HourGlass

The lemniscate of Bernoulli is given by the parametric equations:

$$x = \frac{a \cos(t)}{\sin^2(t) + 1}$$

$$y = \frac{a \sin(t) \cos(t)}{\sin^2(t) + 1}$$

for $-π ≤ t ≤ π$. The parameter "a" describes the size of the lemniscate.

When we rotate a function around the x-axis, we generate a surface of revolution. For the hourglass-shaped surface, we want to rotate the lemniscate about the x-axis such that the axis of revolution passes through the top, center, and bottom of the lemniscate, i.e., the axis of revolution is the x-axis.

The resulting 3D parametric equations for the hourglass-shaped surface of revolution are:

$$x = \frac{a \cos(t)}{\sin^2(t) + 1}$$

$$y = \frac{a \sin(t) \cos(t) \cos(φ)}{\sin^2(t) + 1}$$

$$z = \frac{a \sin(t) \cos(t) \sin(φ)}{\sin^2(t) + 1}$$

where $-π ≤ t ≤ π$ is the parameter for the lemniscate, and $0 ≤ φ ≤ 2π$ is the angle of rotation around the x-axis. The parameter "a" describes the size of the hourglass.

Remember that these equations describe a surface in three-dimensional space, where x, y, and z are coordinates in this space.

The resulting surface will look like an hourglass, with the lemniscate forming the "waist" and the resulting circles forming the "top" and "bottom" of the hourglass.

a = 1;
ParametricPlot3D[{a Cos[t]/(Sin[t]^2 + 1), 
  a Sin[t] Cos[t] Cos[\[CurlyPhi]]/(Sin[t]^2 + 1), 
  a Sin[t] Cos[t] Sin[\[CurlyPhi]]/(Sin[t]^2 + 1)}, {t, 0, 
  2 Pi}, {\[CurlyPhi], 0, 2 Pi}]