Graphics Reference
In-Depth Information
2.2.5. Example.
Let
X
= { (x,y) | x > 0 } Õ
R
2
and define the distance-preserving map
T:
X
Æ
X
by T(x,y) = (x + 1,y). The map T is clearly not onto.
2.2.6. Theorem.
Motions form a group under composition.
Proof.
Exercise.
The idea of a motion as a distance-preserving map is intuitively simple to under-
stand, but it is not very useful for making computations. In the process of deriving a
simple analytical description of motions, we shall not only get a lot of geometric
insights but also get practice in using linear algebra to solve geometric problems. We
begin our study of motions with an approach that is used time and again in mathe-
matics. Namely, if faced with the problem of classifying a set of objects, first isolate
as many simple and easy-to-understand elements as possible and then try to show
that these elements can be used as building blocks from which all elements of the
class can be “generated.”
2.2.1
Translations
The simplest types of motions are translations.
Any map T :
R
n
Æ
R
n
of the form
Definition.
()
T
p=p v
+
,
(2.2)
where
v
is a fixed vector
,
is called a
translation
of
R
n
. The vector
v
is called the
trans-
lation vector
of T.
Writing things out in terms of coordinates, it is easy to see that a map T(x
1
,x
2
,
...,x
n
) = (x
1
¢,x
2
¢,...,x
n
¢) is a translation if and only if it is defined by equations of
the form
¢
xxc
1
=+
1
1
¢
xxc
=+
2
2
2
·
·
·
¢
xxc
n
=+,
(2.3)
n
n
where the c
i
are fixed real numbers. Clearly, (c
1
,c
2
,...,c
n
) is the translation vector of
T in this case.
2.2.1.1. Theorem.
Translations are motions.
Proof.
This is a simple exercise for the reader.
Here are several simple interesting properties of translations.
2.2.1.2. Proposition.
A translation T with
nonzero
translation vector
v
satisfies the
following properties:
