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

AI Thread Summary
A parametric definition for a 3D spiral curve is presented, using equations for z, x, and y. The discussion explores the concept of defining thickness for the spiral, suggesting it could be represented as an inequality, though no 3D parametric graph generators currently support this feature. The analogy of a 2D circle is made, proposing that a similar approach could create a torus-like shape using two parametric curves. The author shares an implicit definition for a system that could represent thickness but admits to challenges with graphing software like Winplot. Ultimately, a parametric representation for a thick spiral is proposed, requiring two parameters to define the surface.
sukharef
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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?
 
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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).

width.png

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.
 
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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.
 
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