The Sphericon

The squircle
The dual-n sphericon where n = 4 and k = 1. It is the dual of the original sphericon, and is known as the squircle.
The octasphericon
The even-n sphericon where n = 8 and k = 1. It is known as the octasphericon.
An Interesting Shape
Not strictly a member of the sphericon family, but interesting nevertheless.