CS685  Homework 1
Due date: September 23
Be as concise as possible
.
1. (5) Consider rigid body transformations in the plane. Draw a right triangle deﬁned by three
points
A
= (
2
,
1
)
,
B
= (
4
,
1
)
,
C
= (
4
,
6
)
.
•
Consider a rotation matrix
T
1
=
±
cos
θ

sin
θ
sin
θ
cos
θ
²
a. What is the determinant of the matrix ?
•
Consider transformation matrix
T
2
=
±
sin
θ
cos
θ
cos
θ

sin
θ
²
a. Is the matrix orthonormal ? What is the determinant of the matrix ?
Matrix is orthonormal when the determinant is
+
/

1
. The determinant of the
above matrix is

1
.
c. Is
T
2
rigid body transformation ? What is the difference between
T
1
and
T
2
, how are
the results different?
T
2
is not a rigid body transformation, since it is not orientation preserving. Rigig
body transformations must have deteminant of the rotation matrix +1. By apply
ing
T
2
in addition to rotation we also get a mirror image of the original triangle.
There is no way how to get from initial to the ﬁnal position by translation and
rotation around origin only.
2. (5) Point
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 Spring '08
 Luke,S
 Linear Algebra, Rotation, yB, v*deltat*cos

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