I am an engineer, so maybe this question is naive.
I study equivalence relations and equivalence classes. An equivalence relation is a binary relation that is reflexive, symmetric, and transitive. Any number is equal to itself (reflexive). If a = b, then b = a (symmetric). If a = b, and b = c, then a=c (transitive). The relation that does not satisfy these conditions is not an equivalence relation.
For example, orthogonality is a non-equivalency relation on a set of lines in $\ R^2$. Because it is not reflexive and transitive.
I want to know: is there any maneuver or algorithm in math that changes a non-equivalency relation to an equivalency relation? For instance, by limitation in space set or adding some conditions.
You can always start with a relation and add to it all the pairs you need to make it an equivalence relation.
For reflexivity, add the pairs $(x,x)$ if they are not there.
For symmetry, add $(y,x)$ if it's not there and $(x,y)$ is.
For transitivity, form the transitive closure.
Whether these additions to the relation $R$ make any sense in the context that led to $R$ in the first place is application specific. For example, if you apply this construction to the orthogonality relation for lines in the plane you end up with the relation that says two lines are equivalent just when they are orthogonal or parallel.