Parametric Curve question

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kankerfist

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Mar 22, 2006
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I have a curve C whose parameterization is:
x = a(Sin[t])(Cos[d])
y = b(Sin[t])(Sin[d])
z = c(Cos[t])
where a,b,c and d are positive constants, and t >= 0.

I want to know how exactly I can show that this curve lies on a plane containing the z-axis. I understand that if I project this curve onto the xy-plane, it will be a straight line if, in R3, it resides on a plane containing the z-axis. I have graphed the projection in mathematica, and it is a straight line on the xy-plane, but I am not sure how to express this projected line's equation on paper. Any help is appreciated!
 
Firstly, you know it's like an ellipsoid with constant azimuthal angle, right? (Just checking.)

Anyway, if x = Ksin(t), y = Lsin(t), (K, L some constants) then sin(t) = x/K = y/L, agreed?
 
The previous question in this section led me to discover that it lies on an ellipsoid, but I did not know that just by looking at it. Also, I see that (x/K) = sin(t) and (y/L) = sin(t). Would the next step be x/K + y/L = 2sin(t)?
 
Remember, we just want to eliminate t. There is no t in x/K = y/L.
 
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