I have been reading a paper regarding Screw Theory and have come across the term "Isometry". A quick Baidu(I'm currently in China) turned up the following:
Given a metric space (loosely, a set and a scheme for assigning distances between elements of the set), an isometry is a transformation which maps elements to the same or another metric space such that the distance between the image elements in the new metric space is equal to the distance between the elements in the original metric space.
I work in computer vision and machine learning. To me, the above statement can be interpreted as
If we project in to another space(potentially a latent space), spatial distance shall be preserved
with the caveat that I understand this to be true up to a scale factor.
In other words, proportionally the distance is the same in the new coordinate system.
Could somebody kindly clarify?
Your understanding is broadly correct in that there exists a function from one metric space to another, that is distance preserving (actually, a bijection.
Not knowing your level of mathematical understanding, I offer the following definition as a prop to my statement.
Let $M_1=(A_1,d_1)$ and $M_2=(A_2,d_2)$ be metric spaces.
Let $\phi:A_1 \rightarrow A_2$ be a bijection such that:
$\forall a,b \in A_1:d_1(a,b)=d_2(\phi(a),\phi(b))$
Then $\phi$ is called an isometry.
That is, an isometry is a distance-preserving bijection.