Hello, everyone

In k-dimensional Euclidean space, (k >= 2)

there are two sets which of each consists of n distinct points.

We say both sets are equivalent (of the same shape)

if one of two translates or rotates in k-dim space, two can become identical.

That is, this is rigid-body motion in k-dim space.

Then, what is the sufficient and necessary condition

for the case when two sets are equivalent ?

Or at least, what is the sufficient condition for this case when two are equivalent?

Actually, I am considering this statement

such that : when there are two n-points sets in k-dim space,

consider for each set, an auxiliary set which contains all the euclidean-distance metrics between each pair of n-points. That is, that set will contain n(n-1)/2 elements.

I guess, if both auxiliary sets have the same elements, both sets equivalent

Is this right?

I am worried that it can happen when two sets are not equivalent, but two set are in mirror symmetry, for example.

Sorry for the long lines of inquiries.

Have a nice day, all of you and Thank you so much !