This is not a calculus question regarding the area of cross section, but it is more of just a geometry question. This must have an easy answer, but I'm sort of confused now.
Suppose you cut out a sphere(say, $x^2+y^2+z^2=16$, a sphere centered at the origin of radius $4$) by a cylinder which has an axis parallel to the z-axis, but not centered at the origin(say, $(x-2)^2+y^2=1$ in the coordinate space). Then, think of the figure which has a boundary made by the cylinder and the sphere.(My English fails me,sorry.) Then, the figure could be explained as a cut of a cylinder which is not parallel to the $xy$-plane, so it must be an ellipse, yet the cut is on the sphere, so it must be a circle. Where does this contradiction come from?
Can anyone explain it to me? Thanks in advance.
Best Answer
If you try to solve the system $x^2+y^2+z^2=16; (x-2)^2+y^2=1$ to find the line of intersection, you will get, subtracting, $4(x-1) = 15 - z^2.$ Note that dependence between $x$ and $z$ is not linear. Rather, taking $z$ as a parameter, we get that the $x$-coordinate draws a parabola while you are moving up or down along your line. In particular, it has a constant "bend". Note that $y$ also does not exhibit a linear dependence on $z$.