Does anyone know how I can determine the equation of the 3D object below? (Maybe there's a program that can do it?) I am looking for a formula to define this 3D object, but am having trouble finding one.
(If you can imagine the 2D object you see revolved about the x-axis, that is the 3D object I'm referring to.) Btw the z-axis can't be seen because the view of the object is head-on and the object is symmetrical with respect to the x-axis. Thank you.
On
Suppose the top of the sketched curve has equation $y=f(x)$ for $0 \le x \le a$ and as in the diagram $f(x) \ge 0$ for $x$ in $[0,a].$ The distance from a 3-d point $(x,y,z)$ to the point $(x,0,0),$ in the plane having constant first coordinate $x,$ is $\sqrt{y^2+z^2},$ and in the revolved figure you want this distance to be $f(x).$
So an equation for the surface of the revolved curve is $$\sqrt{y^2+z^2}=f(x),$$ or more simply, since $f(x) \ge 0,$ it can be written as $$y^2+z^2=f(x)^2,$$ for $x$ satisfying $0 \le x \le a.$
If you are looking for an object similiar to this of your figure, trace in spherical coordinates (see http://en.wikipedia.org/wiki/File:3D_Spherical_2.svg) $$r = a\cdot sin(\phi)cos(\theta),$$ where $a$ is a positive constant, $-\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}$, and $0 \leq \phi \leq \pi.$