How to express a parametrized curve in spherical coordinates Given an helix parametrized by $\lambda$ with
$$ x(\lambda)=\cos \lambda, \quad y(\lambda)=\sin \lambda, \quad z(\lambda)=\lambda$$
How can I express this curve in spherical coordinates $(r,\theta,\phi)$?
For example, I know that the unit circle parametrized by 
$$x(t)=\sin t, \quad y(t)= \cos t$$
Can be expressed in, say, cartesian coordinates, as $x^2 + y^2 = 1$, just by using the identity $\sin^2t + \cos^2t = 1$. But, with the helix, I can't seem to find such correspondence. And I think maybe it doesn't exist, because for the same value of $x$, there are infinite values for $z$. So, I don't know if this question even makes sense. Thanks for the attention.

The same way that you convert any coordinate to spherical
$$\begin{align} r(\lambda) &= \sqrt{x^2+y^2+z^2} = \sqrt{1 + \lambda^2}
\\
\theta(\lambda) &= \arctan(\frac{y}{x}) = \lambda
\\
\phi(\lambda) &= \cos^{-1}(\frac{z}{\sqrt{x^2 + y^2 + z^2}}) = \cos^{-1}(\frac{\lambda}{\sqrt{1+\lambda^2}}) \end{align}$$
