EUCLIDEAN TRANSFORMATIONS

Geometric transformations of the Euclidean space

Regular transformation is a one-to-one mapping of the set M on the set M', in which any point in the set M
( original X Î M ) is attached the only one element of the set M' ( view - image X' Î M' ) due to a given rule (obr. 1.4).
The following relations are valid:

1. If X Y, then X' Y '

Two different points from the set M are mapped
to the two different points in the set M'.

2. If X' Y', then X Y

Two different points from the set M' are images
of two different points in the set M.

If the two sets M, M' are geometric spaces, with points as elements, regular transformations defined on the sets of points in these spaces are called geometric transformations. If M = M', we get the geometric transformation in the space. Image of a geometric figure in a given geometric transformation is a set of images of all figure points. Geometric figures in the space can be mapped to other geometric figures of the same space. Properties of geometric figures, which are preserved under the given geometric transformation are called invariant properties of the given transformation. Geometric transformations preserve incidence as the invariant property. Figures, which are mapped to themselves, are called invariant figures. If all points of the figure are invariant under the given transformation, the figure is called point invariant. Geometric transformation will be denoted as linear, if it satisfies the following property: the image of a line is a line and the image of a plane is a plane. Geometric linear transformations of the space are analytically represented by a regular square matrix, called transformation matrix T with real elements (in the 3 dimensional space it is of rank 4). Coordinates of the image U' of a geometric figure U can be derived from the coordinates of the original figure U points multiplying by the transformation matrix U' = U . T

Geometric transformations can be composed, while the composition of two geometric trasformations T and T' is a new geometric transformation. Its analytic representation is a product of the multiplication of analytic representations of the two composed transformations. Composition of transformations is a non-commutative operation on the set of all transformations (obr. 1.5), it means, it depends on the order in which the transformations are composed

T . T' T' . T

1.4 Euclidean (metric) transformations

A set of Euclidean (metric) transformations called congruences is defined on the set of points in the Euclidean space. Incidence and metric (the length of line segments) are invariants of the metric transformations.

These are: identity, symmetry - reflection in a plane, symmetry - reflection in a line, symmetry - reflection in a point, revolution about an axis, translation, translated reflection in a plane, translated re-flection in a line, revolved reflection in a plane, helical movement.

Some of the metric transformations are studied at the secondary schools, for instance transformations with invariant planes of the space (points in any plane are mapped into the points of the same plane) - symmetry with respect to a point, translation. Symmetry with respect to a line and symmetry with respect to a plane are analogous space transformations to the symmetry with respect to a line as a transformation in the Euclidean plane.

Revolution about an axis is a transformation of the space defined by an axis of revolution o and a direction angle of revolution (rotation angle) j (obr. 1. 6). Point A not located on the axis o of revolution revolves to the point A° according to the following rule:

1. | o A | = | o A°| = | AS^A | = | A° S^A | = r^A, r^A is the radius of revolution

S^A = h Ç o, A Î h ^ o, S^A = h° Ç o, A° Î h° ^ o

Lines h and h° determine the plane a perpendicular to the axis of revolution o, called plane of revolution of the point A.

2. | hh° | = j

Points A, A° are located on the circle k^A (circle of revolution) with the centre S^A and radius r^A.

All points on the axis of revolution are invariant. Axis of revolution is a point invariant line. Positive angle of revolution j is measured in the anticlockwise direction.

Helical movement is transformation composed from revolution about a line o (axis of the helical movement) by the oriented angle j and translation determined by the translation vector parallel to the axis of the helical movement (obr 1.7).
Point A not located on the axis o of the helical movement is transformed to the point A^S according to the following rule:

1. A is revolved by angle j about the axis o to the point A°

2. | o A | = | o A° | = | oA^S | = r^A

2. A° is translated to the point A^S, a = A°A^S.

Points A, A^S are located on the helix with the axis in the axis o of the helical movement and radius r^A. All points on the axis of the helical movement are only translated by the vector a, they are mapped to the other points on the axis of the helical movement.

Axis of the helical movement is an invariant line, but not a point invariant line.

Image of a geometric figure in the metrickej transformation is a geometric figure congruent to its original. relation between coordinates of points of the original and the image in the given transformation are represeneted by the system of transformation equations. Matrix of a linear transformation is a matrix of this system of equations.