Similarities and the hierarchy of invariants

Try

A map that doubles every length changes a 3-by-4 rectangle into a 6-by-8 rectangle. The shape ratios remain the same, but area becomes four times larger. A similarity preserves a fixed positive ratio of all distances; an isometry is the special case with ratio one.

This gives a hierarchy of structures. Euclidean isometries preserve distance; similarities preserve angles and length ratios; invertible affine maps preserve lines, parallelism and division ratios; projective maps preserve incidence and cross-ratios. Enlarging the allowed transformations usually reduces the list of invariants.

Predict

Predict whether a similarity can turn a nonsquare rectangle into a square. Its adjacent side ratio cannot change, so it cannot. An affine map can stretch one coordinate more than the other and accomplish this.

Predict whether unsigned angle preservation also preserves orientation. A reflection preserves the magnitude of each angle but reverses orientation. Specify which meaning of angle a claim uses.

Experiment

Use a unit square to compare transformations.

Map	Adjacent lengths	Angle between adjacent sides	Area
(x,y)→(2x,2y)	2,2	90°	4
(x,y)→(2x,y)	2,1	90°	2
(x,y)→(x+y,y)	1,√2	45°	1

Compute the third-row angle from the dot product of (1,0) and (1,1). The final map keeps area but changes angle. This demonstrates why one preserved quantity cannot be used as a certificate for every stronger kind of equivalence.

Understand

Suppose d(Fx,Fy)=s d(x,y) for a fixed s>0. Define G(x)=(F(x)−F(0))/s. This map fixes zero and preserves distances. The isometry classification gives G(x)=Qx with Q orthogonal, hence F(x)=sQx+b.

For nonzero tangent vectors u,v, the transformed dot product is s²u·v and the transformed norm product is s²||u||||v||. Their quotient is unchanged, proving unsigned angle preservation. The determinant is s²det Q in the plane, so unsigned area scales by s² while oriented area also records the sign of det Q.

This proof identifies exactly what uniform means: one common s works for every pair of points. A point-dependent or direction-dependent scale is not a Euclidean similarity, even if some local angles happen to remain unchanged.

Apply

To prove two shapes are not equivalent under a transformation class, find a class invariant that differs. A rectangle of side ratio three cannot be similar to a square; exchanging corresponding sides only changes the ratio to one third, still not one. In contrast, a suitable coordinate scaling maps it affinely to a square.

Repair “Both figures have the same area, so there is a rigid motion between them.” Equal area is necessary for many transformations but far from sufficient for congruence. Side lengths, angles or even topology can obstruct the map.

For optional depth, Klein’s viewpoint studies a geometry through a transformation group and its invariants. It complements, rather than replaces, an axiomatic approach: one must still define the objects, group action and properties being studied.

Continue

Choose a geometric claim and state the smallest transformation class under which it is invariant. This habit prevents metric assumptions from leaking into affine or projective arguments. Continue to constructible coordinate fields to connect geometric operations with algebraic restrictions.