Curve Intersection of Surfaces in 3D: Solving for the Parametric Equations

[MarcL]

Homework Statement

Show that x=sin(t),y=cos⁡(t),z=sin^2 (t) is the curve of intersection of the surfaces z=x^2 and x^2+y^2=1.

Homework Equations

I don't think there aren't really any equations relevant for this maybe except the unit circle..?

The Attempt at a Solution

I don't know how to equate both equation, I remember one problem very vaguely where we replaced the x^2 variable with s but I doubt this is what I have to do. Any insights on how to start this problem?

 

[Dick]

Homework Statement

Show that x=sin(t),y=cos⁡(t),z=sin^2 (t) is the curve of intersection of the surfaces z=x^2 and x^2+y^2=1.

Homework Equations

I don't think there aren't really any equations relevant for this maybe except the unit circle..?

The Attempt at a Solution

I don't know how to equate both equation, I remember one problem very vaguely where we replaced the x^2 variable with s but I doubt this is what I have to do. Any insights on how to start this problem?

x^2+y^2=1 tells you that you are on the unit circle for x and y. You can certainly parameterize that as x=sin(t), y=cos(t). Why? Then you just have to figure out what z is in terms of t. I really don't think there is much substance to this problem. Why are you confused?

 

[MarcL]

first of all wouldn't it be x=cos(t) and y= sin(t) I mean isn't that what it is on a unit circle? and whoops I just understood what you meant by z, after I figured out x, just plug it in so z=sin^2(t)

 

[Dick]

first of all wouldn't it be x=cos(t) and y= sin(t) I mean isn't that what it is on a unit circle? and whoops I just understood what you meant by z, after I figured out x, just plug it in so z=sin^2(t)

x=sin(t) and y=cos(t) is just as good a parmetrization of the unit circle as x=cos(t) and y=sin(t). Tell me why?

 

[MarcL]

I thought so because x=1 at t=0 well I thought so but you're right it doesn't change anything when I think about it. So i just put it in this form of parametric equation to just satisfy my first equation?

 

[Dick]

I thought so because x=1 at t=0 well I thought so but you're right it doesn't change anything when I think about it. So i just put it in this form of parametric equation to just satisfy my first equation?

Yes, that's the way to show this parametrization works. x=cos(t), y=sin⁡(t), z=cos(t)^2 is also a good (but different) parametrization of the same intersection.