Centre drills are a throw back to early lathe work. All the books like Chapman write about these as rote but seeing as there have been no new books written for centuries all the old rubbish gets regurgitated in any *new* books that come out
Easier to copy / paste than do home work.
The original use for a centre drill was two fold, firstly to put the cone shaped hole in to support a centre and secondly to act as a reservoir for white lead paste that was used as a lubricant for the dead centres of the day.
Now the first reason still applies but seeing as most people today use revolting centres the long pilot is no longer needed, in fact it's a disadvantage as if it breaks the whole tool is scrap and probably the work. A good dodge is to regrind new centres to about half the pilot length, enough to clear the tip of the centre plus a couple of regrinds.
THEN only use these for placing centre holes in work for use with lathe centres.
Anything else you use spotting drills, in industry today you virtually never see centre drills as spotting drills have taken over for one simple reason, they are better or industry wouldn't use them.
However if you do break a centre drill don't throw it away. Usually this is what you get.
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Sooooooooooo.
Make the perpetrator do the repair.
Grind the old broken centre drill up as a trepanning tool as in the two pics below.
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Top view.
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Bottom view, not the clearances all ways round.
Then feed this in round the brokem tip until the broken bit stands proud, then a quick smack leaves it like this.
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Which is now easier to reclaim now it's missing a hardened piece of tool steel