Translation (geometry) Article Index for
Translation 		Website Links For
Translation 		 

Information About

Translation (geometry)
APPAREL
BABY
BEAUTY
BOOKS
CAR TOYS
CELL PHONES
DVD'S
ELECTRONICS
GOURMET FOOD
GROCERIES
HEALTH & PERSONAL
HOME & GARDEN
JEWELRY
MUSIC
MUSIC INSTRUMENTS
OFFICE PRODUCTS
SOFTWARE
SPORTING GOODS
TOOLS & HARDWARE
TOYS
VIDEO GAMES
SHOPPING HOME
MORE SHOPPING...



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 Tv is often written A + v.

Each 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



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 = (wx, wy, wz) using 4 homogeneous coordinates as w = (wx, wy, wz, 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}} =
\begin{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} =
\begin{bmatrix}
1 & 0 & 0 & v_x \
0 & 1 & 0 & v_y \
0 & 0 & 1 & v_z \
0 & 0 & 0 & 1
\end{bmatrix}
\begin{bmatrix}
p_x \ p_y \ p_z \ 1
\end{bmatrix}
=
\begin{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




EXTERNAL LINKS