3D Transformations

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GEOMETRIA

01 TRANSLATION

Moves an object from one position to another without changing its size or orientation in 3D space.

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Example:
If tx = 3, ty = 2, tz = 5 and a point P(x, y, z) = (1, 2, 3), then P' = (x + 3, y + 2, z + 5) = (4, 4, 8)

[ 1  0  0  tx ]
[ 0  1  0  ty ]
[ 0  0  1  tz ]
[ 0  0  0   1 ]

Resizes an object by scaling it along the x, y, and z axes.

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Example:
If sx = 2, sy = 3, sz = 0.5 and P(x, y, z) = (1, 2, 4), then P' = (2, 6, 2)

[ sx  0   0   0 ]
[ 0  sy   0   0 ]
[ 0   0  sz   0 ]
[ 0   0   0   1 ]

Rotates an object around one of the principal axes (X, Y, or Z).

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Example:
Rotate point P(1, 0, 0) by 90° about Z-axis, then P' = (0, 1, 0)

About X-axis

[ 1    0      0    0 ]
[ 0  cosθ  -sinθ   0 ]
[ 0  sinθ   cosθ   0 ]
[ 0    0      0    1 ]

About Y-axis

[  cosθ   0  sinθ  0 ]
[   0     1    0   0 ]
[ -sinθ   0  cosθ  0 ]
[   0     0    0   1 ]

About Z-axis

[ cosθ  -sinθ  0   0 ]
[ sinθ   cosθ  0   0 ]
[  0      0    1   0 ]
[  0      0    0   1 ]

Mirrors an object across a 3D plane (XY, YZ, or ZX).

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Example:
Reflect point P(4, 2, 3) across YZ-plane (x=0), then P' = (-4, 2, 3)

XY-plane (z=0)

[ 1  0  0  0 ]
[ 0  1  0  0 ]
[ 0  0 -1  0 ]
[ 0  0  0  1 ]

YZ-plane (x=0)

[-1  0  0  0 ]
[ 0  1  0  0 ]
[ 0  0  1  0 ]
[ 0  0  0  1 ]

ZX-plane (y=0)

[ 1  0  0  0 ]
[ 0 -1  0  0 ]
[ 0  0  1  0 ]
[ 0  0  0  1 ]

Slants an object parallel to one of the coordinate planes.

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Example:
Shear in X direction with kxy = 1. Point P(1, 2, 3) becomes P'(3, 2, 3)

Shear X (kxy, kxz)

[ 1  kxy kxz 0 ]
[ 0   1   0  0 ]
[ 0   0   1  0 ]
[ 0   0   0  1 ]

Shear Y (kyx, kyz)

[ 1   0   0  0 ]
[ kyx 1  kyz 0 ]
[ 0   0   1  0 ]
[ 0   0   0  1 ]

Shear Z (kzx, kzy)

[  1   0   0  0 ]
[  0   1   0  0 ]
[ kzx kzy  1  0 ]
[  0   0   0  1 ]