I have trouble visualizing what curves are traced out by the intersection of multiple surfaces in $R^3$. for example take the parametric equations $ <cos(t),sin(t),sin(t)$ >
Clearly this would represent X^2 + Y^2 = 1 likewise X^2 + z^2 = 1 and y=z however I don't understand why the curve of intersection is a ellipse, does the two cylinders intersection represent a sphere, I thought it would take on a shape more like a box with smooth corners.
Observe that $(x,y,z)=(\cos t,\sin t,\sin t)$ satisfies $$\frac{x^2}{1^2}+\frac{y^2}{(\sqrt{2})^2}+\frac{z^2}{(\sqrt{2})^2}=1\qquad\text{and}\qquad y=z$$ So, the curve lies in the intersection of an ellipsoid and a plane. Also the curve is closed since $(\cos 2\pi,\,\sin 2\pi,\,\sin 2\pi)=(\cos 0,\,\sin 0,\,\sin 0)$. Then, it looks like an ellipse.