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In Euclidean geometry, translation is a 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. Each translation is an isometry.
Matrix representation
A translation cannot be accomplished using a 3-by-3 matrix, so homogeneous coordinates are normally used.
To translate an object by a vector v = (vx, vy, vz), each homogeneous vector p = (px, py, pz, 1) would need to be multiplied with this translation matrix:
- <math> T_v =
\begin{bmatrix}
1 & 0 & 0 & v_x \\
0 & 1 & 0 & v_y \\
0 & 0 & 1 & v_z \\
0 & 0 & 0 & 1
\end{bmatrix}
<math>
As shown below, the multiplication will give the expected result:
- <math>
T_v 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}
<math>
The inverse of a translation matrix can be obtained by negating the vector:
- <math>T^{-1}_v = T_{-v}<math>
See also
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