Recently I had a conversation about that classical area of geometry, the study of conic sections: ellipses, parabolae, and hyperbolae. Conics are pretty much the simplest plane curves that one can imagine after straight lines, and have some lovely connections to physical motions, the most well-known of which is that planets follow elliptical orbits. This post will quickly cover some of the basics.

Simply put, a **conic section** is the intersection of the **double cone**

*K* := { (*x*, *y*, *z*) ∈ ℝ^{3} | *x*^{2} + *y*^{2} = *z*^{2} }

with a (two-dimensional) plane *P*. The plane *P* is completely described by two things: its **unit normal** *ν* = (*a*, *b*, *c*) ∈ ℝ^{3} (“unit” because we insist that *a*^{2} + *b*^{2} + *c*^{2} = 1) and a **signed distance** *d* ∈ ℝ from the origin in ℝ^{3} to *P* along the direction *ν*. In this notation,

*P* = { (*x*, *y*, *z*) ∈ ℝ^{3} | *a**x* + *b**y* + *c**z* = *d* }.

So, we’re interested in *C* := *K* ∩ *P*, where *K* is the standard cone defined above and *P* is a plane in ℝ^{3}. For the most part, this intersection is a one-dimensional smooth curve in *P*; there are a few degenerate cases that will get pointed out along the way. As it turns out, while the normal *ν* has a great influence on the shape of the curves *C*, the distance *d* is relatively unimportant: you get the same curve *C* simply scaled up or down if you make *d* larger or smaller; the only thing to watch out for is that things *do* change radically at *d* = 0. (Also, I’m restricting attention to the standard cone; if I were being really complete, then I’d consider the possibility that the cone might degenerate into a cylinder.)

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