CSC 461: Computer Graphics I (Fall 2006)
Written Assignment 3
(Chapter 4)
Due November 1, 2006 in class
In Chapter 4, geometric objects and transformations are introduced, focusing
on linear vector space, affine space, coordinate systems and frames,
transformations between coordinate systems, homogeneous coordinates. Also, four
transformations, including translation, rotation, scaling, and shear, are discussed
theoretically and how they are implemented in OpenGL. In this assignment, you
should read the text and course notes (Powerpoint) to
fully understand these topics, and solve the following questions.
- Show
that the following sequences commute:
o A
rotation and a uniform scaling
o Two
rotations about the same axis
o Two
translations
- If we
are interested in only two-dimensional graphics, we can use
three-dimensional homogeneous coordinates by representing a point as p=[x
y 1]T and a vector as v=[a b 0]T.
Find the 3 ´
3 rotation, translation, scaling, and shear matrices. How many degrees of
freedom are there in an affine transformation for transforming
two-dimensional points?
- We
have used vertices in three dimensions to define objects such as
three-dimensional polygons. Given a set of vertices, find a test to
determine whether the polygon that they determine is planar.
- Given
two non-parallel three dimensional vectors u and v, how can we form an
orthogonal coordinate system in which u is one of the basis vectors?