Ref: Marc van Leeuwen’s answer here: Link.

Let ${ E }$ be an ${ n }$ dimensional Euclidean space.

What are the origin and distance preserving maps ${ f : E \longrightarrow E }$?

Obs: Let ${ E }$ be an ${ n }$ dimensional Euclidean space. Let ${ f : E \longrightarrow E . }$ Then the following are equivalent:

• ${ \lVert f(x) - f(y) \rVert = \lVert x - y \rVert }$ for all ${ x, y \in E , }$ and ${ f(0) = 0 . }$

• ${ f(x) \cdot f(y) = x \cdot y }$ for all ${ x, y \in E . }$

Further any such map is linear and bijective.
Pf: Similar to the proof here: Link.

The origin and distance preserving maps ${ f : E \longrightarrow E }$ are called the orthogonal transformations of ${ E . }$
Note that they form a group under composition, called the orthogonal group ${ O(E) . }$

Note that reflections are simple orthogonal transformations.

Def [Reflections]
Let ${ E }$ be an ${ n }$ dimensional Euclidean space. Let ${ F \subseteq E }$ be a hyperplane (that is, an ${ n - 1 }$ dimensional subspace). Note that

\[{ E = F \oplus F ^{\perp} . }\]

The map sending

\[{ x = p _F (x) + p _{F ^{\perp}} (x) \longmapsto p _F (x) - p _{F ^{\perp}} (x) }\]

that is

\[{ x \longmapsto p _F (x) - (x - p _F (x)) }\]

that is

\[{ x \longmapsto 2 p _F (x) - x }\]

is called a reflection about ${ F . }$

Obs: Let ${ E }$ be an ${ n }$ dimensional Euclidean space. Consider a hyperplane ${ F \subseteq E, }$ and the reflection about ${ F }$

\[{ r(x) = 2 p _F (x) - x . }\]

Then ${ r(x) }$ is an orthogonal transformation.
Pf: Note that

\[{ {\begin{aligned} &\, r(x) \cdot r(y) \\ = &\, (p _F (x) - p _{F ^{\perp}} (x)) \cdot (p _F (y) - p _{F ^{\perp}} (y)) \\ = &\, p _F (x) \cdot p _F (y) + p _{F ^{\perp}} (x) \cdot p _{F ^{\perp}} (y) \\ = &\, (p _F (x) + p _{F ^{\perp}} (x)) \cdot (p _F (y) + p _{F ^{\perp}} (y)) \\ = &\, x \cdot y \end{aligned}} }\]

for all ${ x, y \in E . }$
Hence ${ r(x) }$ is an orthogonal transformation, as needed. ${ \blacksquare }$

Let ${ E }$ be an ${ n }$ dimensional Euclidean space. Let ${ f \in O(E) . }$

Note that reflections are simple orthogonal transformations.

Can we express ${ f }$ as a composition of reflections?

Thm [Cartan-Dieudonne]

Let ${ E }$ be an ${ n }$ dimensional Euclidean space. Let ${ f \in O(E) . }$

Consider the space of fixed points of ${ f , }$ that is ${ \ker (f - \text{id}). }$ Consider its dimension ${ d _f = \dim (\ker (f - \text{id}) ) . }$

Then ${ f }$ can be expressed as a composition of ${ n - d _f }$ reflections.

Pf: We will proceed by induction on ${ n - d _f . }$

The base case is clear. Suppose ${ n - d _f = 0. }$ Hence ${ \ker(f - \text{id}) = E }$ that is ${ f = \text{id} . }$ Hence ${ f }$ is a composition of ${ 0 }$ reflections.

Hence let ${ n - d _f > 0 . }$

Hence ${ \ker(f - \text{id}) \neq E . }$ Hence there is a vector ${ v }$ such that ${ f(v) \neq v . }$

Consider the reflection sending ${ f(v) }$ to ${ v . }$ Consider the hyperplane ${ H := \text{span}(v - f(v)) ^{\perp} , }$ and the reflection about this hyperplane ${ r _H . }$

Note that ${ r _H \circ f }$ fixes ${ v . }$ We will study the spaces of fixed points of ${ f }$ and ${ f ^{’} := r _H \circ f . }$

Note that ${ \ker (f - \text{id}) \subseteq H . }$

Suppose ${ f(x) = x . }$ Note that ${ x \cdot (v - f(v)) }$ ${ = x \cdot v - f(x) \cdot f(v) }$ ${ = 0 . }$ Hence ${ x \in H . }$

