Part 21
Isometries
Printed version of the lecture Differential Geometry on 20. November 2009
Tommy R. Jensen, Department of Mathematics, KNU
21.1
Overview
Contents
1
Isometry
1
2
Orthogonal Transformations
3
3
Characterization of Isometries
5
4
Conclusion
6
Isometry (Koji Takei)
21.2
1
Isometry
Isometries in Euclidean space
Definition 1.1
An isometry of R3 is a mapping F : R3 → R3 that satisfies
d(F(p), F(q)) = d(p, q)
for all points p, q of
R3 .
21.3
1
Example 1.2(1): translation
Definition
Let a be a point in R3 and define a mapping T by
T (p) = p + a for all p in R3 .
Then T is called the translation by a.
A translation is an isometry
We check, for any two points p, q :
d(T (p), T (q))
=
d(p + a, q + a)
=
||(p + a) − (q + a)||
=
||p − q||
=
d(p, q).
21.4
Example 1.2(2): rotation
Definition
Let ϑ be a real number in [0; 2π] and define a mapping C by
C(p1 , p2 , p3 ) = (p1 cos ϑ − p2 sin ϑ , p1 sin ϑ + p2 cos ϑ , p3 ).
Then C is called the rotation through the angle ϑ around the z axis.
A rotation around the z axis is an isometry
Some calculations using cos2 + sin2 = 1 prove
||C(p) −C(q)|| = ||p − q|| for all p, q in R3 .
21.5
Composition of isometries
Lemma 1.3
If F and G are isometries of R3 , then the composite mapping G(F) is also an isometry of R3 .
Proof of Lemma 1.3
Let p and q be points in R3 .
Then
d(G(F(p)), G(F(q))) = d(F(p), F(q)) = d(p, q).
21.6
2
Inverse function
Properties of an inverse function
If F : R3 → R3 is one-to-one and onto, then there exists a uniquely determined function F −1 : R3 → R3
with the properties
F(F −1 ) = F −1 (F) = I,
where I : R3 → R3 is the identity mapping: I(p) = p for all p in R3 .
21.7
Composition of translations
Lemma 1.4
• If S and T are translations of R3 , then S(T ) = T (S), and this mapping is also a translation.
• If T is the translation by a, then T −1 exists and is the translation by −a.
• If p and q are points in R3 , then there exists a unique translation T such that T (p) = q.
Special case
If T is a translation and there exists a point p such that T (p) = p, then T is the identity mapping: T = I.
21.8
2
Orthogonal Transformations
Orthogonal transformations
Definition
A mapping F : R3 → R3 is an orthogonal transformation if F is linear, and
F(p) • F(q) = p • q for all p, q in R3 .
Example: The rotation C of Example 1.2 is an orthogonal transformation:
C(p) •C(q)
=
(p1 cos ϑ − p2 sin ϑ , p1 sin ϑ + p2 cos ϑ , p3 )
•(q1 cos ϑ − q2 sin ϑ , q1 sin ϑ + q2 cos ϑ , q3 )
=
p1 q1 (cos2 ϑ + sin2 ϑ )
+p1 q2 (− cos ϑ sin ϑ + sin ϑ cos ϑ )
+p2 q1 (− sin ϑ cos ϑ + cos ϑ sin ϑ )
+p2 q2 (sin2 ϑ + cos2 ϑ ) + p3 q3
=
p1 q1 + p2 q2 + p3 q3 = p • q.
21.9
Orthogonal transformations and isometry
Lemma 1.5
If F : R3 → R3 is an orthogonal transformation, then F is an isometry of R3 .
Proof of Lemma 1.5
First, F preserves norm:
||F(p)||2 = F(p) • F(p) = p • p = ||p||2 .
3
This implies ||F(p)|| = ||p|| for all points p in R3 .
Next, F is an isometry, since for all p, q in R3 :
d(F(p), F(q)) = ||F(p) − F(q)|| = ||F(p − q)|| = ||p − q|| = d(p, q).
21.10
Isometries and orthogonal transformations
Lemma 1.6
If F is an isometry of R3 such that F(~0) = ~0, then F is an orthogonal transformation.
