PROSEMINAR ON WEB 3

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PROSEMINAR ON WEB 3

3-D COMPUTERFUNDAMENTALS

• BY MOKUBE

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TOPICS

• 3-D TRANSFORMATION
• RASTERIZATION OF
• 2-D PRIMITIVES
• 2-D AND 3-D CLIPPING

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CONTENT OF THE TOPICS

• 3-D TRANSFORMATION
• EUCLIDEAN SPACES AND DEGREES OF FREEDOM
  
  
  
  TRANSLATION,SCALING ,ROTATION
• REPRESENTATION ,PROJECTION ,AND
• IMPLEMENTATION.
• RASTERIZATION OF 2-D PRIMITIVES
• -POINTS RASTERIZATION,
• -LINE SEGMENTS RASTERIZATION,
• -POLYGONS RASTERIZATION
• 2-D AND 3-D CLIPPING
• -CLIPPING- STRATEGIES, -POINTS,
• -LINE SEGMENTS,AND POLYGONS

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• Transformation
• Definition this is a function which maps points
  from one coordinate system to another.
• A single point in a 2-D Euclidean
  space is unambiguosly specified through ist
  Cartesian coordinates(x,y) whereas with 3-D
  Euclidean space,a single point is uniguely
  specified by three coordinates(x,y,z).
• Points,scalar values and vectors are used
  to construct the description of a virtual world.A
  rigd body possesses in a plain three degrees of
  freedom and six degrees of freedom in 2-D and
  3-D spaces respectively.
• The degrees of freedom in 2-D spaces are
  translation along the two axes and rotation
  around a point.
• .

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• TRANSLATION
• In 2-D space,translation is the movement of
  points along the x and y
• axes with the distance between the points
  being constant or the moving
• of the coordinate system into the opposite
  direction as in the figurs below.
• Simple Algorithms are used to describes a
  Translation Transformation

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• SCALING
• This is propotional expansion or contraction of
  distances between point or of space itself and
  hence not a rigid body transformation.
• ROTATION
• In 2-D rotation is either clockwise or
  anticlockwise on the X and Yaxes at some angles-
  the angle of rotation.3-D rotation is along the
  x,y and z axes and involves a kind of reference
  system,a positive direction of rotation to be
  applied and in what order,hence involving the
  trigonometric sine and cosine. Functions in its
  calculations.
• The chosen reference system is vertical for
  y,horizontal for x,and z
• diagonally away from us and the order 3-D
  rotation transformation
• is seen as matrix multiplication of a rotation
  matrix by a vector
• producing a result vector.

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• Graphical explanation

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• REPRESENTATION OF TRANSFORMATION
• Scaling and rotation transformatons are linear
  whereas translation is affine
• transformation and are all better expressed by a
  4x4 matrix and somewhat
• adjusted coordinate vectors.This is expressed as
  below

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• Projection transformation
• Definition3-D world coordinates mapping
  onto 2-D screen coordinates.Parallel and
  perspective projection do exist.
• Parallel when one dimension of space is
  collapsed and all the points of the parallel
  lines in a 3-D space mapps into a single point in
  2-D plane.If the projecction line crosses the
  projection plain at angle of 90 degrees, this is
  an orthogonal parallel projection while the rest
  are oblique.The actual size of object remains
  thesame and the parallel lines are preserved.This
  projection is used in CAD applications.
• Diagram

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• Perspective produces an image in which sizes of
  the projections of objects depends on the
  objectsdistance to the viewer.At point of
  intersection of the rays from the object,an image
  seen as an object to a viewer is formed.This
  transformation is non-linear and hence require to
  mainttain homogeneouse coordinats for it to be
  represented in a matrix form.there exist
  one-point and two-point perspectives and one
  whose transformation depend on the angle between
  the cameras view and the normal of the screen.
• The diagrams below perspective transformation.

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Diagrams explains it
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• IMPLIMENTING TRANSFORMATION
• Fixed Point Arithmeticthis is the
  representation of integers such as binary (power
  of two), decimal(power of ten) numbers in
  computer.
• Negative numbers are computed using the twos
  complement
• form which negates the leftmost digit.
• Fixed point easily carries on fast addition and
  subtraction when
• there exist no floating point unit but then has
  the problem of
• overflows
• The performance of basic geometric
  transformation,the techniques describing
  location,orientation and motion of visual objects
  and their visibility to a viewer has been
  discussed so far in this section.

