Translation (geometry)

Encyclopedia : T : TR : TRA : Translation (geometry)

---

In Euclidean geometry, a translation, or translation operator, is an affine transformation of Euclidean space which moves every point by a fixed distance in the same direction. It can also be interpreted as the addition of a constant vector to every point, or as shifting the origin of the coordinate system. In other words, 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.

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(n ) / T ≅ O(n ).

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_} = \begin1 & 0 & 0 & v_x \\0 & 1 & 0 & v_y \\0 & 0 & 1 & v_z \\0 & 0 & 0 & 1 \end. \! ]

As shown below, the multiplication will give the expected result:

[ T_} \mathbf =\begin1 & 0 & 0 & v_x \\0 & 1 & 0 & v_y \\0 & 0 & 1 & v_z \\0 & 0 & 0 & 1\end\beginp_x \\ p_y \\ p_z \\ 1\end=\beginp_x + v_x \\ p_y + v_y \\ p_z + v_z \\ 1\end= \mathbf + \mathbf . \! ]

The inverse of a translation matrix can be obtained by reversing the direction of the vector:

[ T^_} = T_} . \! ]

Similarly, the product of translation matrices is given by adding the vectors:

[ T_}T_} = T_+\mathbf} . \! ]

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

 

---

From Wikipedia, the Free Encyclopedia. Original article here. Support Wikipedia by contributing or donating.
All text is available under the terms of the GNU Free Documentation License See Wikipedia Copyrights for details.

Search Titles

0

1

2

3

4

5

6

7

8

9

A

B

C

D

E

F

G

H

I

J

K

L

M

N

O

P

Q

R

S

T

U

V

W

X

Y

Z

?