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
x= sint, y=sqrt(2)cost, z=sint u can clearly see that x^{2} + y^{2} +z^{2}=2{sin^{2}t +cos^{2}t} =2 x^{2} + y^{2} +z^{2}=2
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 [tex]x=z=(1-y^2)^{1/2}/\sqrt 2[/tex]
Thanks matt grime, i've checked yours is correct. but can you show me how the above equation is reached.
If matt grime will forgive me for sticking in my oar: x= sint, y=sqrt(2)cost, z=sint so obviously x= z. x^{2}= sin^{2}t= (1-cos^{2}t). But y^{2}= 2 cos^{2}t so cos^{2}t= y^{2}/2. That is x^{2}= 1- y^{2}/2 and x= z= √(1- y^{2}/2).