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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.

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  • $\begingroup$ 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. $\endgroup$
    – callculus42
    Sep 15, 2018 at 14:56
  • $\begingroup$ @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. $\endgroup$
    – Vicrobot
    Sep 15, 2018 at 15:00
  • $\begingroup$ @callculus and you can see; it has half hour over; but no reply related to content $\endgroup$
    – Vicrobot
    Sep 15, 2018 at 15:01
  • $\begingroup$ 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. $\endgroup$
    – callculus42
    Sep 15, 2018 at 15:08
  • $\begingroup$ 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 $\endgroup$
    – Vicrobot
    Sep 15, 2018 at 15:09

2 Answers 2

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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.

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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.

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  • $\begingroup$ 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: i.imgur.com/y3Jga6o.png (one has to keep in mind that pairs of antipodal points are identified). $\endgroup$
    – Maxim
    Sep 27, 2018 at 11:44
  • $\begingroup$ @Maxim: why an empty plot ??? $\endgroup$
    – user65203
    Sep 27, 2018 at 11:51
  • $\begingroup$ 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. $\endgroup$
    – Maxim
    Sep 27, 2018 at 12:01
  • $\begingroup$ @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. $\endgroup$
    – user65203
    Sep 27, 2018 at 12:03
  • $\begingroup$ 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? $\endgroup$
    – Maxim
    Sep 27, 2018 at 12:11

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