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The thing that makes quadric surfaces "3D analogs of conic sections" is just that they are defined by a single equation of degree 2. It's not a particularly helpful characterization though, I would say. It strikes me more as something a pedagogue would say in a (poor) attempt to relate a new concept to one already known.

[One could see quadric surfaces as "slices" of a certain geometric object, analogously to conic sections, but only if you are only interested in them up to isomorphism. Then a quadric surface can be regarded as a hyperplane section of the image $V \subset \mathbb{P}^9$ of the Veronese embedding $\mathbb{P}^3 \to \mathbb{P}^9$.]

The thing that makes quadric surfaces "3D analogs of conic sections" is just that they are defined by a single equation of degree 2. It's not a particularly helpful characterization though, I would say. It strikes me more as something a pedagogue would say in a (poor) attempt to relate a new concept to one already known.

[One could see quadric surfaces as "slices" of a certain geometric object, analogously to conic sections, but only if you are only interested in them up to isomorphism (as algebraic varieties). Then a quadric surface can be regarded as a hyperplane section of the image $V \subset \mathbb{P}^9$ of the Veronese embedding $\mathbb{P}^3 \to \mathbb{P}^9$. Note that this is only partially analogous to the representation of a conic section (considered as a plane curve) as the intersection of cone and plane, since the conic section is related to the aforementioned intersection by a projective transformation, which is a lot stronger than saying they are isomorphic as algebraic varieties.]

The thing that makes quadric surfaces "3D analogs of conic sections" is just that they are defined by a single equation of degree 2. It's not a particularly helpful characterization though, I would say. It strikes me more as something a pedagogue would say in a (poor) attempt to relate a new concept to one already known.

[One could see quadric surfaces as "slices" of a certain geometric object, analogously to conic sections, but only if you are only interested in them up to isomorphism. Then a quadric surface can be regarded as a hyperplane section of the image $V \subset \mathbb{P}^9$ of the Veronese embedding $\mathbb{P}^3 \to \mathbb{P}^9$.]

The thing that makes quadric surfaces "3D analogs of conic sections" is just that they are defined by a single equation of degree 2. It's not a particularly helpful characterization though, I would say. It strikes me more as something a pedagogue would say in a (poor) attempt to relate a new concept to one already known.

[One could see quadric surfaces as "slices" of a certain geometric object, analogously to conic sections, but only if you are only interested in them up to isomorphism. Then a quadric surface can be regarded as a hyperplane section of the image $V \subset \mathbb{P}^9$ of the Veronese embedding $\mathbb{P}^3 \to \mathbb{P}^9$.]

The thing that makes quadric surfaces "3D analogs of conic sections" is just that they are defined by a single equation of degree 2. It's not a particularly helpful characterization though, I would say. It strikes me more as something a pedagogue would say in a (poor) attempt to relate a new concept to one already known.

The thing that makes quadric surfaces "3D analogs of conic sections" is just that they are defined by a single equation of degree 2. It's not a particularly helpful characterization though, I would say. It strikes me more as something a pedagogue would say in a (poor) attempt to relate a new concept to one already known.

[One could see quadric surfaces as "slices" of a certain geometric object, analogously to conic sections, but only if you are only interested in them up to isomorphism. Then a quadric surface can be regarded as a hyperplane section of the image $V \subset \mathbb{P}^9$ of the Veronese embedding $\mathbb{P}^3 \to \mathbb{P}^9$.]