6. Geometri c Tra nsformations
Three examples of functions with important geometric consequences
follow from the SMSG postulates. The first two are actually given in the
Ruler Postulate. Recall:
Postulate 3 (Ruler Postulate): The points on the line can be placed in a
correspondence with the real numbers such that
1. To every point of the line there corresponds exactly one real
number.
2. To every real number there corresponds exactly one point on
the line.
3. The distance between two distinct points is the absolute value
of the difference of the corresponding real numbers.
Let l be a given line in the plane and denote the set of real numbers by R .
Then Postulate 3 asserts the existence of a function
ruler : I " R ,
where for any point P on I , ruler(P) = the real number guaranteed by
item 1. First, convince!yourself that this function is one-to-one and onto.
Then observe that item 2 is nothing more than a description of the
inverse to ruler. In a practical sense, 1 and 2 allow us to replace any line
in the plane with a number (or ruled) line, so we don’t really compute
anything, we just use the points on a number line as a replacement for
the points of the given line.
Using the notation I " I to mean the collection of all pairs of points on I ,
item 3 defines another function, the distance function:
!
d : I " I # R,
d (P ,Q ) = ruler (P ) $ ruler (Q ) .
Question: Is d one-to-one? Is d onto?
!
Finally, let l = AB and m = AC be two distinct lines in the plane
intersecting at A. We have seen how the plane can be divided into four
convex regions and the given lines:
!
Region I: All points on the same side of AC as B and all points on the
same side of AB as C.
! side of AC as B and all points on the
Region II: All points on the opposite
same!side of AB as C.
Region III: All points on the opposite!side of AC as B and all points on the
! side of AB as C.
opposite
! of AC as B and all points on the
Region IV: All points on the same side
opposite!side of AB as C .
! relative to the intersection point
Points on each line can be separated
A.
!
Now, let P be a point in the plane not on either line. Let s be the line
through P parallel to m , guaranteed to exist by Postulate 16, and let Pl
be the intersection of s with l . Similarly, let t be the line through P
parallel to l , and let Pm be the corresponding intersection of t and m .
!
!
I I
I
C
Pl
m
s
B
A
P
l
t
Pm
III
IV
Assign P the pair of numbers ( d (P , Pm ), d (P , Pl )) if P is in Region I,
( " d (P , Pm ), d (P , Pl )) if P is in Region II, ( " d (P , Pm ), "d (P , Pl )) , if P is in
Region III, and ( d (P , Pm ), "d (P , Pl )) if P is in Region IV (in the picture above
P is in Region IV).
!
!
!
Question:
! How would you assign a pair of numbers if P is on one or the
other of the two lines?
Once this is done, a correspondence between the points of the plane and
pairs of numbers in R " R # R 2 is been established, and you can check that
this correspondence is a function that is one-to-on and onto. In fact, you
should recognize that this function “coordinatizes” the plane al la
Cartesian coordinates.
!
Geometric Transformations continued
We are now ready to specialize our study of functions. Let " denote the
Euclidean plane.
Definition: A transformation is a function T : " #!" that is both oneto-one and onto.
Note that a transformation always has
! an inverse, and the inverse is itself
a transformation.
Also note that the composition of two transformations is again a
transformation.
The simplest transformation is the identity transformation, defined
previously by I : " # ", I (P ) = P .
Because both the domain and codomain of a transformation is the plane,
we visualize
the effect of a transformation on points of the plane by
!
studying the images of sets under the transformation. For example, the
image of lines, triangles, circles, etc often reveal important properties of
the specific transformation under investigation.
In this spirit we make the following definitions.
Definition: A subset, S, of the plane is said to be invariant under the
transformation T : " # " if T(S) = S.
Definition: A point P of the plane is said to be a fixed point of
T : " # " if T(P) = P.
!
!
Definition: A subset, S, of the plane is said to be a fixed set under the
transformation T : " # " if S is invariant and each point of S is fixed.
Definition: A transformation T : " # " is called and isometry if for any
two points P and Q of the plane, d (P , Q ) = d (T (P ), T (Q )).
!
In other words, distances between points are preserved.
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