|These ellipsoids meet along a space curve.|
The ellipsoids in this system meet at one point, where they have
a common tangent plane.
For one value of t between 0 and 1, the quadric is just one point: the equation ((1 - t) . E1 + t . E2)(x,y,z) = 0 has exactly one solution.
The ellipsoids E1 and E2 are
disjoint, with disjoint interiors. Some of the quadrics in this
linear systems are ellipsoids, but there are two values of t between 0
and 1, where the quadric is just one point. For t between these two
values, the quadric is "empty" because there are no real solutions of
((1 - t) . E1 +
E2)(x,y,z) = 0.
These empty quadrics could be called "imaginary ellipsoids," since their equations have some complex solutions. In the complex projective coordinate system, any quadric equation (with coefficients A, B, C not all zero) has infinitely many solutions, and any two quadric surfaces have a non-empty intersection.