How do I go about attaching legs like this?

[BearTricks]

I'm making a stool with rounded splayed legs. I was wondering how to go about drilling holes for and attaching the legs in a similar way to the attached image.

In my head I was imagining a sort of rounded mortise, with the shoulder cut at an angle to fit flush with the underside of the seat, but I had no idea how to go about making that accurately if I were to turn the legs on the lathe. Some of those stools look as if they have a dowel through the mortise, but the shortest clearly does not.

 

[Racers]

They are usually have tapered legs fitted into tapered holes.
I made a tapered ream and then used it to make a tapered cutter like a big pencil sharpener for the end of the legs.
But of you turn the legs you don't need the leg cutter

My stool.

21st March by Pete Maddex, on Flickr

Pete

 

[BearTricks]

The tapered ream looks like something I could do fairly easily. Thanks!

 

[Racers]

I made the cutter for mine from a rusty carpenters square blade.

Pete

 

[AndyT]

Although Pete's tapered reamer is a lovely, clever thing (I've seen it) you can get a surprisingly strong joint with a straight tenon in a straight hole.

Use a nice sharp bit, as big as you have or can afford, and bore right through the seat. Turn the legs to exactly the same diameter. (You can use an adjustable spanner as a durable caliper, or a non-adjustable if you are lucky with sizes. Or you can make some other sort of gauge or caliper.) Saw a slot for a wedge. Make the wedges, very slender - only slightly fatter than the saw cut.
Glue, assemble, wedge (across not along the grain,) wait, clean up.

You don't need a shoulder. To some extent, you can adjust the lengths afterwards, but try to be as exact as you can. It all gets a bit harder if you don't want through holes and tenons, so it's a good idea to make the joint decorative, with a nice snug fitting wedge.