Find the equation of this surface - The Snowman

Question

Here is a surface that resembles a snowman:

Its equation has the form $f(x,y,z)=0$: if a point $(x,y,z)$ satisfies that equations it is shaded, otherwise it is "left blank". Your goal is to find $f$.

Some constraints:

• The snowman is composed of three spheres: the "body", the "head" and the "hat".
• The body and the head are tangent with respect to each other.
• Half of the hat penetrates in the head.
• The dimensions and position of the snowman in the space do not count...
• ...as long as the relative dimensions between the body, the head and the hat are respected.

I suggest to use surfer to solve this puzzle. Surfer is an open-source program that allows you to draw a surface given its equation. Of course you can use another software (or paper-and-pencil) if you want.

Good luck and enjoy this puzzle!

Answer 1

Assume that the radii of the spheres $\Gamma_1, \Gamma_2, \Gamma_3 $ be $r_1,r_2,r_3$ both from bottom to top respectively. Now, let the sphere $\Gamma_1 $ be centered at the origin with the equation : $$x^2+y^2+z^2-r_1^2=0$$ Now, the sphere $\Gamma_2$ is tangent to the sphere $\Gamma_1$ hence it is centred at $(0,0,r_1+r_2)$. The equation of this sphere will be : $$x^2+y^2+(z-r_1-r_2)^2-r_2^2=0$$ The third sphere will then be centered at $(0,0,r_1+2r_2)$ and hence has the equation : $$x^2+y^2+(z-r_1-2r_2)^2-r_3^2=0$$ Hence the final equation will be $$(x^2 + y^2 + z^2 - r_1^2) (x^2 + y^2+(z - r_1 - r_2)^2 - r_2^2) (x^2 + y^2+(z - r_1 - 2 r_2)^2 - r_3^2)=0$$ The multiplication of all the terms in the final equation is an analogy from the combined equation of hyperbola. To be more clear, let the terms of the equations be $a,b$ and $c$ such that $abc=0$. For this to satisfy, at least one of them must be zero. This condition prevents any point outside the spheres represented by the equations because no sphere will be exits in the region other than their intended region. Contrary to this if we "add" the equations, it will NOT guarantee the existence of spheres only in the intended regions Now, consider a point $(0,r1+k,r1);k>0$. This point lies to the right of the tangent point of spheres $\Gamma_1$ and $\Gamma_2$. Now, no spheres exist here. So, this point will not satisfy the equation $abc=0$ and hence there is no existence of spheres here. If we assume addition of the terms, the other two terms can be adjusted to nullify the whole to zero which cannot and should not occur. Hence, the multiplication preserves the structure by forcing the existence of these spheres only in their intended regions