Convert space curve to cartesian

[yanyin]

if R = sinti+sqrt(2)costj+sintk, 0<=t<=Pi/2
please eliminate t to determine the cartesian equation of R(t). Put limits on the variables and verbally describe the curve

 

[himanshu121]

x= sint, y=sqrt(2)cost, z=sint

u can clearly see that
x² + y² +z²=2{sin²t +cos²t}

=2

x² + y² +z²=2

 

[matt grime]

And eqaully clearly, surely you can see there is more to it than that? You've just replaced a locally 1-d structure (a curve) with a locally 2-d structure, a sphere.

yes, the x, y, and z coordinates necessarily satisfy that, but that isn't sufficient. You need to intersect with the plane x=z (or similar) at the very least.

generally the equation is x=z=(1-y^2)^{1/2}/\sqrt 2

 

[yanyin]

Originally posted by matt grime
And eqaully clearly, surely you can see there is more to it than that? You've just replaced a locally 1-d structure (a curve) with a locally 2-d structure, a sphere.

yes, the x, y, and z coordinates necessarily satisfy that, but that isn't sufficient. You need to intersect with the plane x=z (or similar) at the very least.

generally the equation is x=z=(1-y^2)^{1/2}/\sqrt 2

Thanks matt grime, I've checked yours is correct.
but can you show me how the above equation is reached.

 

[HallsofIvy]

If matt grime will forgive me for sticking in my oar:

x= sint, y=sqrt(2)cost, z=sint so obviously x= z.

x²= sin²t= (1-cos²t). But
y²= 2 cos²t so cos²t= y²/2. That is x²= 1- y²/2 and
x= z= &radic;(1- y²/2).