3D Reflection

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

3D Reflection: mirroring across a plane instead of a line

2D reflected across lines. 3D reflects across entire planes.

🪞 Reflection about Coordinate Planes

about the yz-plane: (x, y, z) → (−x, y, z) With sx = -1, sz = 1, x' = -2, det = -1.

Set sx = −1 and watch the determinant become −1: that negative sign is what says the orientation has been reversed.

About XY plane(x,y,z) → (x, y, −z) — flips the z-coordinate.

About YZ plane(x,y,z) → (−x, y, z) — flips the x-coordinate.

About XZ plane(x,y,z) → (x, −y, z) — flips the y-coordinate.

💡 One-liner: "How many coordinates change sign when reflecting about a coordinate plane?" → Exactly one — the coordinate perpendicular to that plane.

Why exactly one coordinate flips

Reflecting about a coordinate plane negates the coordinate perpendicular to that plane and leaves the other two alone — because the plane is precisely the set of points where that coordinate is zero, and those points must not move.

So reflection about the xy-plane negates z, about the yz-plane negates x, and about the xz-plane negates y. As in 2D, each is just a scaling with one negative factor.

The determinant is −1: magnitude 1 says volume is preserved, and the minus sign says orientation is reversed. This is what makes a reflected object genuinely different from a rotated one — a reflected right hand becomes a left hand, and no amount of rotating will ever turn one into the other.

💡 That determinant test answers a common exam question in one step: given a matrix, is it a rotation or a reflection? Determinant +1 means rotation, −1 means reflection. No need to examine the entries at all.

Two reflections make a rotation

Each coordinate-plane reflection negates exactly one coordinate and has determinant −1. What happens when two are applied together is worth knowing.

Reflect about the yz-plane: (x,y,z) → (−x, y, z) det = −1 Then about the xz-plane: (−x,y,z) → (−x,−y, z) det = −1 Combined: (x,y,z) → (−x,−y, z) Determinant = (−1) × (−1) = +1 → this is a ROTATION

Two reflections restore the orientation, so the result is a rotation — in this case by 180° about the z-axis. That is why determinant is the reliable test: +1 means rotation, −1 means reflection, however the transformation was built up.

💡 The practical consequence: a reflected object can never be rotated back into its original. A left hand cannot be turned into a right hand by any amount of rotating, because rotation preserves determinant +1 and no sequence of rotations will ever produce −1.

Syllabus points

Reflection about coordinate planes

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