Can the Thickness of a 3D Spiral Curve be Defined Parametrically?

[sukharef]

Hello!
There is a parametric way of defining a spiral curve:
z = a*t;
x = r1*cos(w*t)
y = r2*sin(w*t).
Is there a way to define the thickness of spiral?

 

[ellipsis]

Wow, that's an interesting idea. If there is, the thickness is an inequality. I don't know about any 3d parametric graph generators that handle inequality though, so it'd be hard to play with.

Think about a 2d parametric system, defining a circle. Is there a way to give the circle 'width', such that it resembles a flat torus? Ah, in this case what you need is two parametric curves, and then define another system that lies between the earlier curves (Although, I don't think this is what you wanted).

I defined this system implicitly, not parametrically, but only due to how my graphing software works (Winplot)
$$
x^2 + y^2 \leq (1-A/2)^2
$$
$$
x^2 + y^2 \geq (1+A/2)^2
$$

Unfortunately, I'm not that experienced with 3d parametric equations. I know, though, that a spiral curve with thickness... i.e. a spiral 'cord' is possible to be represented parametrically if the torus is possible. But, unfortunately, I've been having trouble getting the torus to work on Winplot.

 

[Simon666]

This seems like a reasonable attempt:

z=a*t + thickness*cos(u)
x=(r+thickness*cos(u))*cos(w*t)
y=(r+thickness*cos(u))*sin(w*t)

For a surface you'll need two parameters u and t of course.