Level surfaces

Level surfaces: For a function $w=f(x,\,y,\,z) :\, U \,\subseteq\, {\mathbb R}^3 \to {\mathbb R}$ the level surface of value $c$ is the surface $S$ in $U \subseteq {\mathbb R}^3$ on which $f\Bigl|_{S} = c\,$.

Example 1: The graph of $z=f(x,\,y)$ as a surface in $3$-space can be regarded as the level surface $w = 0$ of the function $w(x,\,y,\,z) = z - f(x,\, y)$.

Example 2: Spheres $x^2+y^2+z^2 = r^2$ can be interpreted as level surfaces $w = r^2$ of the function $w = x^2+y^2+z^2$. Can you see how to interpret ellipsoids in the same way?

From the earlier example of $w = f(x,\,y,\, z) = x^2 + y^2 - z^2$. we obtain three particularly important surfaces as level surfaces:

$$x^2+y^2-z^2 \,=\,-1$$ Two-sheeted Hyperboloid

$$x^2+y^2 - z^2 \,=\, 0$$ Double Cone

$$x^2+y^2 - z^2 \,=\, 1$$ Single-sheeted Hyperboloid

  all the surfaces have been the graph of some quadratic relation in $x,\, y,$ and $z$ like $z - x^2 + y^2 = 0$ in the case of a hyperbolic paraboloid or $x^2 + y^2 + z^2 = r^2$ for a sphere,

  all the cross-sections of these surfaces have been conic sections like parabolas, hyperbolas etc.

In view of the first of these comments we make the following

Do you see that the circular cylinder to the left is the graph in $3$-space of $x^2+ y^2 = r^2$ for fixed $r$ because every horizontal slice is the same circle of radius $r$? Similarly, the cylinder to the right is parabolic; it's the graph of, say, $z = y^2$, since the intersection with every vertical plane $x = a$ is the same parabola $z = y^2$, say. Not surprisingly, it's called a Parabolic cylinder.