2D Reflection

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

2D Reflection: flipping a shape like a mirror image

A special case of negative scaling.

🪞 Reflection Matrices

About x-axis(x,y) → (x, −y) — flips vertically.

About y-axis(x,y) → (−x, y) — flips horizontally.

About origin(x,y) → (−x, −y) — flips both ways (equivalent to 180° rotation).

About line y=x(x,y) → (y, x) — swaps coordinates.

💡 One-liner: "Is reflection about the origin the same as a 180° rotation?" → Yes — both send (x,y) to (−x,−y), producing an identical result.

Reflection is negative scaling

Every reflection in the table above is a scaling with a negative factor. Reflecting about the x-axis is scaling by (1, −1); about the y-axis, (−1, 1); about the origin, (−1, −1). Recognising this means there is one matrix to remember rather than four.

Reflection about the line y = x is the exception to that pattern — it swaps the coordinates rather than negating one, giving a matrix with zeros on the diagonal and ones off it. It is the same matrix you would use to transpose a 2×2 matrix, which is a useful way to remember it.

💡 Reflection reverses orientation: a shape labelled clockwise becomes labelled anticlockwise, which is why text in a mirror is unreadable. Its determinant is −1 — the magnitude 1 says area is preserved, and the minus sign says the shape has been flipped. Rotation, by contrast, has determinant +1 and can never produce a mirror image however far you turn it.

The exam's favourite equivalence

Reflection about the origin and rotation by 180° both map (x, y) to (−x, −y), so they are the same transformation. But this is a coincidence of 180°, not a general rule — no other rotation equals a reflection, precisely because their determinants (+1 and −1) differ. A question asking whether a given matrix is a rotation or a reflection is answered by taking the determinant.

Syllabus points

Reflection about axes and lines

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.