Surface formed by moving area along curved axis

[Kavorka]

I have never heard of a way to investigate this mathematically but I'm sure there is. How would you describe the surface area or volume of some 3-D surface formed by moving an enclosed area along a curved axis a certain distance? You could easily describe a torus by taking a circle and forming a surface of revolution, but you're also moving that circular area along a circular axis that passes through its center by rotating it. What if you took that circle and moved it instead along a parabola between two values? Surely it would form a continuous smooth surface with a definite volume and surface area, and yet its not a surface of rotation its like a surface of area path integration, but how would you describe it? What if the axis doesn't pass through the center of the circle, but some other point (I guess you could find a new curve that does pass through the center, but that seems impossible in many situations)? What if its not a circle, but any area? What if the axis is not perpendicular to the area? For all these situations I imagine a unique solid that is formed, can calculus help or would this be a numerical problem?

 

[phinds]

I'd call it a worm.

 

[Nidum]

Circle lofted around a parabola :

The general process of generating shapes by sweeping a section profile along a path is called lofting .

The shape shown is a simple one but very much more complex shapes can be generated .

The lofting process has practical applications in engineering .

The term lofting comes from the days when ships hulls and later aircraft fuselages and wings were drawn full size by hand on a large floor area adapted as a drawing board . Usually an upper floor was used - ie the loft .