  The purpose of this paper is to describe a new $3$-dimensional family of
bodies of constant width that we have called peabodies, obtained from the
Reuleaux tetrahedron by replacing a small neighborhood of all six edges with
sections of an envelope of spheres. This family contains, in particular, the
two Meissner solids and a body with tetrahedral symmetry that we have called
Robert's body. Behind the construction of this family lies the classical notion
of confocal quadrics discussed, for example, by Hilbert in his famous book. We
study confocal quadrics and prove that the distances of an alternating sequence
of four points in two confocal quadrics always satisfies a simple equation and
use this equation to prove that our bodies have constant width.
