Geometry · Groups
Homothety and rotational homothety extend motions of the plane to the full family of similarities: in the formula $f(z)=az+b$, the coefficient $a$ can now be anything (not just $|a|=1$, as for motions) — and, as long as $a\ne1$, the transformation still has exactly one fixed point. Below: interactive models instead of long proofs.
A similarity of the plane is a transformation $P$ under which distances between images of points change by one and the same factor: $\rho(P(x),P(y)) = k\,\rho(x,y)$ for some number $k>0$, the same for any $x,y$. The number $k$ is called the ratio of similarity.
When $k=1$, distances are preserved exactly — these are precisely the motions of the plane: translation, rotation, reflection, glide reflection. Everything from here on concerns the case $k\ne1$ — when figures aren't just moved, but also change size.
The composition of similarities with ratios $k$ and $s$ is again a similarity, with ratio $ks$: this follows directly from the definition ($\rho$ is multiplied first by $k$, then by $s$).
Fix a point $O$ — the center of homothety. A homothety with coefficient $k$ sends a point $A$ to the point $H_O^k(A) = O + k\vec{OA}$ — on ray $OA$ (or its extension), at $|k|$ times the distance $|OA|$. Drag the center $O$ and move the $k$ slider.
Note that the homothety's own coefficient can be negative. Since a homothety is a special case of a similarity (obvious from the definition), there is a direct relationship between the homothety's coefficient and the ratio of the corresponding similarity: a homothety with coefficient $k$ is a similarity with ratio $|k|$.
$H_O^k\circ H_O^s = H_O^{ks}$ — the coefficients simply multiply. Homotheties sharing a center, under composition, behave exactly like nonzero numbers under multiplication — just as translations behave like numbers under addition ($T_a\circ T_b=T_{a+b}$).
Let's combine a homothety with a rotation about the same center $O$: first scale by $k$, then rotate by angle $\alpha$ — or the other way around, the result is the same, since a rotation and a homothety sharing a center commute: $R_O^\alpha\circ H_O^k = H_O^k\circ R_O^\alpha$. Placing $O$ at the complex zero, the whole action is simply multiplication by the number $ke^{i\alpha}$.
Placing the center of a rotational homothety at the complex zero, its formula is multiplication by $a=ke^{i\alpha}$. Shifting the center, the formula becomes $f(z)=az+b$, where $a$ is any complex number except zero ($a=0$ collapses the whole plane to a single point — that's no longer a similarity). As long as $a\ne1$ (that is, as long as the transformation doesn't reduce to a plain translation), it has exactly one fixed point: from $f(z_0)=z_0$ it follows that $z_0=b/(1-a)$. The formula can be rewritten more compactly: $f(z)-z_0 = a(z-z_0)$ — the transformation literally "stretches-and-rotates" everything around $z_0$ by the coefficient $a$.
Every similarity of the plane is one of four kinds:
translation (in particular, the identity transformation);
rotational homothety (in particular, the identity transformation, a rotation, or a homothety);
glide reflection (in particular, a reflection);
reflected homothety (in particular, a reflection).
Rotational homothety was already covered in §3. Reflected homothety is its "mirror" counterpart: a homothety $H_{z_0}^k$ combined with a reflection across an axis through its own center $z_0$ (a homothety and such a reflection commute, just like the homothety and rotation in §3). At $k=1$ this is just an ordinary reflection across that axis. It's exactly this that the "Reflected homothety" tab in the calculator (§7) is for.
The boundaries between classes overlap — and that's not a coincidence, but part of the same picture already seen with motions: the identity transformation is at once a "zero" translation and a "zero" rotational homothety ($k=1$, $\alpha=0$); an ordinary reflection is at once a glide reflection with zero shift and a reflected homothety with $k=1$. There are no other similarities of the plane.
Every similarity of the plane, like every motion, is written as a single formula in complex numbers — only now the coefficient $a$ is no longer locked to the unit circle.
$f(z)=az+b$, $a\ne0$ — orientation-preserving similarities: at $a=1$ this is a translation by vector $b$; at $a\ne1$ — a rotational homothety with coefficient $|a|$, rotation angle $\arg a$, and center $z_0=b/(1-a)$ (see §4).
$f(z)=a\bar z+b$, $a\ne0$ — orientation-reversing similarities: at $|a|=1$ this is a glide reflection, in particular a reflection (see "Motions of the Plane"); at $|a|\ne1$ — a reflected homothety with coefficient $|a|$ (see above).
These four cases are exactly the classification theorem from §5 — only written not geometrically, but algebraically.
Set up $g$ (applied first) and $f$ (applied second) — below, $f\circ g$ appears immediately: the class of the result and its formula, computed from the algebra rather than eyeballed. There's deliberately no class table here — instead, explore it yourself: for instance, compose two reflected homotheties with perpendicular axes and different $k$, or a rotational homothety with $k$ and with $1/k$ about the same center.