Please avoid giving in to the temptation to explain to me how necessary it is to be mindful of the different types of conic sections. It's all the same thing, kid. It's just a circle under different transformations. What.
The circle is just a special type of an ellipse; that is, it's the case where both horizontal and vertical dimensions become the same. In math, a circle is something like x² + y² =1. Don't think about what that equation means--just think about what it looks like.
Now look at the equation for an ellipse: x²/a² + y²/b² =1. Looks more complicated, right? Wrong. It's just that 'a' and 'b' stand for the horizontal and vertical radii of the ellipse, and both of those are the same in a circle, so they've been omitted from the circle's equation for simplicity.*
Ok, pocket. But you're telling me that these curves--like the parabola, hyperbola, and ellipse--all come from a circle, so how can that be if the circle is actually a special case of an ellipse?
Wow! You're smarter than you look. I usually just stop there, since I figure you don't know or care about what I'm saying anyway. You got me--almost. See, all of those curves are conic sections, which means that they're what happens when a plane passes through a cone. Think of cutting a birthday hat with a ninja sword. No matter how you slice it, you've got a conic section. And a scared kid.
Still don't see the circle? That's because it's not really there. Welcome to the weirdness that is mathematical reasoning. The circle is there because it is inherent to the cone, because a cone is also a cylinder, which is based on a circle. I know, all of this reasoning seems circular. Wait, in this case, that's a good thing! Just remember that the cylinder is a cone that has its apex at infinity. Whoa. Yeah, that's right, Buzz: stretch out the point of that birthday hat to beyond a light-year, and you'll approach a perfect cylinder! Localized space-time aberrations notwithstanding, of course.
Geometric descriptions, outside of the perfect walled garden that is mathematics, given the nature of the quantum, are as inaccurate as they are arbitrarily designated. They are simply the models that we use to make predictions, and they are useful to us only insofar as they are kept within a reasonable degree of uncertainty. What we call them, or how they are applied, is relative to their purpose. Standardization is a nice ideal for those that suffer from the uncomfortable pangs of uncertainty, but it's really only for textbooks. After you've taken the final, you can pretty much call things whatev u liek long ppl unnstand u
As such, what is the precise name for a table-saw cut's curve when the work-piece is passed over the blade in an oblique manner?
A cove.
~~~
pocket's 11th rule of cynicism (repulsion inversion): a thing can become only so repellent before it starts to attract again. No, 'The pocket guide to cynicism' has not been made public. The script has been written, but I'm just not ready for that level of pushback. There's too many sticks-in-the-mud out there who complain about complaining. Now I can't remember how I was supposed to relate all this back to the ellipse. I was working on other things. Got off the path. Writing is best when formed in continuity. You've got to stay in orbit. Oh well, just pretend that some clever closure or quip was here made about things going around and then coming around. Oh yeah, maybe that was the point--infinity doesn't have one!
*If you care: note that the 1 is a just a placeholder** for the radius in an idealized argument; none of these symbols are necessarily an integer value, but since our numerical system must make at least one assumption, we're stuck with a 1 somewhere. You have to base the argument on something. This one is based on the idea that there's such a thing as 1. You can instead displace this assumption over to a variable if you prefer, weirdo. People untrained in the sort of math that's beyond the requirements of the non-STEM major tend to get their panties in a bunch over this sort of thing. This and fractions. Oh, and expressive systems. lol. But Metric is easier! Like it matters. Given the multi-faceted, multi-layered nuance of some of the more beautiful and abstract mathematical concepts out there, it makes me smile that people actually get hung up on which units we use to describe the idea. A college calc course oughta getcha to shed that aversion to fractions: if you prefer the perfect clarity of a decimal to the messy imbalance of the solidus, for you it'll be like being an arachnophobe who's being thrown into a tank of spiders. Believe it or not though, one can in fact get over it. Life is paradox. Sigh. How else would it be for us? You and I exist only between the infinite and the infinitesimal. 1, as a concept, exists only within our continued ability to rationalize its existence so that we can do stuff with it.
** Note that a 1 can also be written as 1².
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