📦 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)
ComponentCalculationResult
x2 + 46
y3 − 12

Resulting point:

P'(6, 2)

Translation Using Vector Notation

Formula

v⃗′ = v⃗ + t⃗

Where:

  • v⃗ = original vector
  • t⃗ = translation vector

Example

v⃗ =
[ 1
  2 ],

t⃗ =
[ 3
 −1 ]

v⃗′ =
[ 4
  1 ]

Translation in Matrix Form

Not a Linear Transformation

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

  • It does not pass through origin
  • T(0) ≠ 0

✔ Homogeneous Coordinate Representation

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

Formula

[ x′
  y′
  1 ] =
[ 1  0  a
  0  1  b
  0  0  1 ]
[ x
  y
  1 ]

Example

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

[ 1  0  4
  0  1 −1
  0  0  1 ]
[ 2
  3
  1 ]
=
[ 6
  2
  1 ]

Translation in 3D

Formula

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

Homogeneous Matrix (3D)

[ 1  0  0  a
  0  1  0  b
  0  0  1  c
  0  0  0  1 ]
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