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:
Where are these conic/quadric properties used in the context of pure mathematics?
In general: What's the importance of conics/quadrics in a course of pure mathematics?