Transformations

A transformation moves or changes a figure according to a rule.

• Pre-image: the original figure (points $A$, $B$, $C$).
• Image: the new figure after the transformation (points $A'$, $B'$, $C'$, read “$A$ prime”).
• Mapping: the pairing of each pre-image point with its image point. We write $A \to A'$.
• Rigid motion (also called an isometry): a transformation that keeps the same size and shape. Translations, reflections, and rotations are rigid.
• Non-rigid transformation: changes the size. A dilation is non-rigid.

A translation slides every point the same distance in the same direction. If you move $a$ units horizontally and $b$ units vertically:

The pair $\langle a,b\rangle$ is the translation vector. Positive $a$ is right, negative is left; positive $b$ is up, negative is down.

A translation slides.

A reflection flips a figure over a line of reflection, producing a mirror image. Coordinate rules:

A reflection flips across a line.

A rotation turns a figure about a fixed point (the center). Rotating counterclockwise about the origin:

A $90^\circ$ clockwise turn equals a $270^\circ$ counterclockwise turn (and vice versa). A $180^\circ$ turn is the same either direction.

A rotation turns about a point.

A composition applies one transformation and then another, in order. Do the first transformation, write the image, then apply the second transformation to that image.

A dilation centered at the origin multiplies every coordinate by the same scale factor $k$:

• $k>1$: an enlargement (the figure grows).
• $0<k<1$: a reduction (the figure shrinks).
• $k=1$: the figure is unchanged.

A dilation is not a rigid motion, but it keeps the same shape, so the image is similar to the pre-image.

Line (reflectional) symmetry: a figure has line symmetry if a reflection over some line maps the figure exactly onto itself. That line is a line of symmetry.

Rotational symmetry: a figure has rotational symmetry if a rotation of less than $360^\circ$ about its center maps it onto itself. The order is how many times it matches in one full turn; the angle of rotation is $360^\circ \div \text{order}$.

• Rigid motions (translations, reflections, rotations, and their compositions) preserve size and shape. The image is congruent to the pre-image.
• Dilations (scale factor $k\ne 1$) preserve shape but change size. The image is similar, but not congruent, to the pre-image.

Two figures are congruent if a sequence of rigid motions maps one onto the other. They are similar if a sequence of rigid motions and dilations does so.

Reflecting twice in a row is always a single rigid motion. Which one depends on the two mirror lines:

• Parallel mirrors $\to$ a translation. The shift is perpendicular to the mirrors, in the direction from the first mirror to the second, and its distance is $2d$, where $d$ is the gap between the mirrors.
• Intersecting mirrors $\to$ a rotation about the point where they cross. If the angle from the first mirror to the second is $\theta$, the rotation angle is $2\theta$ (same turning direction).

Either way, two flips restore the original orientation, so the result is an “even” (direct) isometry.

A reflection flips across a line.

Rotation about a center $(a,b)$: shift the center to the origin, rotate, then shift back.

For a $90^\circ$ counterclockwise turn about $(a,b)$ this collapses to

Reflection over a general line $y=mx+c$: it still sends each point to the opposite side at equal perpendicular distance. A clean way is to compose known moves --- translate so the line passes through the origin, rotate the line onto the $x$-axis, reflect over the $x$-axis, then undo the rotation and translation.

Every rotation, reflection, and dilation about the origin can be written as multiplication by a $2\times 2$ matrix acting on the column vector $\begin{bmatrix}x\\y\end{bmatrix}$:

For example, the $90^\circ$ rotation of $(3,1)$ is

matching $(x,y)\to(-y,x)$. A translation is not linear, so it needs a separate added vector $\begin{bmatrix}a\\ b\end{bmatrix}$ (or a $3\times 3$ “homogeneous” matrix you will meet later). Composing transformations then corresponds to multiplying their matrices.

A glide reflection is a reflection over a line followed by a translation parallel to that line (the order does not matter). It is the fourth basic rigid motion of the plane, alongside translations, rotations, and reflections.

• It reverses orientation (like a single reflection), so it is an “odd” isometry.
• It has no fixed points and no fixed line pointwise --- the classic footprint pattern of a walking person is a glide reflection.

Here $T$ is reflected over the $x$-axis and slid $2$ units right to land on $T'$.

The set of all rigid motions that map a figure onto itself forms its symmetry group.

• Cyclic group $C_n$: only rotational symmetry --- $n$ rotations (including the $360^\circ$ “do nothing”), and no lines of symmetry. A pinwheel or “S” shape is like this.
• Dihedral group $D_n$: rotational and reflective symmetry --- $n$ rotations plus $n$ reflections, giving $2n$ symmetries in all.

A regular $n$-gon has symmetry group $D_n$: order-$n$ rotational symmetry (smallest angle $360^\circ/n$) and $n$ lines of symmetry, for $2n$ total symmetries. An equilateral triangle is $D_3$ ($6$ symmetries); a square is $D_4$ ($8$ symmetries).