Interpolated particle hair are spread all over the object

Question

I cannot figure out how to use interpolated children with hair particle systems correctly. Simple children behave more or less as expected, appearing near the parents, spread distance being determined by the radius parameter. But when I switch to interpolated, the chldren are distributed randomly all over the mesh, the locations seemingly completely uncorrelated with the ones of the parents. The manual says:

Children are emitted between the Parent particles on the faces of a mesh. They interpolate between adjacent parents.

Whatever that exactly means, to me it doesn't sound like what happens in my example, below. What I am doing wrong?

This is the way the parents are placed:

So it looks with simple children:

And this way with interpolated ones:

Answer 1

After looking at the code for quite a bit, I can answer my own question. Interpolated doesn't work like this. What it interpolates is not the position where the children are placed. It is used to interpolate the form (length, curving...) of the hairs from the closest neighbors, but the placement is random. Confining that seems only be possible by specifying a vertex group for the density and using Weight Paint on it.