3P5.13Euclidean.html

Part 13: Isometries of the Euclidean plane
The Euclidean plane	We do linear algebra in the Cartesian space Rⁿ. It comes equipped with an origin and n axes. Distances between points (vectors) can be determined by an inner product. But the space in which we live has no addition or scalar multiplication, origin or axes, just points, lines and planes and Euclidean distance determined by Pythagoras' Theorem. It is called the Euclidean plane E² or Euclidean space E³.	.
Isometries	We are interested in the symmetries of Eⁿ which are the isometries or length preserving bijections. First we consider the Euclidean plane E². If a is a symmetry and P, Q and R are points, then once we know Pa, we also know that Qa lies on a circle with centre Pa and radius \|PQ\|. Hence if we know both Pa and Qa, there are exactly two possibilities for Ra, one preserving the orientation of the triangle PQR and the other reversing it. Also, a preserves the orientation of some triangle if and only if a preserves the orientation of all triangles. Hence a is completely determined once we know the image of any triangle. a is called a direct isometry if it preserves orientations of triangle and an indirect isometry if it reverses orientation. This result can be stated as a theorem: Theorem 13.1 Every isometry of E² maps every triangle into a congruent triangle. The triangles have the same orientation if and only if the isometry is direct. Conversely, if triangles ABC and A'B'C' are congruent, the mapping A -> A', B -> B', C -> C' determines a unique isometry of E². The set of all isometries is a group under composition, the set of direct isometries is a subgroup, the product of two indirect isometries is direct and the product of a direct and an indirect, in either order, is indirect. We now define a faithful group action of the additive group R² on E², called translation. For any vector A, define T_A by T_A : X -> X + A for all X in E². Here, we define X + A as in linear algebra, i.e. it is the endpoint of the line segment A with initial point at X. T_A is called a translation because it moves every point the same distance in the same direction. It is a direct isometry and equals the identity map if and only if A = 0. Otherwise it has no fixed point and hence does not correspond to a linear transformation in R². The set of translations is a group T under composition with T_A.T_B = T_A+B. The group T is an abelian subgroup of the group of direct isometries of E². It is normal since every conjugate of a translation is a product of three direct or a direct and two indirect isometries; since it has no fixed point it must be a translation. Every element has infinite order, since T_Aⁿ = T₀ implies nA = 0 and hence A = 0. We consider now the reflections in a line in E². Let l be a line and let M_l be the map which sends each point A of E² to its mirror image in the line l. M_l is an indirect isometry of order 2. The subset of E² fixed by M_l is the line l. The next type of isometry are the rotations about a point in E². Let P be a point and let R_P,_q be the map which sends each point A of E² to the point reached by rotating A an angle of q anti-clockwise about the point P. R_P,_q is a direct isometry which has finite order if and only if q is a rational multiple of p. It turns out that every isometry of E² can be expressed as a composition of reflections. Theorem 13.2 An isometry F is direct if and only if F is a product of 2 reflections and indirect if and only if it is a product of 3 reflections. Proof Let ABC be a triangle and A'B'C' its image under the isometry F. If A = A', B = B' and C = C' then F is the identity, which is the product of any reflection with itself. If A = A' and B = B' but C not = C', then F is reflection in the line AB and hence the cube of this reflection. So suppose A = A' but B not=B' and C not= C'. Let l be the perpendicular bisector of BB' and let ABC'' be the image of the reflection of A'B'C' in the line m joining A' and B'. Then M_l is a reflection which maps ABC onto A'B'C' if F is indirect and onto A'B'C'' if F is direct. Thus F = M_l or M_l.M_m. Finally, suppose A not=A', B not= B' and C not=C'. Let n be the perpendicular bisector of AA'. Then M_n maps A onto A'. Suppose it maps B' to B'' and C' to C''. Then F.M_n maps A to A , B to B'' and C to C''. Thus F.M_n is one of the types of isometry considered earlier, so is a product of two reflections if direct, one if indirect. Hence F = F.M_n.M_n is a product of three reflections if indirect, two if direct. To see how this theorem applies to translations and rotations, both direct, just note that the product of reflections in two parallel mirrors, distant r apart, is translation by 2r, while the product of reflections in two mirrors which meet at an angle of a is a rotation about the point of intersection by an angle of 2a. Hence we know: Theorem 13.3 The product of two reflections is a translation if the mirrors are parallel and a rotation if they intersect. Their is a fourth kind of isometry, called a glide. It is the product of a translation T_A along a line l and a reflection M_l in the same line, consequently indirect. If G is a glide, then G maps any point P to a point P' on the opposite side of l such that the midpoint of PP' lies on l. Theorem 13.4 G is a glide if and only if G is the product of a reflection and a rotation by p, and the product of a rotation by p and a reflection. Proof Let G = M_l.T_A = T_A.M_l. T_A is the product of two reflections M_j.M_k in lines j and k perpendicular to l. Let H = M_j.M_l = M_l.M_j and H' = M_k.M_l, so H and H' are rotations by p about the intersections of l and j and l and k respectively. Now T_A = M_j.M_k = M_j.M_l.M_l.M_k = H.H', so G = M_l.T_A = M_l.M_j.M_k = H.M_k; and G = T_A.M_l. = M_j.M_k.M_l = M_j.H. Conversely, the product of a rotation about P by p and a reflection M_m is the glide T_A. M_l where A and l are perpendicular to m and the length of A is twice the distance from P to m ( so equals M_l if P is on m.) We have seen that a direct isometry is either a translation with no fixed point or a rotation with one fixed point. An indirect isometry with a fixed point is a reflection, and hence has a fixed line. The only other possibility is an indirect isometry with no fixed points. Theorem 13.5 F is an indirect isometry with no fixed point if and only if F is a glide. Proof We have seen that a non-trivial glide is indirect and has no fixed point. Conversely, let F be indirect with no fixed point. Say F : A -> A'. Let H be the rotation by p which swaps A and A'. Then H.F is an indirect isometry which fixes A' and hence is a reflection M_m. Hence F = H^-1.M_m = H.M_m since H has order 2. To sum up: A direct isometry in E²is a translation (no fixed point) or a rotation (one fixed point). An indirect isometry is a reflection (one fixed line) or a glide (no fixed point).	.
.	Table of Contents. To Continue . Return to Page 1 . Last update Dec 3 1999 Author: Phill Schultz, schultz@maths.uwa.edu.au	.

