Problem 19: The Group of Rigid Motions

I. Introduction to Rigid Motions

In recent problems we have described certain densities as "invariant under the group of rigid motions". Recall that rigid motions consist of rotations, translations, reflections, and compositions of these. For our purposes we need only study rotations by the angle and translations by the vector (a,b), as we are only considering rigid motions that preserve orientation.

Recall that if a point is defined as , then a motion u is described as , where

Note that a, b, and are real values where , and . If we have a set K, where uK is the image of K under the given motion u, then we can say that the two sets are congruent and have equal measure; this is the idea behind invariance.

II. Motions and Matrices

A motion u can be represented in matrix form as

We will use the notation u to represent both the rigid motion and the matrix representing the motion, although these are different objects. Note that even though u is only a transformation by two factors, a rotation and a translation, its corresponding matrix is a 3x3 matrix. When two motions act upon a set of points by composition, their corresponding matrices are multiplied together to give the matrix of the compostion map. Note that the resulting matrix will also be a 3x3 matrix. It is easy to see that the third row has no effect on either the translation or the rotation, and exists so that we can represent our motions by 3x3 matrices.

As described above, the motion is represented by the product of the matrices . Because the defining operation of a composition of motions is matrix multiplication, order does matter! is NOT equivalent to .

Motions also have inverses, where

You may check that equals the identity matrix, which makes sense since is the identity transformation.

III. Translations

Now for some new definitions. Let M denote the group of motions and suppose . A left translation by s is defined as . Analogously, a right translation by s is defined .

For example, if

then

This can also be expressed as

Similarly, the right translation yields

PROBLEM 19(a):

Find .

IV. Differential Forms and Invariance

A 1-form on a motion is given by the equation

where , , and are infinitely differentiable functions that are defined on M. Addition and scalar multiplication of these forms is defined in the normal way. All of the differentiable 1-forms on M form a three-dimensional vector space, denoted , where da, db and d form its basis. The left and right translations are defined as

Now let's find the all the 1-forms that are invariant under left translations and all those that are invariant under right translations. These 1-forms are called left-invariant and right-invariant, respectively. Another way to look at this is to find all matrices that do not change when multiplied by , denoted by and those who don't change under , . It can be shown that such matrices are expressed as

PROBLEM 19(b):

Show is actually left invariant.

So, the 1-forms we seek are


	Left invariant		Right invariant

We know that ,, and are linearly independent because their determinant equals zero. Therefore any linear combination of these 1-forms is also invariant under left translations. This is also true for ,, and .

PROBLEM 19(c):

Prove that any 1-form on M that is left invariant is a linear combination with constant coefficients of the 1-forms. (this is also true for 1-forms that are right invariant)

Solutions

Now, clearly , where e is the identity matrix; we have the following equality.

References: Santalo