If you cut two pieces of wood at 90 degrees then glued them end to end you would have a 'butt' joint. If you cut them at an angle then glued them together its a 'scarf' joint.

A scarf joint is stronger than a butt joint since the glue surface is over a larger area. For fly rod repair you want a lot of surface area so the scarf angle is very shallow, somewhere between 10 or 15 degrees.

When you plane both ends and overlap the pieces you end up with a shorter rod section. It is possible to do a scarf repair with out effecting the length by building a new piece of matching rod section that is long enough to allow for the joint.

It would take a rod builder almost as much time to craft the replacement section as it would to build a whole new tip. If the rod in question is a valuable antique it might make sense to do the scarf in order to keep the rod as origonal as possible. If it's not particulary special you might just splice in a section from a discarded tip of similar dimensions. This can be done with a 'double scarf' ie. splicing in a 2-3 inch section into the middle of the break. This would leave most of the rod intact and give you the same overall length.

=====/ /===== single scarf
=====/ /=\ \==== double scarf