> & 4 quastions:
> 1) haw can we define tow parallel *curves* in Euclidean geometry .
Pretty much however you wish, as the adjective "parallel" is usually
used to describe straight lines in Euclidean geometry.
Here's one definition that works reasonably well and captures most, if
not all, of the nature of parallelness, which that of constant
separation. Given a curve in Euclidean space, a curve parallel to it
can be constructed as the locus of a point a fixed distance along the
normal through a point as the latter traverses the original curve.
Whether this is the only or best definition I will leave you to think
about, but I've given you a starting point (pun intended).