1 Transformations of the Euclidean space

In the three-dimensional extended Euclidean space E3(R) with the defined homogeneous coordinate system, a group of Euclidean metric transformations can be determined on the subset of all real points E3(R). Any real point A=(x,y,z,1) in the subspace can be attached its image A'=(x',y',z',1), while the relation of coordinates can be expressed in the following way

A ' = A . T M

where TM is the regular square matrix of rank 4.

We recognize these metric transformations of the space: identity, symmetry about a point - central symmetry, symmetry about a line - axial symmetry, symmetry about a plane - plane reflection, revolution about a line, translation, translated symmetry about a line, translated symmetry about a line or a plane, revolved symmetry about a plane, helical movement.

1.1 Identity

Equations and matrix of identity

x ' = x
y ' = y
z ' = z
I = ( 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 )

1.2 Symmetry about a point

Let the centre of symmetry be in the origin O, then

tran8
Figure 1: Symmetry about origin

Equation and matrix of central symmetry about O

x ' = x
y ' = y
z ' = z
T M = ( 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 )

1.3 Symmetry about a line

Let the axis of symmetry be in the coordinate axis x, then

tran17
Figure 2: Symmetry about axis x

Equations and matrix of the axial symmetry about x

x ' = x
y ' = y
z ' = z
T M = ( 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 )

Let the axis of symmetry be in the coordinate axis y, then

tran18
Figure 3: Symmetry about axis y

Equations and matrix of the axial symmetry about y

x ' = x
y ' = y
z ' = z
T M = ( 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 )

Let the axis of symmetry be in the coordinate axis z, then

tran9
Figure 4: Symmetry about axis z

Equations and matrix of the axial symmetry about z

x ' = x
y ' = y
z ' = z
T M = ( 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 )

1.4 Symmetry about a plane

Let the plane of symmetry be in the coordinate plane π=xy, then

tran10
Figure 5: Symmetry about plane xy

Equations and matrix of the symmetry about plane π=xy

x ' = x
y ' = y
z ' = z
T M = ( 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 )

Let the plane of symmetry be in the coordinate plane v=xz, then

tran19
Figure 6: Symmetry about plane xz

Equations and matrix of the symmetry about plane v=xz

x ' = x
y ' = y
z ' = z
T M = ( 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 )

Let the plane of symmetry be in the coordinate plane μ=yz, then

tran20
Figure 7: Symmetry about plane yz

Equations and matrix of the symmetry about plane μ=yz

x ' = x
y ' = y
z ' = z
T M = ( 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 )

1.5 Revolution about a line

Let the axis of revolution be in the coordinate axis z, and angle of revolution be ϕ, then

tran11
Figure 8: Revolution about axis z

Equations and matrix of the revolution about axis z

x ' = x cos ϕ y sin ϕ
y ' = x sin ϕ + y cos ϕ
z ' = z
T M = ( cos ϕ sin ϕ 0 0 sin ϕ cos ϕ 0 0 0 0 1 0 0 0 0 1 )

Let the axis of revolution be in the coordinate axis y, and angle of revolution be ϕ, then

tran22
Figure 9: Revolution about axis y

Equations and matrix of the revolution about axis y

x ' = x cos ϕ + z sin ϕ
y ' = y
z ' = x sin ϕ + z cos ϕ
T M = ( cos ϕ 0 sin ϕ 0 0 1 0 0 sin ϕ 0 cos ϕ 0 0 0 0 1 )

Let the axis of revolution be in the coordinate axis x, and angle of revolution be ϕ, then

tran21
Figure 10: Revolution about axis x

Equations and matrix of the revolution about axis x

x ' = x
y ' = y cos ϕ z sin ϕ
z ' = y sin ϕ + z cos ϕ
T M = ( 1 0 0 0 0 cos ϕ sin ϕ 0 0 sin ϕ cos ϕ 0 0 0 0 1 )

1.6 Translation

Let the translation be given by vector a =(m,n,l,0), then

tran12
Figure 11: Translation

Equations and matrix of the translation

x ' = x + m
y ' = y + n
z ' = z + l
T M = ( 1 0 0 0 0 1 0 0 0 0 1 0 m n l 1 )