What is the Mathematica equivalent(s) of the Wolfram Alpha command to get the center and radius of a sphere given by an expression, e.g. (in Wolfram Alpha);

which returns 6 and $$(-4,3,2)$$.

It turns out if you just minimize the lhs of the equation, you get $$-r^2$$ and the centers immediately, and $$r=\sqrt{|-36|}=6$$:

Minimize[x^2 + y^2 + z^2 + 8 x - 6 y - 4 z - 7, {x, y, z}]
(* result: {-36, {x -> -4, y -> 3, z -> 2}} *)

Why does this work? The value of $$s(x,y,z)=(x-x_0)^2+(y-y_0)^2+(z-z_0)^2-r^2$$ is precisely $$0$$ on the sphere surface, positive on the exterior of the sphere, and negative on the interior. It's easy to see that $$s(x_0,y_0,z_0)=-r^2$$ and the gradient $$\nabla s(x,y,z)=2 (x - x_0)\mathbf{i}+ 2 (y - y_0)\mathbf{j}+ 2 (z - z_0)\mathbf{k}$$ is zero at the center and grows outwards in all directions.

Note: I didn't consider this as your equation has no leading coefficients of $$x^2,y^2,z^2$$ which means the above solution works. However, @yarchik pointed out a problem if the sphere equation is scaled and starts with $$k x^2+ky^2+kz^2+...$$ where $$k\neq1$$ then Minimize will produce the correct centers but gives a minimum value of $$-kr^2$$. So you will need to take this into account, for example here's the same sphere from a proportional equation:

s = Expand[(x^2 + y^2 + z^2 + 8 x - 6 y - 4 z - 7)/100];
m = Minimize[s, {x, y, z}]; (* returns {-(9/25), {x -> -4, y -> 3, z -> 2}} *)
centers = {x, y, z} /. Last[m]; (* {-4, 3, 2} *)
radius = Sqrt[Abs[First[m]/Coefficient[s, x^2]]]; (* 6 *)

Alternatively, if you can find 4 points on the sphere surface (using FindInstance) then that's all you need to get the centers and the radius:

eq = x^2 + y^2 + z^2 + 8 x - 6 y - 4 z - 7;
sphere = (x - c1)^2 + (y - c2)^2 + (z - c3)^2 - r^2;
points = {x, y, z} /. FindInstance[eq == 0, {x, y, z}, Reals, 4];
system = ((sphere - eq) == 0 && r > 0 /. {x -> #[], y -> #[], z -> #[]}) & /@ points;
Solve[system, {c1, c2, c3, r}]
• Your first solution is not correct. The same result should hold for a rescaled equation. But Minimize[(x^2 + y^2 + z^2 + 8 x - 6 y - 4 z - 7)/100, {x, y, z}] yields 3/5 as a radius. Jun 22 '20 at 11:54
• @yarchik you are right - the centers are correct though and the radius easily follows from that - I will correct the answer. This is only relevant if the x^2, y^2,z^2 have leading coefficients Jun 22 '20 at 11:59