Midterm Review

1
Midterm Review

• Ceng 477
• Introduction to Computer Graphics
• Computer Engineering
• METU

2
Review Topics

• The Graphics Pipeline
• Output primitives line drawing, circle drawing
  algorithms
• Filled Area Primitives polygon filling
  algorithms
• Geometric transformation 2D/3D transformation,
  composite transformations
• 2D Viewing Viewing transformations, line
  clipping, polygon clipping algorithms

3
The Graphics Pipeline
Modeling Transformations
Viewing Transformations
M1
V
M2
M3
WCS
VCS
MCS
Normalization Transformation and Clipping
Viewport Transformation
Projection Transformation
Normalized coordinates
Projection
DCS SCS
2D Image
Rasterization
4
Midterm Review Exercise 1

• Using the Cohen-Sutherland line clipping
  algorithm clip the following line

5
Solution

6
Midterm Review Exercise 2

• We want to put a mirror in a 3D scene. The mirror
  is specified as a 3D rectangle specified with its
  corner points and with one of its principle axes
  parallel to the y axis and the other parallel to
  the xz plane (illustrated below).

y
parallel to xz plane
x
z
Give the transformation matrix that will generate
mirror objects of all the objects in the scene
(you may assume all the objects are in front of
the mirror). And show the results as a product of
matrices.
7
Solution
y

• Basically what we are asked is reflection with
  respect to a plane.
• We can formulate this with a composite
  transformation
• Align the mirror plane with one of the planes,
  e.g., yz plane
• Compute reflection wrt to yz plane
• Restore the mirror planes original position and
  orientation

x
z
8
Solution

• Align the mirror plane with one of the planes,
  e.g., yz plane
• Translate P1 onto the y axis.
• Rotate the plane so that P2 is on the z axis.

y
x
z
y
x
z
9
Solution

• Reflection around yz plane is just inversion of x
  coordinate
• Reflection around yz.
• Do the reverse transformations rotate 45 degrees
  counterclockwise and translate P1 back to
  (3,10,-1) to finish the composite transformation.

y
x
z
Final Solution T(3,0,-1)Ry(45)Refyz(
)Ry(-45)T(-3,0,1)
10
Midterm Review Exercise 3

• Write the contents of the sorted edge table for
  the given polygon. Assume the scan conversion
  starts from the bottom scan line.
• Which edges are in the Active Edge List at scan
  line 4?
• Show the double or single counted vertices. But
  you do not need to consider their updated values
  when filling in the edge table

11
Solution

• The vertex A is the bottommost vertex. Starting
  from vertex A fill in the Sorted Edge table.
• E is the only vertex that need to be counted
  once, so we need update the edge FE to FE. The
  other vertices are not need to be updated.

12
Solution
The sorted edge table
Y E1 E2
1 AB AG
2 FG FE
3 CB CD
4 ED
13
Solution
The contents of the sorted edge table
Y Xinit, Ymax,1/m Xinit, Ymax,1/m
1 AB 1,6,0 AG 1,3,1
2 FG 6,3,-3 FE 6,4,2
3 CB 6,6,-5/3 CD 6,6,2/3
4 ED 10,6,-1
At scanline 4 AB,CB,CD,ED are active (FE is
just dropped from the list)