Pipe-fitting conditions in 3D

Question

Let's we have smooth (continuous and infinitely differential) curve $f(x(t), y(t), z(t)) = 0$ in 3D. Now I want to build a tube of diameter $D$ around it.

Questions:

1. What are the set of conditions this curve has to satisfy to make sure tube is not self intersecting and surface area is still smooth (differential)? Checking for curvature at every point to be $\lt \frac{D}{2}$ is not sufficient.
2. How to estimate volume of the tube between given points on the curve?

Answer 1

Regarding your second point, I am quoting from the book:

ABBENA, Elsa; SALAMON, Simon; GRAY, Alfred. Modern differential geometry of curves and surfaces with Mathematica. CRC press, 2006.

A tube about a curve $\gamma$ in $\mathbb{R}^3$ has the following interesting property: the volume depends only on the length of $\gamma$ and radius of the tube. In particular, the volume of the tube does not depend on the curvature or torsion of $\gamma$. Thus, for example, tubes of the same radius about a circle and a helix of the same length will have the same volume. For the proofs of these facts and the study of tubes in higher dimensions, see [Gray]¹.

¹_{A. Gray , Tubes, 2nd Edition, Progress in Mathematics 221, Birkhäuser Verlag, Basel-Boston, 2004.}

So, if I properly understood the above, the volume of the tube between two points $A\in \gamma$ and $B \in \gamma$ is

$\pi r^2 L(A,B)$

where $r$ is the radius of the tube and $L(A,B)$ is the length of the curve between $A$ and $B$.

A full proof is from page 5 to 7 of the above mentioned [Gray]; I found pages 5-7 available at books.google.it

Check also page 11 of Curvature and Convexity I by Erin Pearse.