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I am studying the first semester BSc(Mathematics).

I have searched the whole web for asymptotes; but didn't found anything other than horizontal and vertical asymptotes and a little bit talks about oblique asymptotes. There was nothing at even a basic level.

I really beg y'all to provide some source or such things which could make me understand that topic vastly and clearly.

Things such as asymptotes for general algebric curve; curvilinear asymptotes; total number of asymptotes etc. are the topics that i wanna learn about. My course book is making me so confused; thus tryna find some really good and deep concept source. Questions such as how the curve can have double root at infinity?(and how to visualize it) and many such questions are there.

Vicrobot
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  • Your intuition is right. Your question is too broad. What should I do now? Btw, if the question is put on hold someone can still post comments. – callculus42 Sep 15 '18 at 14:56
  • @callculus I know but this may diverge the attention of crowd from this question. Also you can even vote to delete if you want; but let me get some source please. i need to understand this topic thoroughly. – Vicrobot Sep 15 '18 at 15:00
  • @callculus and you can see; it has half hour over; but no reply related to content – Vicrobot Sep 15 '18 at 15:01
  • My intention is not to fight against questioners. And I don´t want that the question will be deleted. If you get the required information than I´m fine. But I think only links to websites or book recommendations can help you. This all can be done in the comments. – callculus42 Sep 15 '18 at 15:08
  • ok; but if you know some good books that might help me; then please sugest me. You see; i am really in need of it – Vicrobot Sep 15 '18 at 15:09
  • You have to wait some time for an appropiate answer/comment. Just give the others some time. – callculus42 Sep 15 '18 at 15:13
  • A suggestion- asking for help with a worked example from your course would not be closed, and should be helpful for you, even if its not as useful as a good book – Calvin Khor Sep 18 '18 at 05:35
  • I'm having a hard time understanding what your specific question is. What do Taylor series have to do with any of this? – Antonio Vargas Sep 19 '18 at 05:19
  • Your title is much more specific than the post. You should fix that. –  Sep 19 '18 at 12:15
  • @YvesDaoust I edited the title. I wrote a new question regarding what i need in particular for now(https://math.stackexchange.com/q/2921097/547918). But if you can help me with some good resource for study; it would be really helpful. – Vicrobot Sep 27 '18 at 08:27

2 Answers2

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This article is a good introductory text. In practice it might be more convenient to use the Taylor expansion after finding the points at infinity and dehomogenizing the curve to map them to affine points, see Ch. 2.1 in Sendra, Winkler, Perez-Diaz, Rational Algebraic Curves: A Computer Algebra Approach.

Curvilinear asymptotes require computing the Puiseux expansion, see Ch. 2.5 in the same book.

Maxim
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What about stretching space to observe points at infinity, for example with a transformation in polar coordinates

$$\rho\to\frac{\rho}{\sqrt{\rho^2+1}}$$ i.e.

$$(x,y)\to\frac1{\sqrt{x^2+y^2+1}}(x,y).$$

Below, the plot of an equilateral hyperbola $xy=1$. (The vertical segment is a plotting artifact.)

enter image description here

Points on the unit circle are at infinity.

  • But $x^2 y^2 - 1$ will result in an empty plot. You can plot the projection of the curve onto a hemisphere, then $x^2 y^2 - 1$ will look like this: https://i.imgur.com/y3Jga6o.png (one has to keep in mind that pairs of antipodal points are identified). – Maxim Sep 27 '18 at 11:44
  • @Maxim: why an empty plot ??? –  Sep 27 '18 at 11:51
  • Perhaps I'm misunderstanding your change of variables. With the substitution $(x, y) = (x, y)/\sqrt {x^2 + y^2 + 1}$, $x y - 1 = 0$ becomes $x^2 - x y + y^2 + 1 = 0$. There are no real solutions. – Maxim Sep 27 '18 at 12:01
  • @Maxim: $(t,\pm1/t)\to(t,\pm1/t)/\sqrt{t^2+1/t^2+1}$, which is the same as on my figure, plus the symmetrical. By the way, I can't understand how you get this polynomial. –  Sep 27 '18 at 12:03
  • I see, first you take a point on the curve and then you apply the transformation. But what if you cannot solve for $x$ or $y$ explicitly? – Maxim Sep 27 '18 at 12:11
  • @Maxim: then you cannot use your projection technique. –  Sep 27 '18 at 12:12
  • Actually you can, but you would have to use some variable elimination technique. But I agree, your method is exactly the same as taking the projection onto the unit hemisphere that I suggested and then looking at it from above (taking the parallel projection onto the $x y$ plane in the $z$ direction). I was trying to plug $(x, y)/\sqrt {\rho^2 + 1}$ into the equation of the curve. – Maxim Sep 27 '18 at 12:20
  • In other words, I utterly confused the forward and inverse transformations. One remark about the planar projection though is that you won't be able to see the tangency between the curve and the asymptote, because, unless the asymptote is the line at infinity, the tangency will be visible only in the $z$ direction. – Maxim Sep 27 '18 at 13:17