Part 13: Isometries of the Euclidean plane

The Euclidean plane

We do linear algebra in the Cartesian space Rⁿ. It comes equipped with an origin and n axes. Distances between points (vectors) can be determined by an inner product.

But the space in which we live has no addition or scalar multiplication, origin or axes, just points, lines and planes and Euclidean distance determined by Pythagoras' Theorem. It is called the Euclidean plane E² or Euclidean space E³.

Isometries

We are interested in the symmetries of Eⁿ which are the isometries or length preserving bijections.

First we consider the Euclidean plane E². If a is a symmetry and P, Q and R are points, then once we know Pa, we also know that Qa lies on a circle with centre Pa and radius |PQ|. Hence if we know both Pa and Qa, there are exactly two possibilities for Ra, one preserving the orientation of the triangle PQR and the other reversing it. Also, a preserves the orientation of some triangle if and only if a preserves the orientation of all triangles. Hence a is completely determined once we know the image of any triangle. a is called a direct isometry if it preserves orientations of triangle and an indirect isometry if it reverses orientation.

This result can be stated as a theorem:

Theorem 13.1 Every isometry of E² maps every triangle into a congruent triangle. The triangles have the same orientation if and only if the isometry is direct.

Conversely, if triangles ABC and A'B'C' are congruent, the mapping A -> A', B -> B', C -> C' determines a unique isometry of E².

The set of all isometries is a group under composition, the set of direct isometries is a subgroup, the product of two indirect isometries is direct and the product of a direct and an indirect, in either order, is indirect.

We now define a faithful group action of the additive group R² on E², called translation. For any vector A, define T_A by
T_A : X -> X + A for all X in E².
Here, we define X + A as in linear algebra, i.e. it is the endpoint of the line segment A with initial point at X. T_A is called a translation because it moves every point the same distance in the same direction. It is a direct isometry and equals the identity map if and only if A = 0. Otherwise it has no fixed point and hence does not correspond to a linear transformation in R². The set of translations is a group T under composition with T_A.T_B = T_A+B.

