It is said that curves of the second order which we usually refer to as ellipse, parabola and hyperbola, i. e. conics, are all represented on projective plane by closed curves (oval curve), which means there is no distinction between them. Why is it?(adsbygoogle = window.adsbygoogle || []).push({});

Projective space can, in principle, be thought of as Euclidean space supplemented with points at infinity. Say, we have a hyperbola on projective plane. It has two asymptotes, one for each branch. And those asymptotes are not parallel lines (in projective geometry they themselves may cross at any angle, but zero). It means that they can not cross at the same point of infinity, which would give us a nice closed curve. Instead, they both cross the same (and the only) line at infinity at different points. That makes a closed hyperbola consisting of two parts: the curved segment and the line segment at infinity. How can it be considered as a qualitative equivalent of ellipse that is curved everywhere?

The same is true for parabola, there are no asymptotes here, but tangent lines are not parallel anywhere, which means a parabola can not be closed without some kind of similar linear construction, can it?

**Physics Forums - The Fusion of Science and Community**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# A Why are conics indistinguishable in projective geometry?

Tags:

Have something to add?

Draft saved
Draft deleted

Loading...

Similar Threads - conics indistinguishable projective | Date |
---|---|

A Fundamental group of Project Plane with 2 points missing | Oct 12, 2017 |

I Euler Characteristic of the Projective plane and sphere? | Oct 6, 2017 |

I Pole in Stereographic Projection | Sep 2, 2017 |

B Stereographic projection and uneven scaling | May 10, 2017 |

**Physics Forums - The Fusion of Science and Community**