Given a 
and 
,
the translate of S by origin shift p
is defined to be the set
.
When the origin shift p is 
, we call
p+S the trivial translate .
Note that all of the points in S are
displaced by p. We can think of this
as a mapping of points of 
which moves
all points and
point
sets in a rigid motion and, in particular,
moves the origin to p and -p to the origin.
One often encounters expressions such as
a-c and b-c in . It is often
useful to view this from the viewpoint of
a translation of by -c,
particularly in interpreting inner products such as
(a-c)(b-c) in terms of the cosine of an angle
determined by points a and b with vertex c,
which translate to a-c, b-c, and the origin,
respectively.
The following theorem summarizes several basic results which are based on simple linear algebra and set theory.
Proof Left as an exercise.
Proof Left as an exercise.
Proof
Let 
be given. Assume 
.
[Show 
.]
Let r be an arbitrary member of 
r = q+s for some 
,
since r was an arbitrary member of .
[Show 
.]
Let u be an arbitrary member of 
.
Thus 
.
