Parameterise the circle that results from intersecting the unit sphere in centered at the origin with the plane
I tried
however the last equation is an ellipse????
It's definitely going to be a circle, because really: how could a plane intersect a sphere and make an ellipse! You're right that it's absurd.
Have you tried drawing a sketch? Try visualizing the picture with the y axis pointing directly out of the paper, and that may give you some insight. I solved the problem by first coming up with an equation based on the sketch, then proving that it was right. So, like I said. Start with the picture.
ahaok is wrong, a sphere cut by a plane gives a circle, not an ellipse. xxp9 is right- a circle, projected onto a plane at an angle to a line perpendicular to the circle gives an ellipse. That is why you get the equation of an ellipse- you are projecting the circle onto the xy-plane, not parameterizing it.
Here, your sphere is the unit sphere, and the plane is given by x= z. The "standard" parameterization of the unit sphere can be derived from spherical coordinates:
by taking the radius variable, equal to 1:
The fact that z= x means that we must have so that .
So one way of getting the parameterization of the circle is to replace by , reducing from two parameters to one:
Of course, . And, we can use to say that
and so have
.