Let be the standard euclidean metric on defined by
where and are any points in .
a) a translation is a map given by
for some fixed given point . Prove, that the Euclidean metric d on is translation-invariant, ie, for any translation T it follows:
b) A rotation is a map given by
for some fixed real number . Prove, that the Euclidean metric d on is rotation-invariant, ie, for any rotation it follows:
Translation and rotation of a Euclidean metric are investigated. The solution is detailed and well presented.