Asymtotes of a general algebraic curve ( basically of the form of polynomial)

Question

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.

Answer 1

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.

Answer 2

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.)

Points on the unit circle are at infinity.