I tried to get a equation that can represent a spiral like spring:
$$z=\frac{height}{2\pi }\cdot \arctan \left ( \sqrt{\frac{radius^{2}-x^{2}}{radius^{2}-y^{2}}} \right )$$
I am not sure about the equation I got, so I am confused about...
whether this equation really represent a spiral like spring?
whether an equation can represent a curve or a line in 3D space?
whether any other equation can also represent a curve or a line in 3D space?
Assume that a curve $\gamma$ can be parametrized in the form $$\gamma:\quad z\mapsto\bigl(f(z),g(z),z\bigr)\qquad(-\infty<z<\infty)$$ with two functions $f$ and $g$. Then the curve $\gamma$ as a point set can be defined by a single equation as follows: $$\bigl\{(x,y,z)\in{\mathbb R}^3\ \bigm|\ \bigl(x-f(z)\bigr)^2+(y-g(z)\bigr)^2 =0\bigr\}\ .$$ But this is highly artificial. E.g., you will not be able to compute tangent directions, length, etc., from this representation of $\gamma$.
"Generically" $r$ equations involving $n$ variables define a submanifold $M\subset{\mathbb R}^n$ of dimension $d=n-r$. A single equation, say $$\cos(x+y)- z\sin(x-y)-4=0\ ,$$ defines a two-dimensional surface $S\subset{\mathbb R}^3$. But only a careful analysis of this equation will reveal whether $S$ is actually a bona fide surface, or whether singularities are present.