What is the equation for a helix?

[Bronta]

Is there a way to describe a helix not by its parametric form
x=Rcos(t) , y=Rsin(t) , z=ht
, but by a single equation like you can for a sphere with
r² = x²+y²+z² ?

Also, if anyone knows, the same question for a 3-dimensional tube that follows a helixcurve.

 

[LCKurtz]

Is there a way to describe a helix not by its parametric form
x=Rcos(t) , y=Rsin(t) , z=ht
, but by a single equation like you can for a sphere with
r² = x²+y²+z² ?

Also, if anyone knows, the same question for a 3-dimensional tube that follows a helixcurve.

You are comparing "apples to oranges". A helix is a one dimensional object in that it needs only one parameter in order to calculate a location on it. It is simply a curve in space. A sphere is a surface, requiring two variables to describe it. Your equation of a sphere expresses implicitly a function of two variables. For example, given x and y, you can calculate z on the surface. Generally, a surface would be parameterized by two variables.

You can read about a two variable parameterization of a tubular helix here:'

https://math.stackexchange.com/questions/461547/whats-the-equation-of-helix-surface