In a multivariable calculus course I'm required to give the equation of an elliptic cone with a centre of (1,-2,1) in the direction of the x-axis in both rectangular and cylindrical coordinates. I've determined the rectangular equation to be: $$(x-1)^2 = \frac{(y+2)^2}{a^2}+ \frac{(z-1)^2}{b^2}$$
using the standard cylindrical transformations of: $x=r\,cos\theta$, $y=r\,sin\theta$ and $z = z$ makes the resulting equation quite messy. In a solution I was able to find the following conversions were used: $$x=x$$ $$y=b\,r cos(t)-2$$ $$z=a\,r sin(t)+1$$ I understand why these were used from an algebraic standpoint, to cancel out the +2, -1 and use the $cos^2x+sin^2x=1$ identity to eliminate $a$ and $b$ and ultimately arrive at: $$\frac{x-1}{a}=r$$
From my understanding the change of coordinates typically makes the equation "nicer" and easier to work with. What I do not understand is:
a) Why and how is this allowed?
b) What does this translate to geometrically? I believe it is a shift of the origin, in which case are we not representing a different cone?
From a geometric standpoint, the shapes involved exist prior to and independent of the coordinate system. Coordinates are just our way of referring to them.
a) There are infinitely many possible coordinate systems, and you can always make switches between them, though algebraically this will usually be practically impossible. That's why we usually stick to a few standard types.
b) The location of the origin is an artifact of the coordinate system (it's just a convenient reference point for us to use in referring to figures). So "shifting the origin" (i.e., switching to a coordinate system with a different reference point) doesn't change the figure.