Standard Plane Mappings: Translations, Rotations, Reflections

Part 8, Chapter 8: Plane Isometries, Rigid Motions

Learning objectives

Define translation, rotation, and reflection as maps of the plane and apply them to specific points
Use the rotation matrix to rotate a point about the origin by a given angle
Reflect a point across the x-axis, y-axis, and the line y = x
Recognise these mappings as the building blocks of rigid motion in plane geometry

Geometry stops feeling like a list of theorems and starts feeling like a language the moment you can move shapes around. A triangle sitting in one place is just a triangle; the same triangle slid, spun, or flipped is the same triangle wearing a different costume. The three motions in this section, translation, rotation, reflection, are the verbs of that language, and every congruence theorem you ever learned is really a statement about which combinations of these verbs map one figure onto another.

Translation: slide everything by the same vector

A translation by a vector $v = (a, b)$ moves every point of the plane by the same displacement:

$T_v(x, y) = (x + a,\; y + b).$ v(x,y)=(x+a,;y+b).

No point is special; the whole plane shifts as a rigid sheet. Geometrically, you pick up the figure and slide it. The shape, size, and orientation are untouched.

Rotation: pivot around a point

A rotation $R_{\theta, C}$ theta,C pins down one point (the centre $C$ ) and spins everything else around it through angle $\theta$ , counterclockwise being the positive convention in mathematics. For a rotation about the origin,

$R_\theta(x, y) = (x \cos\theta - y \sin\theta,\; x \sin\theta + y \cos\theta).$ theta(x,y)=(xcostheta−ysintheta,;xsintheta+ycostheta).

The formula looks busy but is forced by one requirement: the point $(1, 0)$ must land on $(\cos\theta, \sin\theta)$ , and rotations are linear, so the rest follows.

Reflection: flip across a mirror

A reflection across a line $\ell$ sends each point $P$ to the point $P'$ that is the same distance from $\ell$ on the opposite side. Three workhorse cases:

$\sigma_{x\text{-axis}}(x, y) = (x, -y), \qquad \sigma_{y\text{-axis}}(x, y) = (-x, y), \qquad \sigma_{y=x}(x, y) = (y, x).$ xtext−axis(x,y)=(x,−y),qquadsigmaytext−axis(x,y)=(−x,y),qquadsigmay=x(x,y)=(y,x).

Reflections do something the other two motions cannot: they flip handedness. Trace an “F” on a piece of paper, hold it up to a mirror, and the cross-strokes now point the wrong way. That orientation flip will be the central diagnostic in §6.5.

Where this shows up

Computer Graphics: Every 3D model is positioned in a scene via a translation followed by a rotation (and sometimes a reflection); the same three primitive maps you are learning here are the building blocks of the GPU's transformation pipeline.
Robotics: Robot-arm kinematics is a chain of rotations about each joint axis plus a translation along each link; the entire end-effector position is a composition of these elementary maps.
Crystallography: Crystal lattices are classified by which translations, rotations, and reflections leave them invariant, the 230 space groups are exactly the distinct ways these three operations can combine in 3D.

Drag the handle to set a translation vector, a rotation angle and centre, or a mirror line. The faded F is the original; the solid F is the image. The dashed lines between corresponding vertices confirm that distances are preserved.

Try it

Before dragging: a translation by $(3, 4)$ moves the image how far from the original? Set translation mode and drag the vector tip to $(3, 4)$ to verify it lands $5$ units away by Pythagoras.
Before rotating: under a $90^\circ$ rotation about the origin, where does the point $(1, 0)$ go? Switch to rotation mode, set the centre at the origin, rotate by $90^\circ$ , and check each corner of the F.
Predict first: under reflection across the $y$ -axis, which way should the F face? Switch to reflection mode, make the mirror the $y$ -axis, and confirm the orientation flips.
Compute $R_{180^\circ}$ applied to $(3, -2)$ both algebraically (using the formula) and geometrically (visualise rotating through a half-turn). They should agree.

Pause: a translation has direction (the vector). A rotation has direction (the sign of the angle). A reflection has, what, exactly? Why is the “direction” question different for reflections than for the other two?

A trap to watch for

Beginners often write “rotate by $90^\circ$ ” without saying about what centre. That sentence is incomplete. Rotating the point $(3, 0)$ by $90^\circ$ about the origin gives $(0, 3)$ ; rotating the same point by $90^\circ$ about $(1, 0)$ gives $(1, 2)$ ; rotating about $(3, 0)$ itself leaves it fixed. A rotation is always specified by angle and centre together. A related trap: software graphics packages use clockwise as positive (because the $y$ -axis points down on a screen). Mathematics uses counterclockwise. If your widget rotation looks backwards from your hand-calculation, this is usually the reason.

What you now know

You have the three rigid motions of the plane in both algebraic and geometric form: translation slides by a vector, rotation spins around a centre, reflection flips across a mirror line. The next section gives the unifying definition, all three are isometries, distance-preserving maps, and shows that this single property captures exactly what it means for a transformation to keep a figure's shape and size intact.

Quick check

Mark section complete →

References

Lang, S. (1971). Basic Mathematics. Springer. Chapter 6, §1, the canonical treatment of plane mappings.
Coxeter, H. S. M. (1969). Introduction to Geometry, 2nd ed. Wiley. Chapter 3 develops the same three motions and shows their interplay with similarity.
Hartshorne, R. (2000). Geometry: Euclid and Beyond. Springer. Chapter 17 places these motions inside the broader axiomatic Euclidean framework.
Greenberg, M. J. (2008). Euclidean and Non-Euclidean Geometries, 4th ed. W. H. Freeman. Chapter 9 contrasts these motions with the corresponding motions in non-Euclidean planes.