Mathematics 138
Representing Isometries in the Euclidean Plane
The key results:
Geometric version
Every isometry can be represented as the product of three or fewer reflections.
Algebraic version:
Every isometry can be represented as a 3x3 matrix

where the 2x2 matrix R has columns that are orthogonal unit vectors.
Proof:
To show that "if T is an isometry, it can be represented by such a matrix,"
we can prove the contrapositive, "if the matrix does not have this form, it is not an isometry."

First look at the bottom row.  If it is not ( 0 0 1), then there is some point in the plane
z = 1 whose image does not lie in the plane z = 1.

Now look at the 2x2 matrix R.
If either column is not a unit vector, the distance from the origin to (1,0) or (0,1) changes.
If the columns are not orthogonal, we can show that the distance from (1,0) to (0,1) changes.
Call the columns u and v.
d(u,v)2 = <u-v, u-v> = <u,u> + <v,v> - 2 <u,v> = 2 - 2 <u,v>
This has the correct value 2 if and only if <u,v> = 0.
An enumeration of all possibilities with three or fewer reflections.
Case 1: 0 reflections: the identity
Fixed point: all
Fixed lines: all
Matrix R is the identity, and = 
Case 2: 1 reflection: a reflection
Fixed points: an entire line of them (the axis of reflection)
Fixed lines: the axis of reflection and all lines perpendicular to it
Matrix R has determinant -1.
 is normal to the axis of reflection, along the eigenvector with eigenvalue -1.
The eigenvector with eigenvalue +1 is along the axis of reflection.
The eigenvector with eigenvalue -1 is normal to the axis of reflection.
The axis of reflection passes through the point .

Case 3: 2 reflections in parallel lines: a translation
Fixed points: none.
Fixed lines: all lines with direction .
Matrix R is the identity.
Case 4: 2 reflections in orthogonal lines: a half-turn

Fixed points: one point P = 
Fixed lines: every line through P
The matrix R is minus the identity, and every vector is an eigenvector with eigenvalue -1.
This is a special kind of rotation because it has fixed lines and because the order of the
two reflections does not matter.  In general, changing the order of two reflections will
replace the angle of rotation by its negative, but in this case the angle is p, and -p is the
same as +p.
An algebraic proof:
The square of a half-turn is the identity.
Write the half-turn as AB, where A and B are reflections in orthogonal lines.
Then ABAB = I.
Multiply by A on the left, B on the right.
AABABB = AB
But the square of any reflection is the identity; so AA = I and BB = I.
Thus BA = AB and the order does not matter.
Case 5: 2 reflections in non-orthogonal, non-parallel lines: a rotation
Fixed points: one point P, the center of rotation
Fixed lines: none
The matrix R has determinant 1 and no real eigenvalues or eigenvectors
Case 6: 3 reflections in lines that are neither parallel nor concurrent: a glide reflection
Fixed points: none
Fixed line: the axis of the glide reflection
A reflection in this axis is preceded (or followed) by a translation along this axis.
The matrix R has determinant -1, but  is not an eigenvector with eigenvalue -1
The eigenvector with eigenvalue +1 is parallel to the axis of the glide reflection, but in general this axis does not pass through the origin. It passes through P =  (proof to follow)
We could choose axes so that R is the matrix  and the isometry is 
This is a reflection in the line x = b/2 followed (or preceded) by a translation by a along that line.

Previous results for reflections:
Parallel lines:
Three reflections in parallel lines are equivalent to a single reflection in a certain line of the same pencil.
ref(a) ref(b) ref (c) = ref (a - b +c)
where a, b, and c are the (signed) distances from the origin to the three lines.

A given translation can be represented as the product of two reflections, one of which may be in any specified line perpendicular to the axis of the translation.
Intersecting lines:
Three reflections in lines that intersect at a point P are equivalent to a single reflection in a certain line through P.
ref(q) ref (f) ref (y) = ref (q - f + y)

A given rotation can be represented as the product of two reflections, one of which may be in any specified line through P.
Combining translations and rotations.
This makes it possible to reduce a product of four reflections to a product of two reflections.
All the diagrams are on the course Web site.
Case 1 -- a rotation about P followed by a translation to a rotation about some other point:
a. Express the translation as the product of two reflections, the first of which is about a line through P.
b. Express the rotation as the product of two reflections, the second of which is about a line perpendicular to the axis of translation.
c. Now the second and third of the four reflections are the same and their product is the identity.
d. This leaves just two reflections, in lines that intersect at the center of the product rotation.

