TRANSLATION (GEOMETRY)

In Euclidean geometry, a 'translation' is moving every point a constant distance in a specified direction. It is one of the rigid motions (other rigid motions include rotation and reflection). A translation can also be interpreted as the addition of a constant vector to every point, or as shifting the origin of the coordinate system.
If 'v' is a fixed vector, then the translation ''T''_'v' will work as ''T''_'v'('p') = 'p' + 'v'.
If ''T'' is a translation, then the image of a subset ''A'' under the function ''T'' is the 'translate' of ''A'' by ''T''. The translate of ''A'' by ''T''_'v' is often written ''A'' + 'v'.
In an Euclidean space, any translation is an isometry. The set of all translations form the translation group ''T'', which is isomorphic to the space itself, and a normal subgroup of Euclidean group ''E''(''n'' ). The quotient group of ''E''(''n'' ) by ''T'' is isomorphic to the orthogonal group ''O''(''n'' ):
:''E''(''n'' ) ''/ T'' ≅ ''O''(''n'' ).

Contents

Matrix representation

See also

External links

Matrix representation

Since a translation is an affine transformation but not a linear transformation, homogeneous coordinates are normally used to represent the translation operator by a matrix. Thus we write the 3-dimensional vector 'w' = (''w''_''x'', ''w''_''y'', ''w''_''z'') using 4 homogeneous coordinates as 'w' = (''w''_''x'', ''w''_''y'', ''w''_''z'', 1).
To translate an object by a vector 'v', each homogeneous vector 'p' (written in homogeneous coordinates) would need to be multiplied by this 'translation matrix':
: $T\_\{mathbf\{v\}\}\; =$
egin{bmatrix}
1 & 0 & 0 & v_x \
0 & 1 & 0 & v_y \
0 & 0 & 1 & v_z \
0 & 0 & 0 & 1
end{bmatrix}
. !
As shown below, the multiplication will give the expected result:
: $T\_\{mathbf\{v\}\}\; mathbf\{p\}\; =$
egin{bmatrix}
1 & 0 & 0 & v_x \
0 & 1 & 0 & v_y \
0 & 0 & 1 & v_z \
0 & 0 & 0 & 1
end{bmatrix}
egin{bmatrix}
p_x \ p_y \ p_z \ 1
end{bmatrix}
=
egin{bmatrix}
p_x + v_x \ p_y + v_y \ p_z + v_z \ 1
end{bmatrix}
= mathbf{p} + mathbf{v} . !
The inverse of a translation matrix can be obtained by reversing the direction of the vector:
: $T^\{-1\}\_\{mathbf\{v\}\}\; =\; T\_\{-mathbf\{v\}\}\; .\; !$
Similarly, the product of translation matrices is given by adding the vectors:
: $T\_\{mathbf\{u\}\}T\_\{mathbf\{v\}\}\; =\; T\_\{mathbf\{u\}+mathbf\{v\}\}\; .\; !$
Because addition of vectors is commutative, multiplication of translation matrices is therefore also commutative (unlike multiplication of arbitrary matrices).

See also

★ Translation (physics)

★ Translational symmetry

External links

★ Translation Transform at cut-the-knot

★ Geometric Translation (Interactive Animation) at Math Is Fun