Helix - Variable Diameter with constant Pitch

[wacman]

Im trying to find an equation for a helix that gets wider and thinner yet the angle of all the coils remains constant.

Is this possible? Any ideas?

Thank you!
PS - I am not a math expert, but throughly enjoy the process!
walt

 

[arildno]

Well, you might look at parametrizations of the form:
$$x=r(t)\cos\theta(t),y=r(t)\sin\theta(t),z=k\theta(t)$$
where k is a constant, and $r(t),\theta(t)$ are functions of t, r(t) being non-negative, and $\theta(t)$ a strictly increasing function.

 

[Nabeshin]

Well, you might look at parametrizations of the form:
$$x=r(t)\cos\theta(t),y=r(t)\sin\theta(t),z=k\theta(t)$$
where k is a constant, and $r(t),\theta(t)$ are functions of t, r(t) being non-negative, and $\theta(t)$ a strictly increasing function.

This certainly describes a helix with constant spacing along the z-axis with variable radius, addressing the concern of a helix that gets wider, but what about the OP's question of "thinness"? I don't really know what I mean by this, perhaps he is envisioning a physical 3-dimensional coil rather than the curve you suggested.

 

[arildno]

Well, in that case, he's after a helical surface, rather than a helix.

He didn't ask about that.

 

[lstellyl]

Actually, I think arildno answered the question anyway.

Just replace $$r(t)=r(k \theta (t) )$$ where $$r(z)$$ is any positive function describing the radius of the helix (or "thinness" of it) as related to its height (z)