3D Translation

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

3D Translation: the same idea, one dimension deeper

Everything from 2D transformation returns here — just with a z-coordinate added everywhere.

3D translation matrix (homogeneous, 4×4): [x'] [1 0 0 tx] [x] [y'] = [0 1 0 ty] [y] [z'] [0 0 1 tz] [z] [1 ] [0 0 0 1 ] [1] Worked example: translate (1,2,3) by (tx=4, ty=1, tz=2) x'=1+4=5, y'=2+1=3, z'=3+2=5 New point: (5, 3, 5)

x' = x + tx, y' = y + ty, z' = z + tz With tx = 2, tz = 0, x' = 3, z' = 3.

Set both to 0 and the solid cube lands exactly on the dashed one — the identity again.

💡 Notice the pattern: 3D matrices are always 4×4 (vs 2D's 3×3) — one extra row/column for the z-axis, plus the same homogeneous "1" trick from 2D.

Why 4×4 and not 3×3

The reasoning is identical to 2D, one dimension along. Rotation, scaling and shear all multiply coordinates and can be written as a 3×3 matrix in 3D. Translation adds, and addition cannot be expressed as matrix multiplication — so a fourth homogeneous coordinate is appended, the offsets sit in the fourth column, and they get multiplied by that 1.

The payoff is the same too: once translation is a matrix like the others, an entire chain of transformations collapses into one matrix that can be applied to thousands of vertices. That is what makes real-time 3D graphics possible at all.

💡 Exam trap: translation is still the only transformation here that moves the origin. Rotation, scaling and shear all leave (0,0,0) exactly where it is — which is precisely why transforming about any other point needs the translate-do-translate-back sandwich.

Properties worth stating

Translation preserves lengths, angles, areas, volumes and orientation — a rigid transformation. Its inverse is translation by (−tx, −ty, −tz).

Syllabus points

3D translation matrix (homogeneous)

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