Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [isometry]

An isometry is a map between metric spaces that preserves the distance. This tag is for questions relating to isometries.

1136 questions
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Isometry between vector spaces with different dimensions

Is it possible to have an isometry between two spaces of different dimension? Suppose we have $f : \mathbb{R}^d \rightarrow \mathbb{R}$ with $d \neq 1$ such that $|f(x) - f(y)| = \|x -y\|_2$ for all $x,y \in \mathbb{R}^d$. Does a mapping like this…
James Arten
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Property of Isometry that leaves no fixed points.

I've been asked to prove the following theorem, but I seem to be able to prove more than the theorem requires. I would like to check if what I proved is justified. Let $F$ be an arbitrary isometry of the plane. If $F$ does not leave any point fixed,…
Arial
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Isometries of the reals

We have been learning about isometries and how reflections, translations etc. and how they can affect a function. I was wondering if someone could help me with this proof by using the definition of isometries which I am struggling to understand.…
user253595
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Does invertible linear mapping, fixing the origin, imply isometry?

I'm trying to prove that the linear transformation in an affine mapping always is an isometry, when the affine mapping is an element of a space group. I have a proof of this from one paper, but I don't quite understand it. (Paper here:…
Carl Tudén
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Isometry from l∞ to C[0,1]

Is it possible to define an isometry from l∞ to C[0,1] using disjoint functions? I can define an isometry from l-p space to L-p space using disjoint functions but I am having difficulties with l∞ to C[0,1].
taupi
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Properly discontinuous action and Fundamental Domain

Does properly discontinuous action by a discrete group G on R^n necessarily have compact fundamental domain? Definitions: Properly discontinuous: For each x in R^n there is open neighborhood U of x such that gU and U does not intersect for all g in…
ardhajya
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Surface patch that is an isometry proof

Question: Let $σ: R2 ->S$ be a surface patch that is also an isometry. Prove that $σ_u$ and $σ_v$ are perpendicular. So essentially for this proof I'm thinking I want to relate the dot product to the fact that if $σ_u . σ_v=0$ Then that means…
Bugcatcher123
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$(\mathbb{E}^n,+)$ is embedded in the Group of Isometries as its Normal subgroup.

$(\mathbb{E}^n,+)$ is embedded in the Group $(M(n),\circ)$ as its Normal subgroup. Define $T:\mathbb{E}^n \to M(n)$ be the map $T(v):=T_v$, the Translation defined by the point $v$. i.e. $T_v(x)=x+v$ for all $x \in \mathbb{E}^n.$ Claim 1: $T$ is a…
Saikat
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Isometry of translation (does $x+v \mapsto y+v$)

Consider $f:\mathbb R^n\rightarrow \mathbb R^n$ where $f\in E_n$ and $E_n$ is the group of isometries of $\mathbb R^n$. Given $x,v\in \mathbb R^n$ arbitrary vectors of $\mathbb R^n$ and $y=f(x)$. Does $f(x+v)=y+v$? If so, why?
Khal
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What does it mean for a set to be isometric with another set?

If a set $A$ is isometric with a set $B$, does that mean there exists an isometry from $A$ to $B$? Or is it the other way around, that there exists an isometry from $B$ to $A$? Edit: I forgot that there were multiple definitions of an isometry, but…
user281997
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What kind of isometry is a composition of four reflections?

Suppose that l,m,k,t differnt lines (Like this illustration) Describe accurately this composition: $S_k\circ\ S_t\circ\ S_m\circ\ S_l$ Well, composition of isometry is a group and i know that composition of two different reflections is a rotation.…
Dan
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isometries in 3 dimensions

The 2-dimensional surface of a square can be isometrically transformed into a 3-dimensional cylinder or a cone. In what other shapes can it be isometrically transformed? More generally, what is the underlying theory concerning isometric…
exp8j
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How to interpret the listings of a rotation matrix?

Assume $\phi: \Bbb R^2 \rightarrow \Bbb R^2$ is an isometrie with $\det(\phi) = 1$. Then, $\phi$ is a rotation and its matrix can be denoted by So, I know that the unit circle is parameterized by the trigonometric functions $\sin$ and $\cos$, but…
Julian
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Proof of a theorem about isometries of $\mathbb{R^n}$

I am looking for a thorough proof of the following theorem Theorem: If an isometry of $\mathbb{R^n}$ fixes a non-empty set of points F, then it fixes the smallest affine subspace of $\mathbb{R^n}$ which contain F.
aribaldi
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Prove if $f(A)=g(A)$, $f(B)=g(B)$, and $f(C)=g(C)$, then $f=g$.

Suppose that $f$ and $g$ are isometries and $A, B, C$ are non-collinear. Prove that if $f(A) = g(A)$, $f(B) = g(B)$, and $f(C) = g(C)$ then $f = g$. I was thinking that because every isometry has an inverse which is also an isometry, then…
AndroidFish
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