Parametric representation of a plane inside a cylinder

[ProPatto16]

Homework Statement

find parametric representation for the part of the plane z=x+3 inside the cylinder x²+y²=1

The Attempt at a Solution

intuitively... the cylinder is vertical with the z axis at its centre. and the plane is the whole surface inside the cylinder where y=0... visually cutting the cylinder into 2 half cylinders. also intuitively this means we only have to restrict the value of x, since y=0 at all point in the given plane and all values of z are in the given plane and in the cylinder. so y=0 and z=z where z is an element of R and x+3 needs to be between -1 and 1 of the x coordinates of the cylinder so x needs to have restriction -4<x<-2... those signs are meant to be equal to as weel as greater/less than.

but how do i make up the parametric equations? and is that even right?

 

[Metaleer]

You have to think that when looking straight into the cylinder, that is, at the XY plane from above, what you see is the very outline of the cylinder, so for this curve,

$$x(t) = \cos t$$

$$y(t) = \sin t$$​

How do you get $$z(t)$$? From the equation of the plane, $$z=x+3$$; yet we already have $$x$$ as a function of the parameter $$t$$, so

$$z(t) = x(t) + 3 = \cos t + 3$$​

so the parametric representation of your curve should be

$$\mathbf{r} = \cos t \mathbf{i} + \sin t \mathbf{j} + (\cos t + 3)\mathbf{k},$$​

where $$t \in [0, 2 \pi]$$.

If you meant the surface inside the cylinder, all you have to is think of a trivial parametrization for a surface given in rectangular coordinates:

$$\mathbf{S}(x, y) = S_1(x, y)\mathbf{i} + S_2(x, y)\mathbf{j} + S_3(x, y)\mathbf{k}$$​

For us, $$S_1(x, y) = x$$, $$S_2(x, y) = y$$ and $$S_3(x, y) = x + 3$$, where it's easily seen that the parameters $$x$$ and $$y$$ must vary such that $$x^2 + y^2 \leq 1$$, to stay within the bound determined by the cylinder.

Hope this helps.

 

[ProPatto16]

what made you look at it from that angle? like why should i know to do that? any particular reason?

and to me it looks like youve found the parametric equations for inside the cylinder?
the question says find the parametric equations for the part of the place z = x +3 that is inside the cylinder. on that plane there is no value of y other than zero. so I am a bit lost?

 

[LCKurtz]

what made you look at it from that angle? like why should i know to do that? any particular reason?

and to me it looks like youve found the parametric equations for inside the cylinder?
the question says find the parametric equations for the part of the place z = x +3 that is inside the cylinder. on that plane there is no value of y other than zero. so I am a bit lost?

You have z = x + 3 in terms of x and y. Since the x and y variables are to describe a circle, you might normally use polar coordinates r and θ. That gives you the standard equations for x and y:

x = r cos(θ)
y = r sin(θ)

for the domain. And it is easy to express z in terms of r and θ since you have z = x + 3 so

z = 3 + r cos(θ)

with appropriate r and θ limits to describe the circular domain.

 

[ProPatto16]

the parametric representaion... is it simply [rcos$$\theta$$, rsin$$\theta$$, rcos$$\theta$$+3] where 0 < r < 1 and 0 < $$\theta$$ < 2pi ??

those signs would equal to also.

 

[LCKurtz]

the parametric representaion... is it simply [rcos$$\theta$$, rsin$$\theta$$, rcos$$\theta$$+3] where 0 < r < 1 and 0 < $$\theta$$ < 2pi ??

those signs would equal to also.

Yes. That is a parametric representation of the surface. It isn't the only one but it is a very natural one.