Isometries and Rigid Motions
An isometry is a transformation that preserves distances (and therefore angles).
In Euclidean space \( \mathbb{R}^n \), a common form is the affine map
\[
x' = A x + t,
\quad A \in \mathbb{R}^{n\times n},\; t \in \mathbb{R}^n.
\]
Translations \(t\) do not change distances, so the key question becomes: what kind of matrix is \(A\)?
Orthogonal Matrices: the Isometry Test
A linear map \(x' = A x\) preserves dot products if and only if \(A\) is orthogonal:
\[
(Ax)\cdot(Ay) = x\cdot y \;\;\forall x,y
\quad \Longleftrightarrow \quad
A^T A = I.
\]
Since distances satisfy \( \|x-y\|^2 = (x-y)\cdot(x-y) \), preserving dot products implies preserving distances:
\[
\|Ax - Ay\| = \|A(x-y)\| = \|x-y\|.
\]
Therefore, the practical test is:
\[
\boxed{\text{Isometry (rigid motion)} \;\Longleftrightarrow\; A^T A \approx I}
\]
where “\(\approx\)” uses a tolerance because of numerical rounding.
Determinant and Orientation
For an orthogonal matrix \(A\), the determinant is always \( \det(A)=\pm 1 \):
-
\(\det(A)\approx +1\): proper isometry (rotation-type; orientation preserved).
-
\(\det(A)\approx -1\): improper isometry (reflection-type; orientation flips).
Examples
Distance Verification (Geometric Check)
Even if you do not trust the algebra, you can verify distance preservation directly:
choose points \(P_i\) and compare pairwise distances:
\[
d_{ij} = \|P_i - P_j\|,\qquad d'_{ij} = \|P'_i - P'_j\|.
\]
For an isometry, all differences \( |d'_{ij}-d_{ij}| \) should be near zero (up to numerical tolerance).
This is exactly what the calculator’s distance table reports.
Practical Notes on Tolerance
-
With exact symbolic entries, orthogonality can be exact. With decimals (like 0.7071), small error is normal.
-
Use a stricter tolerance (e.g., \(10^{-10}\)) when using exact expressions like \(\cos(\pi/4)\).
-
Use a looser tolerance (e.g., \(10^{-6}\) or \(10^{-4}\)) when inputs are rounded.