Note that

\[{ {\begin{aligned} &\, \ker(f ^{’} - \text{id}) \\ = &\, \lbrace x \in E : (r _H \circ f) (x) = x \rbrace \\ = &\, \lbrace x \in E : f(x) = r _H (x) \rbrace \\ \supseteq &\, \lbrace x \in E : f(x) = x = r _H (x) \rbrace \\ = &\, \ker(f - \text{id}) \cap H . \end{aligned}} }\]

Hence

\[{ \ker(f ^{’} - \text{id}) \supseteq \ker(f - \text{id}) \cap H . }\]

Note that even ${ f = r _H \circ f ^{’} . }$ Hence

\[{ \ker(f - \text{id}) \supseteq \ker(f ^{’} - \text{id}) \cap H . }\]

Hence we have the observations

• ${ \ker (f - \text{id}) \subseteq H . }$

• ${ \ker(f ^{’} - \text{id}) \supseteq \ker(f - \text{id}) \cap H = \ker(f - \text{id}) .}$

• ${ \ker(f - \text{id}) \supseteq \ker(f ^{’} - \text{id}) \cap H . }$

Note that the second observation can be strengthened to

\[{ \ker(f ^{’} - \text{id}) \supseteq \ker(f - \text{id}) + \text{span}(v) }\]

where ${ v \not \in \ker(f - \text{id}) . }$

Note that taking dimension of the third observation

\[{ {\begin{aligned} &\, d _f \\ \geq &\, \dim(\ker(f ^{’} - \text{id}) \cap H) \\ = &\, d _{f ^{’}} + (n-1) - \dim(\ker(f ^{’} - \text{id}) + H) \\ = &\, d _{f ^{’}} + (n-1) - n \\ = &\, d _{f ^{’}} - 1 \end{aligned}} }\]

that is

\[{ d _{f} \geq d _{f ^{’}} - 1 .}\]

Note that taking dimension of the second observation

\[{ d _{f ^{’}} > d _f. }\]

Hence

• ${ d _f + 1 \geq d _{f ^{’}} . }$

• ${ d _{f ^{’}} \geq d _f + 1 . }$

Hence

\[{ d _{f ^{’}} = d _f + 1 . }\]

Hence by induction hypothesis on ${ f ^{’} , }$ ${ f ^{’} = r _H \circ f }$ can be expressed as a composition of ${ n - (d _f + 1) }$ many reflections.

Hence ${ f = r _H \circ f ^{’} }$ can be expressed as a composition of ${ n - d _f }$ many reflections, as needed. ${ \blacksquare }$

Note that the number of reflections ${ n - d _f }$ is optimal.

Obs: Let ${ E }$ be an ${ n }$ dimensional Euclidean space. Let ${ f \in O(E) . }$

Then ${ f }$ cannot be expressed as a composition of fewer than ${ n - d _f }$ reflections.

Pf: Suppose ${ f }$ can be expressed as a composition of ${ \ell }$ many reflections

\[{ f = r _{\ell} \circ \ldots \circ r _1 . }\]

Note that the space of fixed points

\[{ \ker( f - \text{id}) \supseteq \bigcap _{i=1} ^{\ell} \ker(r _i - \text{id} ) . }\]

Note that each ${ \ker(r _i - \text{id}) }$ is a hyperplane, and hence is the kernel of a functional ${ f _i : E \longrightarrow \mathbb{R} . }$

Note that

\[{ \dim \left( \bigcap _{i=1} ^{\ell} \ker(r _i - \text{id} ) \right) = \dim \left( \bigcap _{i=1} ^{\ell} \ker (f _i) \right) \geq n - \ell . }\]

Consider the linear map ${ F : E \longrightarrow \mathbb{R} ^{\ell}, }$ ${ x \longmapsto (f _1 (x), \ldots, f _{\ell} (x)) . }$

Note that ${ \ker (F) = \cap _{i=1} ^{\ell} \ker (f _i) . }$

Note that by rank-nullity

\[{ \dim(\ker(F)) = n - \dim(\text{im}(F)) \geq n - \ell }\]

as needed.

Hence taking dimension

\[{ d _f \geq n - \ell . }\]

Hence

\[{ \ell \geq n - d _f . }\]

Hence ${ n - d _f }$ is the optimal number of reflections, as needed. ${ \blacksquare }$