Exercise 1

Proseminar Computer Graphics
Summer Semester 2015
Theoretical Exercise Sheet 1
Hand-out: March 28th, 2015
Discussion: April 17th, 2015
Transformations and Projections
General Remarks
These theoretical exercises provide an opportunity to rehearse the theoretical concepts
taught in the lecture. Try solving the exercises on your own. The appropriate solution to the
exercises will be discussed during a following Proseminar.
You will have the opportunity to volunteer for presenting your solution in front of the class.
Please prepare accordingly. You can explain your solution on the white board; it is not
required to prepare slides. If necessary have a copy of your solution available for your
presentation. These presentations of solutions to the theoretical exercises will count
towards the active participation component in the final marking.
There will be four exercise sheets during the semester with 4-5 individual exercises each.
The sheets will be made available on the lecture webpage. Each sheet will cover specific
topics addressed in the previous lectures. Note that an online submission of your solution is
not required.
Interactive Graphics and Simulation Group, Institute of Computer Science, University of Innsbruck
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Exercise 1
Given is a rectangle in 2D, specified by its four vertices A, B, C, and D. Determine a set of
individual 2D transformation matrices in homogeneous representation that transform all
vertices, and thus the rectangle (A,B,C,D) into rectangle (A’, B’, C’, D’), as shown below. Note
that the transformations should in the end transform vertex A into A´, B into B´, etc.
y
5
B’
A’
-5
C’
D’
0
5
D
C
A
B
x
-5
Exercise 2
Provide a formal proof that the 2D operations of translation and rotation in homogeneous
representation are not commutative, except for trivial cases such as the zero or the identity
matrix. Start by determining the general format of the respective transformation matrices.
Interactive Graphics and Simulation Group, Institute of Computer Science, University of Innsbruck
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Exercise 3
A 3D object in homogeneous representation is transformed by the transformation matrix M
shown below. The matrix is composed of the following transformations, carried out in this
order: a uniform scaling, a rotation, and a transformation. Note that the scaling is applied
first. The target is to return the object into its original configuration. Determine and write
down the three inverse submatrices in correct order. Start by considering the structure of
the respective transformation matrices.
0
0
M
 3

0
0 3 3
3 0 6
0 0 1

0 0 1
Exercise 4
A plane P is given in Hesse normal form as n·p – d = 0. The plane is transformed by the
concatenated transformations M1, M2 given below into a new plane P’. Provide the 44
transformation matrix M that transforms n into the appropriate normal vector n’ of the new
plane P’. Consider the type of transformations carried out.
Transformations:
1
0
M 2 M1  
0

0
0 0 0 1
1 0 0 0
1 1 0  0

0 0 1  0
0
1 0 0
0  1 0

0 0 1
0
0
Exercise 5
Show for the perspective transformation of the perspective viewing frustrum into the
canonical view volume that points on the near plane map to the plane z = 1. Start with the
perspective transformation matrix and examine how the z-coordinate of vertices is affected
by the transformation. Also consider the final mapping from homogeneous coordinates to
3D Cartesian representation.
Interactive Graphics and Simulation Group, Institute of Computer Science, University of Innsbruck
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