos é) z
•     0      0    0      1
• And the matrix for rotating around the zaxis by
  angle é is

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Underlying geometry

• Vertices
• Lines
• Faces
• Transforms