The group T is an abelian subgroup of the group of direct isometries of E². It is normal since every conjugate of a translation is a product of three direct or a direct and two indirect isometries; since it has no fixed point it must be a translation. Every element has infinite order, since T_Aⁿ = T₀ implies nA = 0 and hence A = 0.

We consider now the reflections in a line in E². Let l be a line and let M_l be the map which sends each point A of E² to its mirror image in the line l. M_l is an indirect isometry of order 2. The subset of E² fixed by M_l is the line l.

The next type of isometry are the rotations about a point in E². Let P be a point and let R_P,_q be the map which sends each point A of E² to the point reached by rotating A an angle of q anti-clockwise about the point P. R_P,_q is a direct isometry which has finite order if and only if q is a rational multiple of p.

It turns out that every isometry of E² can be expressed as a composition of reflections.

Theorem 13.2 An isometry F is direct if and only if F is a product of 2 reflections and indirect if and only if it is a product of 3 reflections.

Proof Let ABC be a triangle and A'B'C' its image under the isometry F. If A = A', B = B' and C = C' then F is the identity, which is the product of any reflection with itself. If A = A' and B = B' but C not = C', then F is reflection in the line AB and hence the cube of this reflection.

So suppose A = A' but B not=B' and C not= C'. Let l be the perpendicular bisector of BB' and let ABC'' be the image of the reflection of A'B'C' in the line m joining A' and B'. Then M_l is a reflection which maps ABC onto A'B'C' if F is indirect and onto A'B'C'' if F is direct. Thus F = M_l or M_l.M_m.

Finally, suppose A not=A', B not= B' and C not=C'. Let n be the perpendicular bisector of AA'. Then M_n maps A onto A'. Suppose it maps B' to B'' and C' to C''. Then F.M_n maps A to A , B to B'' and C to C''. Thus F.M_n is one of the types of isometry considered earlier, so is a product of two reflections if direct, one if indirect. Hence F = F.M_n.M_n is a product of three reflections if indirect, two if direct.

To see how this theorem applies to translations and rotations, both direct, just note that the product of reflections in two parallel mirrors, distant r apart, is translation by 2r, while the product of reflections in two mirrors which meet at an angle of a is a rotation about the point of intersection by an angle of 2a. Hence we know:

Theorem 13.3 The product of two reflections is a translation if the mirrors are parallel and a rotation if they intersect.

Their is a fourth kind of isometry, called a glide. It is the product of a translation T_A along a line l and a reflection M_l in the same line, consequently indirect. If G is a glide, then G maps any point P to a point P' on the opposite side of l such that the midpoint of PP' lies on l.

Theorem 13.4 G is a glide if and only if G is the product of a reflection and a rotation by p, and the product of a rotation by p and a reflection.

Proof Let G = M_l.T_A = T_A.M_l. T_A is the product of two reflections M_j.M_k in lines j and k perpendicular to l. Let H = M_j.M_l = M_l.M_j and H' = M_k.M_l, so H and H' are rotations by p about the intersections of l and j and l and k respectively.

Now T_A = M_j.M_k = M_j.M_l.M_l.M_k = H.H', so G = M_l.T_A = M_l.M_j.M_k = H.M_k; and G = T_A.M_l. = M_j.M_k.M_l = M_j.H.

Conversely, the product of a rotation about P by p and a reflection M_m is the glide T_A. M_l where A and l are perpendicular to m and the length of A is twice the distance from P to m ( so equals M_l if P is on m.)

We have seen that a direct isometry is either a translation with no fixed point or a rotation with one fixed point. An indirect isometry with a fixed point is a reflection, and hence has a fixed line. The only other possibility is an indirect isometry with no fixed points.

Theorem 13.5 F is an indirect isometry with no fixed point if and only if F is a glide.

Proof We have seen that a non-trivial glide is indirect and has no fixed point.

Conversely, let F be indirect with no fixed point. Say F : A -> A'. Let H be the rotation by p which swaps A and A'. Then H.F is an indirect isometry which fixes A' and hence is a reflection M_m. Hence F = H^-1.M_m = H.M_m since H has order 2.

To sum up: A direct isometry in E²is a translation (no fixed point) or a rotation (one fixed point). An indirect isometry is a reflection (one fixed line) or a glide (no fixed point).

Table of Contents.

To Continue .

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Last update Dec 3 1999

Author: Phill Schultz, schultz@maths.uwa.edu.au