Reflection
Geometry & MeasurementA reflection flips a figure across a line of reflection, producing a mirror image that is congruent to the original.
Formula
\text{Reflection across x-axis: } (x, y) \to (x, -y)
Definition
A reflection flips a shape across a line, called the line of reflection or mirror line. Every point of the shape is flipped to the opposite side of the line, staying the same distance from the line.
Example
Reflecting the point $(3, 4)$ across the x-axis: the x-coordinate stays the same, and the y-coordinate changes sign. $(3, 4)$ becomes $(3, -4)$. Reflecting across the y-axis: $(3, 4)$ becomes $(-3, 4)$.
Key Insight
The reflected image is a mirror image: it looks the same but is flipped. If you write a word on paper and hold it up to a mirror, the mirror shows a reflection. Left and right are swapped, but up and down stay the same.
Definition
A reflection across line $l$ maps each point $P$ to point $P'$ such that $l$ is the perpendicular bisector of $PP'$. Reflecting across the x-axis: $(x,y) \to (x,-y)$. Across the y-axis: $(x,y) \to (-x,y)$. Across $y=x$: $(x,y) \to (y,x)$. Across $y=-x$: $(x,y) \to (-y,-x)$. Reflections are isometries that reverse orientation.
Example
Reflect triangle with vertices $A(1,2)$, $B(4,2)$, $C(3,5)$ across the y-axis: $A'(-1,2)$, $B'(-4,2)$, $C'(-3,5)$. The image is congruent to the original but is a mirror image (orientation reversed). The y-axis is equidistant from each original and reflected point.
Key Insight
Reflections reverse orientation: a clockwise-oriented triangle becomes counterclockwise after reflection. Two reflections compose to a rotation (if the lines intersect) or a translation (if the lines are parallel). This is why every rotation can be decomposed into two reflections.
Definition
Reflection across a line $l$ through the origin with unit direction vector $u = (\cos\theta, \sin\theta)$ is given by the matrix $M = 2uu^T - I = \begin{bmatrix}\cos(2\theta) & \sin(2\theta)\\\sin(2\theta) & -\cos(2\theta)\end{bmatrix}$. Reflections are involutions ($M^2 = I$) with determinant $-1$, distinguishing them from rotations ($\det = +1$). Every isometry of $\mathbb{R}^2$ is a composition of at most $3$ reflections.
Example
Composing reflections across lines at angle $\theta$ apart: the result is a rotation by $2\theta$ about their intersection. Reflecting across x-axis then y-axis (angle $\pi/2$ apart) gives rotation by $\pi$ ($180$ degrees), consistent with the composition $\begin{bmatrix}1&0\\0&-1\end{bmatrix} \times \begin{bmatrix}-1&0\\0&1\end{bmatrix} = \begin{bmatrix}-1&0\\0&-1\end{bmatrix}$.
Key Insight
The decomposition of every isometry into reflections is the key to understanding the symmetry group structure. The minimal number of reflections needed ($1$, $2$, or $3$) classifies the isometry: $1$ for reflections, $2$ for rotations and translations, $3$ for glide reflections.