Rotation

1. Rotation transformation

A rotation transformation is defined by giving the position of its center C and the rotation-angle ω.

The transformation is applied to a shape and rotates it about its center by the oriented angle ω.
This is an example of an isometric transformation of the plane (or isometry) i.e. a transformation that does not modify metric relations of a shape.

2. Coordinates description

In cartesian coordinates a rotation is determined by the location x₀=(x₀, y₀)^t of its center C and the angle ω.
The transformation equations can be expressed through a matrix:

3. Symmetry at a point

The symmetry at a point (or half-turn) is a special rotation by an angle equal to π.

The product of two symmetries with different centers {C,C'} is a translation by the double of CC'.

This is the reason why the set of all possible rotations of the plane is NOT a group (see also section-5).
The set of rotations though about the same point C is indeed a group.

4. Rotation as a product of two reflexions

Every rotation is the product of two reflexions on lines intersecting at its center by and angle equal to ω/2.

The important remark is that the two lines {CA,CB} can be rotated about C and give the same rotation as product of reflexions. For this it suffices to maintain the same oriented angle ω/2 between them.
By reversing the orientation of the angle i.e. taking the reflexions in the opposite order we define the inverse rotation.

5. Generic rotation-product

In most cases the product of two rotations is indeed a rotation. The new center C'' of the resulting rotation can be determined from the other two centers {C, C'} and the corresponding rotation-angles {ω₁, ω₂}.

In the above figure the first rotation R₁ is the product of reflexions {F₁, F₂}, R₁=F₂*F₁, on lines {CA, CB} and the second rotation R₂ is the product of reflexions {F₃, F₄}, R₂ = F₄*F₃ on lines {C'B, C'D}. Line CB is identical to C'B, thus giving as product R = R₂*R₁ = F₄*F₃*F₂*F₁ = F₄*F₁, since a reflexion taken twice is the identity.
Thus the result of the two rotations is the rotation about C'' by an angle equal to the sum of the two rotation angles.
This happens though in the generic case in which the two lines {CA, C'D} intersect.
In case ω₁+ω₂=2k*π the composition R = R₂*R₁=F₄*F₁ is easily seen to be a translation.
The analysis gives a geometric procedure to construct the center (or translation vector) of the composition R.

6. Remarks

1) The set of all rotations together with the set of all translations builds the group of the orientation preserving isometries of the euclidean plane.
2) The set of all reflexions is a coset of the previous group.
3) Every isometry can be expressed as the product of one, two or three reflexions. Hence the set of all reflexions generates the whole group of isometries. Products of three reflexions on lines in general position i.e. lines which can be taken as sides of a triangle build the so-called glide-reflexions (see GlideReflexion.html ).
4) A thorough discussion of the group of isometries can be found in [Yaglom, vol. I].

Bibliography

[Yaglom] Yaglom, I. M. Geometric Transformations (3 vols.) Washington DC, Math. Assoc. Ammer., 1962

Return to Gallery

http://users.math.uoc.gr/~pamfilos/