I am new to multivariable calculus and am just get my head around the parameterization of surfaces. After research I found that a cylinder entered on the z - axis with radius R has a parameterisation $C(\theta, z) = Rcos\theta i + Rsin\theta j + z k $
But, what I don't understand is how this parameterization is derived in the first place. Like what is the method to arrive at that. I researched on several websites, including Khan Academy, but couldn't understand. Please help.
On
Some ideas for you:
As the cylinder has center on the $\;z\,-$ axis, if we "cut" the cylinder with the plane $\;z=0\;$ (the $\;xy\,-$ plane), then we get a circle $\;C:\;x^2+y^2=R^2\;$ , which is easily parametrizable by
$$C:\;r(\theta):=(R\cos\theta,\,R\sin\theta,\,0)\;$$
Now, just repeat the above argument for any plane of the form $\;z=k\;,\;\;k\;$ a constant (this means: a plane parallel to the $\;xy\,-$ plane) , and you'll get with that cut a circle $\;C':\;(R\cos\theta,\,R\sin\theta,\,k)\;$ .
From the above it is easily seen, imo, why the given cylinder has the parametrization it has.
The coordinate pairs $$ (x,y) = (R \cos \theta, R \sin \theta) $$ describe the points on a circle with center at the origin and radius $R$, see e.g. the polar coordinates construction, for a constant radius $R$.
If you then add a variable $z$ coordinate you get all such circles along the $z$-axis. $$ (x,y, z) = (R \cos \theta, R \sin \theta, z) \quad (z = \text{const}) $$ These circles form the cylinder surface. If $z$ is unbound it is a cylinder of infinite length, if $z \in [a,b]$ for real $a, b$ it is a finite cylinder.
Usually one adds the top and bottom discs for a finite cylinder.