Proof of Lemma 1.6
The proof has two parts:
• proof that F(p) • F(q) = p • q for all p, q in R3 , and
• proof that F is linear.
21.11
Proof that F(p) • F(q) = p • q for all p, q in R3 .
We know that F is an isometry, and F(~0) = ~0. Therefore:
||F(p)|| = d(~0, F(p)) = d(F(~0), F(p)) = d(~0, p) = ||p||.
If p, q are two points in R3 , then ||F(p) − F(q)||2 = ||p − q||2 :
(F(p) − F(q)) • (F(p) − F(q)) = (p − q) • (p − q),
so linearity of the dot product implies
||F(p)||2 − 2F(p) • F(q) + ||F(q)||2 = ||p||2 − 2p • q + ||q||2 .
Because of ||F(p)|| = ||p|| and ||F(q)|| = ||q|| we get
F(p) • F(q) = p • q.
21.12
Proof that F is linear
Let u1 = (1, 0, 0), u2 = (0, 1, 0) and u3 = (0, 0, 1).
We know that F preserves dot products, so
F(ui ) • F(u j ) = ui • u j = δi j for 1 ≤ i, j ≤ 3.
Let p = (p1 , p2 , p3 ) be any point in R3 .
Orthonormal expansion gives:
F(p) = ∑(F(p) • F(ui ))F(ui ) = ∑(p • ui )F(ui ) = ∑ pi F(ui ).
If p, q are points and a, b are real numbers, then F(ap + bq) =
∑(api + bqi )F(ui ) = a ∑ pi F(ui ) + b ∑ qi F(ui ) = aF(p) + bF(q).
This shows that F is a linear function.
21.13
4
3
Characterization of Isometries
Characterization of isometries
Theorem 1.7
Let F be an isometry of R3 .
There exists a unique translation T and a unique orthogonal transformation C such that
F = T (C).
Proof of Theorem 1.7
The proof has two parts:
• proof of existence of T and C, and
• proof of uniqueness of T and C.
21.14
Proof of existence of T and C with F = T (C)
We know that F is an isometry.
Let T be the translation by F(~0).
Then T −1 is the translation by −F(~0), by Lemma 1.4.
It follows from Lemma 1.3 that T −1 (F) is an isometry.
Because of T −1 (p) = p − F(~0) for all points p in R3 , we get
T −1 (F)(~0) = T −1 (F(~0)) = F(~0) − F(~0) = ~0.
Lemma 1.6 implies that T −1 (F) is equal to an orthogonal transformation C.
From C = T −1 (F) follows T (C) = T (T −1 (F)) = F.
21.15
Proof of uniqueness of T and C with F = T (C)
We assume that T is a translation and C is an orthogonal transformation such that F = T (C).
We will prove that T is the translation by F(~0), so T = T, and C = T −1 (F) = C.
We have
T (C) = T (C) =⇒ C = T −1 (T (C)).
Both C and C are linear functions, therefore C(~0) = C(~0) = ~0.
Now ~0 = C(~0) = T −1 (T (C))(~0) = T −1 (T )(C(~0)) = T −1 (T )(~0).
But T −1 (T ) is a translation, so if it maps ~0 to ~0, then T −1 (T ) is the identity, T −1 (T ) = I.
This implies T = T.
It follows from C = T
−1
(F) = T −1 (F) that C is equal to C.
21.16
5
Decomposition of an isometry
Definition
Let F be an isometry such that F = T (C) where T is a translation and C is an orthogonal transformation.
Then T is called the translation part of F.
And C is called the orthogonal part of F.
In matrix notation, F can be calculated as

 
a1
c11
F(p1 , p2 , p3 ) =  a2  +  c21
a3
c31
c12
c22
c32


c13
p1
c23   p2  ,
c33
p3
where (ci j ) is an orthogonal matrix, with its inverse equal to its transpose, and the translation part of F is
the translation by a = (a1 , a2 , a3 ).
21.17
4
Conclusion
The End
END OF THE LECTURE!
(Never give up. . . )
21.18
Next time:
Tangent Map of an Isometry
21.19
6