Computer Graphics Raster Devices Transformations

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Computer GraphicsRaster DevicesTransformations

• Areg Sarkissian

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Raster Devices

• Most displays used for computer graphics are
  raster displays.
• The surface of raster displays has a certain
  number of pixels that it can show, such as 640 X
  480 307,000 pixels
• All raster displays have a built-in coordinate
  system that associates a given pixel in an image
  with a given physical position on the display
  surface.
• The horizontal coordinate Sx increases from left
  to right, and the vertical coordinate Sy
  increases from top to bottom. They have an
  upside-down coordinate system. (0,0)
  (639,0)
• (0,479)
• Consist of three main parts 1) Frame Buffer 2)
  Scan Controller 3) Digital to Analog Converter
  (DAC)

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Frame Buffer

• A region of memory sufficiently large to hold all
  of the pixel values for display.
• A graphic card that is installed in a personal
  computer actually houses the memory required for
  the frame buffer.
• The display memory can be thought of a two
  dimensional array memxy

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Scan Controller DAC

• The scan controller causes the frame buffer to
  send each pixel through a converter to the
  appropriate physical spot on the display surface.
• The scan controller addresses one pixel value,
  memxy, in the frame buffer at the same time
  it addresses one position, (x, y), on the face of
  CRT.
• The converter takes a pixel value such as 010010
  and converts it to the corresponding quantity
  that produces a spot for each red, green, and
  blue colors on the display.
• The dots are so close together that eye sees one
  composite dot and a single color that is the sum
  of the three component colors. Thus, the
  composite dot can be made glow in a total of
  4x4x464 different colors.

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Scan Controller DAC

• Some of the more expensive systems have a frame
  buffer that support 24 planes of memory. Each of
  the DACs has eight input bits, so there are 256
  level of red, green, blue. (16 million colors)
• Monochrom
• A single DAC converts pixel values in the frame
  buffer to voltage levels, which drive a single
  electron-beam gun. Note that 6 planes in frame
  buffer give 64 levels of gray.

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Indexed Color and the Lookup Table

• A color lookup table (LUT), which offers a
  programmable association between a pixel value
  and the final displayed color
• The color depth is again six, but the 6 bits
  stored in each pixel go through an intermediate
  step before they drive the CRT.
• These bits are used as an index into a table of
  64 values (LUT0LUT63)
• Each LUTi contains a 15-bit value, which every
  five bits drives corresponding DAC. (Red, Green,
  Blue)
• The set of 215 possible colors that the system
  is capable of displaying is called its palette,
  so the palette for this system is 32K colors
• Since LUT has only 64 bits index, so it can only
  show 64 different color at a time on the screen.

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Indexed Color
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Coordinates Translation

• Usually the upside-down coordinates of raster
  displays are translated to a more familiar
  coordinates (Cartesian Coordinates) in
  programming languages, such as OpenGL.

y
x
(0, 0)
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Transformations

• Transformations are very useful in computer
  graphics in a number of situations
• A scene can be fashioned by placing a number of
  instances of an object at different places and
  with different sizes using proper transformation.
• A single motif can be designed and then fashion
  the whole shape of an object by reflection,
  rotation, and translation of the motif. (ex.
  Snowflake)
• Helps a designer to view an object from different
  vantage points and make a picture from each one.
• In a computer animation, several objects must
  move relative to one another from frame to frame.

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Transformations

• A transformation alters each point P in space (2D
  or 3D) into a new point Q by means of a specific
  formula or algorithm.
• An arbitrary point P in the plane is mapped to
  another point Q. We say that Q is the image of P
  under the mapping T.
• An object is being transformed by transforming
  each of its points using the same function T(P)
  for each point.
• Points are represented as

y
QT(P)
y
QT(P)
P
P
x
x
z
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Affine (or Linear) Transformations

• Types of Affine Transformations
• Translation
• Scaling
• Reflection
• Rotation
• An affine transformation is represented by
  matrices.

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Translation

• Translates a picture into a different position on
  a graphics display.
• The translation part of the affine transformation
  arises from the third column of the matrix for 2D
  and forth column for 3D.
• 2D 3D

Example 2D translation
?
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Scaling

• Changes the size of a picture and involves two
  scale factors Sx and Sy for the x- and
  y-coordinates
• (Qx, Qy) (SxPx, SyPy)
• Thus, the matrix for scaling is simply

3D
2D
Example
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Reflection

• If a scale factor is negative, then there is also
  a reflection about a coordinate axis.

About x-axis
About y-axis
P
Example reflection about x-axis
x
QT(P)
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2D Rotation

• The rotation of a figure about a given point
  through some angle.
• The matrix for 2D rotation about origin
• Example P(x, y) and

QT(P)
60
P
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3D Rotation

• The 3D rotation matrices about each coordinate

About x-axis (x-roll)
About y-axis (y-roll)
About z-axis (z-roll)
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Rotation about an arbitrary point

• The result of this rotation will be product of
  three matrices.
• Step 1) Translation (a, b) to (0, 0)
• Step 2) Rotation point P about origin (0, 0)
• Step 3) Translation (0, 0) to (a, b)
• Example (a, b) is an arbitrary point and P(x, y)

T(P)Q
P
(a, b)
(0,0)
Step 1
Step 2
Step 3
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References

• Computer Graphics Using OpenGL
• F.S Hill, Jr