Calculating with isometries
This benefits from column-vector notation and matrix multiplication. Writing x =
(
x
y
)
=
r cos ϕ
r sin ϕ
for
the position vector of X
r
= (x, y) = rS
ϕ
and applying the multiple-angle formulæ, rotation becomes
g(x) = r
cos(θ + ϕ)
sin(θ + ϕ)
= r
cos θ cos ϕ −sin θ sin ϕ
sin θ cos ϕ + cos θ sin ϕ
=
cos θ −sin θ
sin θ cos θ
x
For reflections, the sign of the second column is reversed:
cos θ sin θ
sin θ −cos θ
. Every isometry therefore has
the form f (x) = Ax + c where A is an orthogonal matrix.
16
Examples 3.13. 1. We revisit Example 3.10 in matrix format:
f (x) =
1
5
3x + 4y
4x − 3y
+
3
1
=
1
5
3 4
4 −3
x
y
+
3
1
Since
sin θ
cos θ
=
4/5
3/5
=
4
3
, we see that its effect is to reflect across the line through the origin making
angle
1
2
tan
−1
4
3
≈ 26.6° with the positive x-axis, before translating by (3, 1).
2. △
a
has vertices (0, 0), (1, 0), (2, −1) and is congruent to △
b
, two of whose vertices are (1, 2) and
(1, 3). Find all isometries transforming △
a
to △
b
and the location(s) of the third vertex of △
b
.
Let f = Ax + c be the isometry. Since d
(1, 2), ( 1, 3)
= 1 these points must be the images
under f of (0, 0) and (1, 0). There are four distinct isometries:
Cases 1, 2: If f ( 0, 0) = (1, 2) and f (1, 0) = (1, 3), then c = f
0
0
=
1
2
and
A
1
0
+ c =
1
3
=⇒ A
1
0
=
0
1
=⇒ A =
0 a
12
1 a
22
for some a
12
, a
22
. Since A is orthogonal, the options are A =
0 ∓1
1 0
and
we obtain two possible isometries:
• f
1
( x) =
0 −1
1 0
x +
1
2
rotates by 90°, then translates by
1
2
.
• f
2
( x) =
0 1
1 0
x +
1
2
reflects across y = x, then translates by
1
2
.
The third point of △
b
is f
1
(2, −1) = (2, 4) or f
2
(2, −1) = ( 0, 4).
Cases 3, 4: f (0, 0) = (1, 3) and f (1, 0) = (1, 2) results in two further
isometries f
3
and f
4
. The details are an exercise.
All four possible triangles △
b
are drawn in the picture.
In 1872, Felix Klein suggested that the geometry of a set is the study of its invariants: properties
preserved by its group of structure-preserving transformations. In Euclidean geometry, this is the
group of Euclidean isometries (Exercise 9). Klein’s approach provided a method for analyzing and
comparing the non-Euclidean geometries beginning to appear in the late 1800s. By the mid 1900s, the
resulting theory of Lie groups had largely classified classical geometries. Klein’s algebraic approach
remains dominant in modern mathematics and physics research.
16
An orthogonal matrix satisfies A
T
A = I. All such have the form
cos θ ∓sin θ
sin θ ±cos θ
=
a ∓b
b ±a
where a
2
+ b
2
= 1.
42