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I've studied conics and quadrics in the past (specifically, in a course of analytic geometry). For these courses, we usually learn the basics of linear algebra and we apply these to conics and quadrics. And then, there is something interesting:

  • There are simple forms of a conics/quadrics in which every other conics/quadrics accross the euclidean space are isometric to it. By applying this isometry, it becomes easy to classify the conic/quadric and to obtain the coordinates of several important points as center, focal points, etc.

This is interesting because it gives the idea that It's possible to simplify a lot of calculations on their quadratic forms to obtain their essential features and it could be very complicated to do it without these simplifications. The problem here is that this doesn't seems to be about conics, it seems to be about isometries/congruence.

For the conics, we are introduced to certain properties of these curves, for example:

  • If you pass a light through the focus of an ellipse, then it will reflect to the other focus.

  • For every reflection comming to the interior of a parabola, it will always reflect to Its focus.

But this seems to be more interesting to engineering/physics than to mathematics, this property is used to build better antennas, for example. I'm curious about two things:

  1. Where are these conic/quadric properties used in the context of pure mathematics?

  2. In general: What's the importance of conics/quadrics in a course of pure mathematics?

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Conics, quadrics and their higher-dimensional equivalents (hyperquadrics) are the geometric representation of the solution sets of algebraic equations of degree 2. Many of the geometric properties translate to properties of the solution sets, such as symmetries, connectedness, presence or absence of straight lines inside the solution set etc. In this way conics and quadrics are relevant in whatever branch of mathematics that uses quadratic equations (number theory springs to mind).

Of course, if the primary purpose is to study properties of algebraic equations then one should not be limited to degree 2: the branch of mathematics known as algebraic geometry starts with the study of solution sets to (systems of) higher-degree algebraic equations. The first higher-degree example is given by elliptic curves.

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