The Classification of Isometries of the Plane
Notes for Math 371 by M. Flashman
Proposition: Any plane
isometry is either a reflection or the product of two or three
(i) The product of two
reflections that have the lines of reflection intersect at a point O is
a rotation with center O through an angle twice the size of the angle between
the two lines of reflection.
(ii) The product of two reflections that have parallel lines of
reflection is a translation in the direction perpendicular to the two lines
and by a length twice the distance between the two lines of reflection.
(iii)The product of three reflections is either a reflection or
a glide reflection.
Proof: We can do this geometrically or using the matrices!
(i) and (ii): Suppose S and T are the reflections and P' = ST(P) for any point P.
For (i): Recall that O denotes the point of intersection of the lines of reflection
of S and T.
It suffices to show that PO is congruent to P'O and
that <POP' is congruent to the double of the angle between the lines
For (ii) It suffices to show that PP' is
perpendicular to both lines of reflection and that if QQ' is a line segment
with Q on one reflection line and Q' on the other and QQ"
perpendicular to the line, then PP' is congruent to the double of
(a)Suppose all three lines have a point in common, O.
Holding the angle
between the first and second lines of reflection fixed, rotate
the pair about the point O until the second line coincides with the
third line of reflection.
So the product on the three lines is the same
as the reflection on the rotated first line.
(b) Suppose no pair of the three line is parallel but they have no
single point in common.
So the three lines form a triangle.
first and second lines with the angle between them held fixed on their
until the second line is perpendicular to the third line.
Now rotate the second and third lines about the point where
they now intersect at a right angle
until the third line is
perpendicular to the first line.
So the transformed first and second lines are
and the translation determined by the product of these reflections is parallel to the third line,
so the product of the three
reflections is a glide reflection.
(c) Suppose the three lines form a pair of parallel lines with a
transverse line. Then rotate the transverse line and one of the
parallels with which it is composed, holding the angle between them
fixed about the point of their intersection, so that the three
resulting lines are organized as in b. Thus this product is also a
(d) Suppose all three lines are parallel. Holding
the distance between the first and second lines of reflection fixed, we
translate the pair by a vector perpendicular to them both until the
second line coincides
with the third line of reflection. So the product on the three lines is
the same as the reflection on the first line (translated).
Algebraic proofs of (i) and (ii):
(i) Assume that the point of interection of the two lines is the origin
and the first line of reflection is the X Axis , while the
second line has equation AX +BY =0. So, ST(x,y) = S
(x, -y) = ?
Suppose the line makes an angle t with the X-axis. Then
S(x,-y) = Rt RX R-t (x,-y)
=Rt RX ( cos(-t)x+ -sin(-t)(-y), sin(-t) (x) + cos(-t) (-y))
=Rt RX (x cos(t) - ysin(t), -xsin(t) - ycos(t))
=Rt (x cos(t) - ysin(t), xsin(t) + ycos(t))
= (cos(t)[x cos(t) - ysin(t)] + -sin(t)[ xsin(t) + ycos(t)],
sin(t)[x cos(t) - ysin(t)] + cos(t)[ xsin(t) + ycos(t)])
= (x cos2(t) - ycos(t)sin(t) - xsin2(t) - ysin(t)cos(t), x sin(t) cos(t) - ysin2(t) + xcos(t)sin(t) + ycos2(t))
= (x cos(2t)) - ysin(2t) ,x sin(2t) + ycos(2t))
= R2t (x,y)
(ii) Assume the first line is the X axis and the second line has
equation Y = c. Then
ST(x,y) = S (x,-y) = (x, -y +
2(c+y) ) = (x, y +2c)
But if W is vertical translation by 2c, then W(x,y) = (x,y+2c).
So ST(x,y) = W(x,y).