Process of finding vector function of an intersection between a plane and a cylinder

Question

The example I'm talking about :

Find the vector function that describes the intersection between a cylinder whose equation is $x^2 + y^2 = 1$ and a plane whose equation is $y + z = 2$.

The solution given is to substitute $x = cos(t)$, $y = sin(t)$ and find that $z = 2 - sin(t)$.

From there, just find the vectorial function with those parametric equations.

My question is purely theoretical, I gave the intersection between a cylinder and a plane as an example.

I don't understand why we can automatically assume that $x = cos(t)$ and $y = sin(t)$. Is it just good mathematical instincts? In a lot of examples I found online, they make substitutions like these, that seem to be just off of the top of their head.

You could set $x$ and $y$ as whatever you'd like, right?

Then that means there is an infinity of solutions?

Answer 1

This is just a convenient way to parametrize by the paramater $t\in[0,2\pi)$ the equation for the cylinder since in this way its equation is satisfied

$$x^2+y^2=\cos^2 t + \sin^2 t =1$$

As an alternative we can also set, for example, $x=t\in[-1,1]$ but in this case we would have $y=\pm \sqrt{1-t^2}$ which is not so good since we have to consider two distinct parts for the cylinder, notably the vector equation would be

• for $t\in[1,-1]$: $\left(t, \sqrt{1-t^2}, 2-\sqrt{1-t^2}\right)$
• for $t\in(-1,1)$: $\left(t, -\sqrt{1-t^2}, 2+\sqrt{1-t^2}\right)$

or also we could set $y=t\in[-1,1]$ and so on.

So of course we have (ideally) infinitely many ways to parametrize the vector function but they are all equivalent as a solution, meaning that they represent the same unique intersection.