The Tangle of Western Mathematics regarding Isometry.

 Pythagoras → Descartes → Euler/Lagrange/Gauss → Grassmann/Hamilton → Cayley/Sylvester → Gram/Schmidt → modern linear algebra.

PYTHAGOREAN THEOREM

c² = a² + b²

│

│  establishes the special relationship

│  between perpendicular components and length

▼

ORTHOGONAL COORDINATE COMPONENTS

v = (x₁,x₂,...,xₙ)

│

│  generalizes Pythagoras

▼

EUCLIDEAN NORM

||v||² = x₁² + x₂² + ... + xₙ²

│

│  can be written

▼

DOT PRODUCT

v·v = ||v||²

│

│  and for two vectors

▼

ORTHOGONALITY

u·v = 0

│

│  allows an orthonormal basis

▼

column-stacking vectors → column matrix → Gram matrix MᵀM → orthonormal set → orthonormal basis → square Q → orthogonal matrix → orthogonal group O(N).


CARTESIAN / ORTHONORMAL BASIS

eᵢ·eⱼ = δᵢⱼ

│

│  transform the basis:

│  e₁→q₁, e₂→q₂, ... eₙ→qₙ

▼

MATRIX

Q = [q₁ | q₂ | ... | qₙ]

│

│  ask whether the new basis

│  retains the old dot products

▼

GRAM MATRIX

QᵀQ

│

│  if those dot products remain

│  exactly those of the Cartesian basis:

▼

QᵀQ = I

│

▼

ORTHOGONAL MATRIX

Q ∈ O(N)

│

│  therefore

▼

||Qv||²

= (Qv)ᵀ(Qv)

= vᵀQᵀQv

= vᵀIv

= vᵀv

= ||v||²

│

▼

LENGTH IS PRESERVED

│

▼

DISTANCE IS PRESERVED

||Qx-Qy|| = ||x-y||

│

▼

ADD AN ARBITRARY TRANSLATION t

f(x) = Qx + t

│

│  because t cancels when comparing points

▼

||f(x)-f(y)|| = ||x-y||

│

▼

EUCLIDEAN ISOMETRY



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