General equation for a cylinder of thickness T

Question

I know the equation for a cylinder of radius $R$ centered on the $z$-axis is $F(x,y,z) = x^2 + y^2 - R^2$, which makes intuitive sense by considering the z-axis as the line $x^2 + y^2 = 0$ (only satisfied in $\mathbb{R}$ by $x=y=0$) and then taking a circle of radius $R$ away from the $z$-axis.

However I don't know how to construct an equation when it's a cylindrical "shell" of thickness "t". I think the equation for the inner surface of the shell is $F(x,y,z) = x^2 + y^2 - (R-t)^2$ but I am not sure how to proceed. I can also work with parametric equations if that's available.

Answer 1

You have surface $F_1=x^2+y^2-R^2=0$ and $F_2=x^2+y^2-(R-t)^2=0$. You want to find an equation $F = 0$, which is true when either $F_1=0$ or $F_2=0$. That can be done with a simple multiplication, because the result is zero when any of the multiplicants is zero: $$ F=F_1F_2 = \Big(x^2+y^2-R^2\Big)\Big(x^2+y^2-(R-t)^2\Big)=0 $$

Answer 2

Probably the most convenient thing for subsequent use is to parametrize the solid cylindrical shell which is easily done in cylindrical coordinates: $z \in [0,h]$, $\theta \in [0,2\pi]$ and $r \in [R-T,R]$. Then the (side of the) inner cylinder is obtained with $r=R-T$ and the (side of the) outer one is obtained with $r=R$.