Given a two surfaces say: $z=1-y$ and $ x^2+y^2+z^2=1$, we find that they intersect at: $$x^2-2yz=0$$
How is the above a space curve? Is it not just another surface?
And why do we need to introduce a parameter, say $y=t$ , for it to be a space curve? I really don't understand the purpose of parametrization.
Best Answer
You are right. $$x^2-2yz=0$$ is not a space curve. Is it not just another surface but one that shares a commonalty. It is new a cone, a surface similar to the plane and sphere given, with which it shares the same curved intersection. It is one among a set of possible infinitely many surfaces with a common track or locus.
$ z=1-y $ and $ x^2+y^2+z^2=1$, do not intersect at $$x^2-2yz=0.$$
In fact no two surfaces intersect along another surface!
Just as three curved lines can be concurrent, three surfaces can have a ( I don't know if a proper word exists .."concur-line" :) for lack of one).
If any two of the surfaces is given, the third and indefinitely many more can be set up or found.
$$ (x,y,z) = (\pm \sqrt {2 t ( 1-t)} , t , (1-t) ) \tag{1}$$ is an easy parameterization of the common curved intersection. It has fixed or unique position in 3-space. Here also we can have several possibilities of changing the velocity vector. It is like a race-track. The track is fixed but so many speed / gradient variations are possible prescribed at any point on common line. The purpose of such parameterization is to find embedding in 3-space for this common line of cutting.
This line gives (x,y,z) coordinates as a function of a needed single parameter.
A simpler example. You can imagine all spheres passing through $ x^2 + y^2 = 1$ with their centers on z-axis and varying radii.
EDIT1:
2 errors corrected. $ x^2- 2 y z = 0$ is a cone, not a hyperboloid. But my remarks do not suffer. Other is typo about $\sqrt (..) $ in parameterization.
The red circle is the "concur-line" of the cone, plane and sphere of parameterization (1).
EDIT2:
Picturization of all possible coniciods passing through this line (including the above three) is indicated in code given (please make the correction):
https://mathematica.stackexchange.com/questions/87639/manipulated-surfaces-to-include-parametric3d-plot-of-concurrent-line
The animation shows successive changes, as morphing parameter $lambda$ ( zero for circular cone) is varied:
Double plane $\rightarrow$ oblate ellipsoid $\rightarrow$ hyperboloid of 1 sheet $\rightarrow$ cone $\rightarrow$ sphere $\rightarrow$ hyperboloid of 2 sheets $\rightarrow$ prolate ellipsoid $\rightarrow$ Double plane
This I hope completes an answer to a question " How is the parameterization of all surfaces inducing a common single space curve of intersection "?