Consider a point in the plane that has two perpendicular lines through
it that pass through two given points. In the figure below, *A* and
*B* are the given points and *P* and *Q* are points such
that angle *APB* and angle *AQB* are right angles. It is
well-known that the set of all such points is a circle with the segment
between the two given points as diameter.

Consider a point in space that has three mutually perpendicular lines
through it that each pass through the circle
*x*^{2} + *y*^{2} = 1 in the *xy*-plane.
In the figure below, *A*, *B*, and *C* are points on the
circle and angles *APB*, *APC*, and *BPC* are all right
angles. Find the locus of all such points *P*.