simple geometry question- equation of cylinder A cylinder is $(x-a)^2+(y-a)^2=r^2$ with axis at $z$. I don't see where the '$z$' is in the equation. The book (calc 3) I'm using mentions the equation works for any $z$, but I don't see where the $z$ output is in the equation

Here is the excerpt from the book, the only part referencing to the basic equation of the cylinder as I've read 1-2 chapters beyond this point (it is a 4 chapter vector calc text)
The equation of an object is a way of telling whether a point is part of an object -- if you substitute the coordinates of the point into the equation and the equation is true, then the point is on the object; if the equation is not true for that point, then the point is not on the object. There is no $z$ because the z-coordinate is not part of the decision of whether a point is on the cylinder.

The fact that there is no z tells you that all points where the x- and y-coordinates satisfy the equation are part of the cylinder, regardless of the value of z. For example, if the equation is $(x-1)^2 + (y+2)^2 = 4$, then one of the points on the cylinder is (1,0,0), but so is (1,0,1) and (1,0,-1) and (1,0,5) and (1,0, -789) and so on. This means that the cylinder goes on forever both up and down.
An equation for a smoothed cylinder can be :

$$ x^2+y^2+z^{2n} = 1 $$

with $n\in\mathbb{N}$, $n\neq0$. 

The greater the $n$ the smoother the top and bottom edges resulting from the "cut" of the infinite cylinder.

See for example https://www.wolframalpha.com/input/?i=plot+x%5E2%2By%5E2%2Bz%5E20+%3D+1

smoothed cylinder