2D Translation

Theory of Computation & Computer Graphics — Two-Dimensional Transformation, NEC licence examination syllabus (Nepal Engineering Council).

2D Translation: sliding a shape without changing it at all

The simplest transformation — and the reason every 2D transformation uses a slightly odd-looking 3×3 matrix instead of a 2×2 one.

Translation moves every point of an object by the same fixed offset (tx, ty) — the shape doesn't rotate, resize, or distort, it just moves.

Translation matrix (homogeneous coordinates): [x'] [1 0 tx] [x] [y'] = [0 1 ty] [y] [1 ] [0 0 1 ] [1] Worked example: translate point (2,3) by tx=5, ty=4 x' = 1(2) + 0(3) + 5(1) = 7 y' = 0(2) + 1(3) + 4(1) = 7 New point: (7, 7)

x' = x + tx, y' = y + ty With tx = 2, ty = 1, x' = 4, y' = 1.

Set tx and ty to 0 and the solid shape lands exactly on the dashed one — that is the identity, and it is why translation needs the 3×3 matrix.

Why the extra "1" row/column (homogeneous coordinates)? Plain 2×2 matrices can only represent rotation/scaling — they can't add a constant offset through matrix multiplication alone. Homogeneous coordinates add a third dimension purely as a trick to make translation ALSO expressible as one matrix multiplication, so all transformations can be combined uniformly.

💡 Practice translating a full triangle (3 points) by a given offset — a very standard first-numerical format in this topic.

What the matrix is actually doing

Translation is the odd one out. Rotation, scaling and shear all multiply coordinates, and multiplication can be written as a 2×2 matrix. Translation adds, and addition cannot — which is the entire reason for homogeneous coordinates.

By appending a 1 to every point, the added constants tx and ty sit in the third column and get multiplied by that 1. Addition is disguised as multiplication, and now all four transformations are matrices — so they can be composed by multiplying them together. That single trick is what makes the whole graphics pipeline possible.

💡 Exam trap: translation is the only one of the four that does not have the origin as a fixed point. Under rotation, scaling and shear the origin never moves; under translation everything moves, origin included.

Properties worth stating

Translation preserves everything about the shape — lengths, angles, areas and orientation. It is a rigid transformation. The inverse is simply translation by (−tx, −ty), which is worth remembering because inverse transformations appear constantly in composite problems.

Syllabus points

Translation matrix; homogeneous coordinates

Create a free account to tick topics off, take notes as you read, watch the video lessons and get a day-by-day study plan built around your exam date.