The following are animations of the ,, and traces being taken of various surfaces.
- This is the sphere, defined by . Notice that all three traces are families of circles. For the traces, the radius of a circle in a given trace is . This explains why there is no trace when , then grow from to along and then shrink from back to along . By symmetry, identical properties hold when is replaced by or .
- This is the hyperboloid of one sheet, defined by . Notice that in the and traces, one sees a family of hyperbolas whereas in the traces, one sees a family of circles.
- This is the hyperboloid of two sheets, defined by . By contrast to the hyperboloid of one sheet, the and traces are families of hyperbolas, whereas the traces are a family of circles. Notice that there is a region of values for which the trace does not intersect the hyperboloids, corresponding the the non-existence of a solution to when is between and .