📦 Translation Transformation

Last Updated: Jan 2026

A Translation Transformation moves a vector or object from one position to another without changing its size or direction.

• Shape remains same
• Orientation remains same
• Only position changes

🗣 Hinglish Tip: Translation = object ko utha ke kahin aur rakh dena, bina ghumaaye

Translation shifts every point of a vector/object by a fixed amount.

• Right / Left → x-direction
• Up / Down → y-direction

Translation in 2D (Coordinate Form)

Formula

If a point (x, y) is translated by (a, b):

$(x',\,y') = (x + a,\,y + b)$

Where:

• a = shift along x-axis
• b = shift along y-axis

Example

Given:

$P(2, 3),   Translation vector (4, -1)$

Resulting point:

$P'(6, 2)$

Translation Using Vector Notation

Formula

$\vec{v}' = \vec{v} + \vec{t}$

Where:

• $\vec{v}$ = original vector
• $\vec{t}$ = translation vector

Example

$\vec{v} =
\begin{bmatrix}
1 \\
2
\end{bmatrix},
\qquad
\vec{t} =
\begin{bmatrix}
3 \\
-1
\end{bmatrix}$

$\vec{v}' =
\begin{bmatrix}
4 \\
1
\end{bmatrix}$

Translation in Matrix Form

Not a Linear Transformation

Translation cannot be represented using a normal 2x2 matrix because:

• It does not pass through origin
• $T(0) \ne 0$

$\color{yellowgreen} \checkmark$ Homogeneous Coordinate Representation

To represent translation in matrix form, we use homogeneous coordinates.

Formula

$\begin{bmatrix}
x' \\
y' \\
1
\end{bmatrix}
=
\begin{bmatrix}
1 & 0 & a \\
0 & 1 & b \\
0 & 0 & 1
\end{bmatrix}
\begin{bmatrix}
x \\
y \\
1
\end{bmatrix}$

Example

Translate point (2,3) by (4, -1)

$\begin{bmatrix}
1 & 0 & 4 \\
0 & 1 & -1 \\
0 & 0 & 1
\end{bmatrix}
\begin{bmatrix}
2 \\
3 \\
1
\end{bmatrix}
=
\begin{bmatrix}
6 \\
2 \\
1
\end{bmatrix}$

Translation in 3D

Formula

$(x', y', z') = (x + a, y + b, z + c)$

Homogeneous Matrix (3D)

$\begin{bmatrix}
1 & 0 & 0 & a \\
0 & 1 & 0 & b \\
0 & 0 & 1 & c \\
0 & 0 & 0 & 1
\end{bmatrix}$

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