Case 2 -- a rotation about P followed by a rotation about Q: 
a. Construct the line m that passes through P and Q.
b. Represent the first rotation as a product of two rotations, the second one about m.
c. Represent the second rotation as a product of two rotations, the first one about m.
d. Now the second and third of the four reflections are the same and their product is the identity.
e. Either the first and fourth reflections are about parallel lines and their product is a translation,
or they are about intersecting lines and their product is a reflection.

Case 3 -- a translation parallel to line r followed by a translation parallel to line s.
Point P is the point where lines r and s intersect.
a. Represent the translation along r as the product of two reflections, the second one about 
a line through P.
b. Represent the translation along s as the product of two reflections, the first one about 
a line through P.
c. Rotate the lines for the second and third reflections through p/2 (still the same rotation)
d. Now the first and second reflections combine to a half-turn about a point R, while the third and fourth combine to a half-turn about a point S. Let t denote the line RS.
e. Represent the first half-turn as a reflection about a line perpendicular to t followed by a reflection in t. Represent the second half-turn as a reflection in t followed by a reflection about a line perpendicular to t. The second and third reflections cancel; the first and fourth combine to 
give a translation along t.

At this point we have proved the general result that the product of any number of reflections can
be reduced to a product of three or fewer reflections.  If the number of reflections n is even, reduce
it to 2 by combining rotations and translations as specified above (plus possibly dealing with the easy special cases of two rotations about the same point or two translations along the same direction.)
If the number of reflections n is odd, reduce n-1 of them to 2, leaving the result as a product of at most 3 reflections

It is still useful, though, to be able to reduce the product of three reflections to a standard form, a glide reflection that consists of a reflection in a line n followed (or preceded) by a translation along n.
Special case: the second and third reflections are about parallel lines.
a. The first and second reflections combine to form a rotation.  Rotate both of these lines by p/2; their product is still the same rotation.  Denote by n the line for the new first reflection.
b. Now the second and third rotations are about perpendicular lines that intersect in a point F.  Their product is a half-turn about F.
c. Re-express this half-turn as the product of a new second reflection about a line parallel to n, followed by a new third reflection about a line perpendicular to n.
d. The first and second reflections combine to a translation along n.  This is followed by a third reflection in n.

General case: the second and third reflections are about lines that intersect at P.  They combine to form a rotation about P
Denote by g (Ryan's notation) the axis for the first reflection.
a. Rewrite the rotation about P as the product of two reflections.  The new second reflection is about a line l perpendicular to g.  The new third reflection is about a line m.
b. Lines l and g intersect at a point F, and since they are perpendicular, the product of the first reflection and the new second reflection is a half-turn about F.
c. Rewrite this half-turn as a reflection in a line n through F perpendicular to m, followed by a reflection in a line parallel to m.
d. The second and third reflections are now both about lines perpendicular to n; so they combine into a translation along n.  This is preceded by a reflection in line n.  So line n is the axis of the glide reflection.

A glide reflection T with axis n followed by a reflection in line a is a rotation or translation.

Case 1:
n intersects a at P.
Write the translation in the glide reflection as Wa Wb,where a is chosen to pass through P.
Now Wa T = Wa WnWa Wb
But the lines a, n, and a are concurrent at P, so the three reflections through them are equivalent to a single reflection. and Wa T is the product of just two reflections.

Case 2:
n is parallel to a
Wa T = Wa WnWa Wb
Since a and n are parallel, Wa Wn is a translation. So is Wa Wb. The product of these two translations is another translation. 
Conclusion: the product of any finite number of reflections can be reduced to the product of three or fewer reflections and hence is the identity, a reflection, a translation, a rotation, or a glide reflection.

An algebraic approach to glide reflections:
The matrix for a glide reflection is T=, where the 2x2 matrix R = 
has eigenvectors v with eigenvalue +1 (the axis of the glide reflection) and w with eigenvalue -1 (perpendicular to the axis)
The vector s =   can be written as s = p v + q w.
The point P =  is transformed into a point whose first two coordinates are
 the vector R s +  = p v - q w + 2p v + 2q w = p v + q w + 2p v = s + 2p v.
So the point P lies on the axis of the glide reflection and is translated through 2p along this axis.

A couple of similar results for rotations:
If R =  so that the rotation is a half-turn, then P =  is the center of rotation.
If R has determinant 1 but is not a half-turn, the center of rotation lies on the perpendicular bisector of the segment joining the origin to the point  , since this point and the origin must be equidistant from the center of rotation.