Everybody knows that when a plane intersects a cone at different angles and positions, we get conic sections. But, I wanted to know that if the same was possible in higher dimensions. If we take the 4 dimensional equivalent of a cone, and make it intersect with the 3 dimensional equivalent of a plane (cube or a cuboid) in 4 dimensional space, will we get 3 dimensional shapes like spheres or ellipsoids?
[Math] Sections of cones in higher dimensions
3dconic sectionsgeometry
Best Answer
Yes, you get ellipses and spheres. There are also 3D equivalents to a parabola and hyperbola. There are actually two types of 4D conic surfaces: one is a single connected surface, and the other has two separate surfaces. Here are some animations I just made on the topic:
3D View of Conic Sections from $x^2+y^2=z^2$
2D View
4D Conic Sections of $x^2+y^2=z^2+w^2$
4D Conic Sections of $x^2+y^2+z^2=w^2$