# Animations of traces

The following are animations of the $x$,$y$, and $z$ traces being taken of various surfaces.

• This is the sphere, defined by $x^2+y^2+z^2=1$. Notice that all three traces are families of circles. For the $x$ traces, the radius of a circle in a given trace is $\sqrt{1-x^2}$. This explains why there is no trace when $x^2>1$, then grow from $0$ to $1$ along $-1\leq x \leq 0$ and then shrink from $1$ back to $0$ along $0\leq x \leq 1$. By symmetry, identical properties hold when $x$ is replaced by $y$ or $z$.
• This is the hyperboloid of one sheet, defined by $x^2+y^2-z^2=1$. Notice that in the $x$ and $y$ traces, one sees a family of hyperbolas whereas in the $z$ traces, one sees a family of circles.
• This is the hyperboloid of two sheets, defined by $x^2-y^2-z^2=1$. By contrast to the hyperboloid of one sheet, the $y$ and $z$ traces are families of hyperbolas, whereas the $x$ traces are a family of circles. Notice that there is a region of $x$ values for which the trace does not intersect the hyperboloids, corresponding the the non-existence of a solution to $y^2+z^2=k^2-1$ when $k$ is between $0$ and $1$.