I am interested in learning more about what has best been described as
a "spherical cone" in the following website (Figure 11.16):
http://www.math.cornell.edu/~dwh/books/eg99/Ch11/Ch11.html

A shape comprised by taking a spherical lune and combining the edges
to produce a "spherical cone".  Noting that different lunes can create
different "spherical cones," I am specifically interested in one whose
tangential lines at the tip of the combined lune edges produce a 90deg
angle (similar to the Figure 11.16 in the above link).

I am interested in any studies related to this shape. Has anyone ever
studied it, or used any part/section of it in practical applications?
Assuming the shape is hollow, a discussion on the shape's interrior
and its practical application would be most welcome.

Happy hunting...