This animation represents level surfaces of the function f(x,y,z) = x^2 +y^2 -z^2. The graph of this function is a 3-dimensional solid in a 4-dimensional space - a hypersurface!

Although one often thinks of a gentleman's hat, a cylinder is any surface generated by dragging a curve in the 3 dimensional x, y ,z coordinate frame along the x, the y, or the z-axis.
In this particular example, the cylinder is generated...

The paraboloid z = x^2 + y^2 has a much simpler formula in polar coordinates. This formula is z = (r cos t)^2+(r sin t)^2 = r^2.

The surface z = x^2 -y^2 is generated by sliding the curve z = -y^2 from the y-z plane along the curve z = x^2 in the x-z plane.

The surface z = (x^2 - y^2)/(x^2 + y^2) has the simple equation z (r, t) = Cos (2 t) in cylindrical coordinates. The surface is therefore obtained by wrapping the graph of the curve z = Cos (2 t) on a circular cylinder and then expanding this...