3D Scaling

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

3D Scaling

Same idea as 2D, now with a z-factor too.

3D scaling matrix: [x'] [sx 0 0 0] [x] [y'] = [0 sy 0 0] [y] [z'] [0 0 sz 0] [z] [1 ] [0 0 0 1] [1] Worked example: scale (2,3,4) by (sx=2, sy=1, sz=3) x'=4, y'=3, z'=12 New point: (4, 3, 12)

x' = x·sx, y' = y·sy, z' = z·sz With sx = 1.5, sz = 1.5, x' = 3, vol = 2.25.

Set sx = sz = 2 and the volume multiplies by 4 here — with all three factors doubled it would be 8, not 2.

💡 One-liner: "What does sx=sy=sz produce?" → Uniform scaling — the object grows/shrinks proportionally in all directions, keeping its original shape.

What the three factors do

Each factor stretches along its own axis, and as in 2D, scaling about the origin also moves the object — every coordinate is multiplied, so distant points travel further than near ones.

The determinant is sx·sy·sz, and that is the factor by which volume multiplies. Double all three and the volume goes up eightfold, not twofold — the classic exam question, and the reason a "double the size" instruction is ambiguous until someone says whether they mean each dimension or the volume.

💡 Uniform scaling (sx = sy = sz) preserves shape and angles. Any mismatch distorts: a cube becomes a cuboid and a sphere becomes an ellipsoid. A zero factor collapses the solid into a flat plane and has no inverse — that information is gone for good.

Scaling about a fixed point

To scale an object in place rather than about the origin, use the same three-step sandwich as everywhere else: translate the fixed point to the origin, scale, translate back. The next topic works this through in full.

Syllabus points

3D scaling matrix

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