Dilations, Reflections, and Other Transformations

Part 10, Chapter 10: Vectors and Operations on Points

Learning objectives

Apply dilation (scalar multiplication) to a point in the plane
Compute reflections of points across the axes and through the origin
Understand the geometric effect of dilation and reflection
Combine dilation and reflection operations

Coordinates let us move points around with algebra. A dilation rescales a figure by a constant factor; a reflection flips it across an axis. Both look like simple coordinate operations on the surface, but together they generate every shape transformation you will see in geometry, physics, and computer graphics. This section formalises the two operations and shows that they sit at the foundation of linear maps, a topic that runs through the rest of the textbook.

Dilation, scaling toward or away from the origin

For a scalar $c \in \mathbb{R}$ and a point $P = (x, y)$ , the dilation of $P$ by factor $c$ is

$D_c(P) = (cx, cy).$ c(P)=(cx,cy).

The point slides along the ray from the origin through $P$ . When $c > 1$ it moves farther away (expansion); when $0 < c < 1$ it moves closer (contraction); when $c = 0$ every point collapses to the origin. When $c < 0$ the point flips through the origin and then scales.

Where this shows up

Computer Graphics: Scaling a 3D model uniformly is a dilation $\vec{v} \mapsto c\vec{v}$ ; reflecting it for a mirror surface is a reflection across the mirror plane. Both are matrix-vector products, performed millions of times per frame.
Physics: Doubling the velocity of a particle is a dilation; reversing it for an elastic collision is a reflection. Conservation of momentum constrains which compositions of dilation and reflection are physically allowed.
Photography: Zoom is a dilation of the visible field; horizontal flip (for selfies) is a reflection. The image-editing operations in every photo app are exactly the maps you are studying here, composed.

Drag $P$ to position it. Slide the scale factor $c$ . The orange point $cP$ moves along the ray from the origin; the distance from the origin scales by $|c|$ .

Three reflections

A reflection flips a point across a line (an "axis of reflection") or through a point. The three reflections most useful in coordinate geometry are:

Reflection across the $x$ -axis: $(x, y) \mapsto (x, -y)$ , only the $y$ -coordinate flips sign.
Reflection across the $y$ -axis: $(x, y) \mapsto (-x, y)$ , only the $x$ -coordinate flips sign.
Reflection through the origin: $(x, y) \mapsto (-x, -y)$ , both flip. This is the same as $D_{-1}$ −1.

An important identity: reflecting across the $x$ -axis and then across the $y$ -axis gives the same result as reflecting through the origin, which equals dilation by $-1$ .

What dilations and reflections preserve

Both operations preserve the geometric structure in different ways:

Dilation by $c \neq 0$ scales every distance from the origin by $|c|$ ; preserves angles between lines; takes lines through the origin back into themselves.
Reflection preserves all distances (it is an isometry); preserves angles in magnitude but reverses orientation.
Both preserve collinearity: if three points lie on a common line, their images do too.

These invariance properties are why dilations and reflections appear in classifications of "rigid" and "similar" figures.

Try it

Predict first: dilating $P = (2, -5)$ by $c = 3$ from the origin lands where? Place $P$ in the widget, set $c = 3$ , and verify the image is $(6, -15)$ .
Before adjusting: a dilation by $c = -1$ should send $(2, -5)$ where? Set $c = -1$ and verify the point moves to $(-2, 5)$ , reflected through the origin.
Predict first: with $c = 1/2$ , where should $P = (2, -5)$ contract to? Set the slider and verify the image is $(1, -2.5)$ , half the original distance from the origin.

A trap to watch for

The most common slip is reflecting across the wrong axis. "Reflect across the $x$ -axis" means the $x$ -axis is the mirror; the $y$ -coordinate is what changes sign, not the $x$ -coordinate. Likewise, "reflect across $y$ -axis" flips $x$ , not $y$ . The mnemonic: the coordinate opposite to the axis you are reflecting across is the one that flips. Reflecting $(3, 5)$ across the $x$ -axis gives $(3, -5)$ , not $(-3, 5)$ . If you ever produce $(-3, -5)$ from a single reflection, you have actually done two reflections (a rotation by $180^\circ$ ).

What you now know

You can dilate a point by any scalar, reflect it across either coordinate axis or through the origin, and read off which geometric invariants survive each operation. The next section combines these with a third operation, vector addition, to show that points in the plane satisfy a complete algebra known as a vector space.

Quick check

Mark section complete →

References

Lang, S. (1971). Basic Mathematics. Springer. Chapter 9, §1, dilations and reflections of points in the plane.
Coxeter, H. S. M. (1969). Introduction to Geometry, 2nd ed. Wiley. §3 covers isometries and similarities, the broader framework.
Artin, M. (2011). Algebra, 2nd ed. Pearson. Chapter 4 introduces linear transformations as a generalisation of dilation.
Stewart, J. (2015). Calculus, 8th ed. Cengage. Appendix B reviews reflections and scalings in the coordinate plane.