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• Rasterization of 2-D primitives
• The process of transforming 2-D analytical
  description of geometric
• primitive into a proper set of descrete
  pixels irasterization.
• Point rasterization setting specific cell of
  bitmap memory to a color
• value.Difficulties arise with realization
  as imag bitmap is inherently
• descrete.2-D space tries to mimic contains
  continuoseranges of real
• numbers.
• Assumptionto reduce round-off errors the
  coordinates of the point to
• be plotted are integers and the area of
  point equals that of the pixel
• Line rasterizationfinding all pixels
  intersected by or closed enough
• to a given lines path.For a line forming
  two similar triangles with
• points(Xstart,Ystart),(Xend,Yend) and the
  point (X,Y) on the line.
• A forward difference technque is applied to
  iteratively compute
• coordinates of multiple points.It describes
  the rate of change of the
• function with respect to the change of the
  argument.
• J.E.Bresenham iterative method involves
  taking a wider range of
• coordinates assuming the change in the x
  coordinate being wider,

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• Polygon rasterization simple polygon consists of
  convex and concave and uses the scan-line
  approach for their rasterization i.e letting the
  scan-line to travel verticaly along the
  polygon,intersectin at each heights at some line
  of pixels.convex is differentiated from concave
  its connecting line at any two points never
  leaves the polygons boundaries.And only one
  continuos span of pixel is needed for any
  horizontal scan-line whereas multiple is needed
  for concave as in the diagram below.
• verteces
• -preset the start and end values
• -determine the scan-line from the edges and
• -rasterize the current scan-line.

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• The algorithm for the rasterization of concave
  polygon in finding the
• Intersection of a scan-line with edges is as
  follows
• -pre-sort all edges(excluding useless
  horizontal ones)according to their least Y
  coordinate
• -maintain a list of active edgesby adding the
  edges of the pre-sorted list to it.
• -examine if y of the pre-sorted list is equal
  to that of the current scan-line.
• -use current X coordinates for the edges in
  the activeedge list to fill the span and repeat
  the iteration.

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• MEISTERS THEOREM
• Meisters theorem which states that any simple
  polygon with
• A tleast three vertices has atleast two
  non-intersecting
• ears,fascilitates polygon triangulation.An ear is
  determinedby a
• polygons vertex such that ist two neighbor
  vertices can
• connected by a diagonal.

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• Rendering interpolatively shaded polygons
• Here we integrate rasterization and
  computation of light intensity by keeping a color
  intensity value or values in every vertex of the
  polygon and linearly interpolating these values
  at the time of pixel line computation in finding
  the color for each pixel inside the polygon
• Rendering textured polygon
• -A texture associates some color values to two
  parameters (u,v) of polygons area.Procedural
  textures can be distinguishedwhere some function
  (a procedure)computes a color for arbitrary (u,v)
  coordinates.
• Commonly textures are stored as a bitmap
  whereby colors are explicitly stored for each
  (u,v) pair inside a 2-D array.

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• Anti-aliasing
• There exist some degree of inperfections,for
  instance when drawing polygons a pixel partially
  covered by the analytical area of the polygon was
  not plotted.Also when picking a color value from
  the texturemap,only one texture cell was
  considered.
• -the textures are precomputed at different
  scales
• -smaller textures are used for polygons
  further away from the viewer
• -MIP mapping method are employed.
• Stochastic sampling is another approach
  used-leaves visual noise and not regular
  patterns in the filtered image.

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• 2-D and 3-D Clipping
• Clipping the process to locate primitives
  part or parts which
• satisfy some spatial constraints.
• 2-D Clipping strategies
• -clipp before the rasterization phase for
  polygons restricted by simple clipping area
  (rectangle).
• -clipp during rasterization stage for convex
  primitivescheck if a pixel is within the proper
  bounds,and then plotting.
• -the primitive is rasterized into some bigger
  buffer selecting from the buffer only those
  pixels inside the complex clipping area.

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• Clipping points
• -rectangular clipping described by infinite
  four lines which limit the least and greatest X
  coordinates and the least and greatest Y
  coordinates.
• Clipping line segments
• -simultaneous clipping with rasterization
  and checking if the pixel is within the
  boundaries or not.
• - with line segment were finding
  intersection between segment and boarders of
  clipping areas.
• -trivial acceptance or rejection based on
  region outcodes speeds up comparism.
• -use of bitwise and and or operation.
• -An algorithmic procedure describes the
  binary search clipping

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• clipping polygons
• -clipping the polygon iteratively along all
  edges
• -searching intersection between polygons
  edges and edges of the clipping area.
• -polygon changes ist shape as a result of
  clipping.
• 3-D Clipping strategies
• -ensures transfornation of all valid points
• -limited numbers of primitives and overflow
  due to big coordinates
• There is no 3-D effect for far objects and
  has a perspective angle of
• 90.

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To sum up

• Transformation techniques allow to describe
  location, orientation and motion of virtual
  objects and also their visibility to the viewer
• The rasterization routines are used to picture
  geometric primitives. Shading and texture mapping
  can be added to improve visual realism
• Clipping discards the objects in the world space
  which would not contribute to the generation of
  the image

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• ANY QUESTIONS
• THANKS A BUNCH