Parabolic Asymptote

Summary: When the highest member factorises as $P=(ay-bx{)}^{2}M$, the curve coincides at infinity with a parabola whose axis is the asymptote direction $y/x=b/a$. After rotating coordinates to align with the asymptote ($t=(ax+by)/g$, $u=(ay-bx)/g$, with $g=\sqrt{a^2+b^2}$), the asymptotic equation reads $u^2+At/g+B/g^2=0$ — a parabola, generically. Three degenerations arise when $A=0$: parallel-line asymptotes, a bounded curve, or a hyperbolic curvilinear asymptote $u^2=-C/(g^{3}t)$ requiring deeper case analysis.

Sources: chapter7 (§§178-184)

Last updated: 2026-04-28

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Setup

§178. Suppose the highest member has a double real linear factor:

$P=(ay-bx{)}^{2}M,M\text{ homogeneous of degree }n-2.$

For the curve to satisfy $P+Q+R+S+\cdots =0$ at infinity, the same logic as in rectilinear-asymptote-from-equation requires $(ay-bx{)}^2$ to stay bounded relative to the lower terms. But now $(ay-bx{)}^2$ is degree-zero in $x,y$ along the asymptotic direction, while $M$ is degree $n-2$, $Q$ is degree $n-1$, etc. The bookkeeping changes.

§178. Solving formally,

$(ay-bx{)}^2=\frac{-Q-R-S-\cdots }{M}.$

Now $S,T,\dots$ are of strictly lower degree than $Q$ and $R$ relative to $M$, so $S/M,T/M,\dots \to 0$ at infinity. Only $Q/M$ and $R/M$ contribute to leading order:

$(ay-bx{)}^2=-\frac{Q}{M}-\frac{R}{M}=-\frac{Q}{M(\mu y+\nu x)}(\mu y+\nu x)-\frac{R}{M},$

where $\mu ,\nu$ are arbitrary so that $\mu y+\nu x$ has degree 1. Setting $y/x=b/a$ (equivalently $y=b$, $x=a$) makes both fractions $Q/[M(\mu y+\nu x)]$ and $R/M$ constants, since both numerator and denominator have the same degree in each.

The asymptotic equation

§179. Define

$A:=\frac{Q}{M(\mu y+\nu x)}{|}_{y=b,x=a},B:=\frac{R}{M}{|}_{y=b,x=a}.$

Then at infinity

$(ay-bx{)}^2=-A(\mu y+\nu x)-B.$

The free choice $\mu =b$, $\nu =a$ gives $\mu y+\nu x=by+ax$. Now substitute $u=(ay-bx)/g$, $t=(ax+by)/g$ with $g=\sqrt{a^2+b^2}$ — this rotates the axes so that $u$ measures perpendicular displacement from the asymptotic direction and $t$ runs along it. Note $by+ax=gt$ and $ay-bx=gu$, so

$g^2{u}^2=-A(gt)-B⟺\boxed{u^2+\frac{At}{g}+\frac{B}{g^2}=0.}$

This is the equation of a parabola in the $(t,u)$-plane (with axis along $t$), and it represents the curve to which $P+Q+R+\cdots =0$ tends at infinity. Hence the asymptote is not a straight line — it is a curve, the parabola itself. The original curve has two branches that, as they go to infinity, more and more closely coincide with this parabola.

§179 closes by noting “the curve is therefore like a hyperbola as it goes to infinity” — meaning two branches escape to infinity (cf. the hyperbola’s asymptotic structure) — but its asymptotes are not straight lines.

Degenerate sub-cases when $A=0$

§180-184. If $Q$ vanishes or is divisible by $ay-bx$, then $A=0$. The asymptotic equation reduces to $u^2+B/g^2=0$, and three sub-cases arise:

(i) $B<0$ — two parallel-line asymptotes

§180. Write $B/g^2=-f^2$. Then $u^2=f^2$ factors as $(u-f)(u+f)=0$, two parallel straight lines. Each is an asymptote, with the curve approaching one or the other; four branches in total.

(ii) $B>0$ — no infinite branch

§181. Then $u^2+f^2=0$ has no real solution: the curve does not go to infinity through this factor, even though $P$ has a real linear factor. This is one place where the converse to the criterion in branches-at-infinity fails to be a clean iff: real linear factors are necessary for infinite branches but a double factor with $A=0,B>0$ produces no branch.

(iii) $B=0$ — degenerate, recurse on next member

§182-184. The leading asymptotic equation collapses entirely. To see what really happens we have to bring in the next member $S$ in the series. Write the analogue of $A,B$:

$\frac{S(by+ax)}{M}{|}_{y=b,x=a}=C,\frac{T(by+ax{)}^2}{M}{|}_{y=b,x=a}=D,\dots$

(higher-degree shifts in the denominator). After rotating axes, the asymptotic equation becomes

$u^2+\frac{C}{g^{3}t}+\frac{D}{g^4{t}^2}+\frac{E}{g^5{t}^3}+\cdots =0.$

If $C\ne 0$, the leading non-vanishing term is $C/(g^{3}t)$, so

$u^2=-\frac{C}{g^{3}t},u=\pm \sqrt{-\frac{C}{g^{3}t}}.$

This is a hyperbolic curvilinear asymptote: the curve has two branches in the upper-and-lower-half-plane sense (one for each sign of $u$), each tending to the line $u=0$ along the $t$-axis like $1/\sqrt{t}$. The asymptote-line is $u=0$ but the curve to which the original curve coincides at infinity is $u^2=-C/(g^{3}t)$, an explicit non-line.

If $C=0$ as well, recurse to $D$, then $E$, etc. — the asymptotic curve becomes $u^2=-D/(g^4{t}^2)$, $u^2=-E/(g^5{t}^3)$, and so forth, each with the same kind of three-way split (real branches / no real branches / further degeneration) on the sign of the leading constant.

What kind of asymptote does a “double factor” really give?

The chapter’s lesson is that the asymptote dictated by a double factor is generically a parabola, not a straight line. Four cases summarise (§§178-184):

Sub-case	Asymptotic curve	# infinite branches
Generic ($A\ne 0$)	Parabola $u^2+At/g+B/g^2=0$	2
$A=0$, $B<0$	Two parallel lines $u=\pm f$	4
$A=0$, $B>0$	None (curve bounded in this direction)	0
$A=B=0$, $C\ne 0$	Hyperbolic curve $u^2=-C/(g^{3}t)$	2
$A=B=C=0$, $D\ne 0$, $D<0$	$u=\pm f/t$ — two pairs of branches	4
$A=B=C=0$, $D>0$	None	0
$\dots$	recurse	$\dots$

The recursion always terminates because the equation has only finitely many members.

Connection to the parabola of parabola

The parabola of chapter 6, $y^2=2cx$, is precisely the second-order line $(P=\gamma y^2)$ whose highest member is the double factor $\gamma y^2=\gamma (y{)}^2$ (linear factor $y$ with multiplicity 2). The asymptotic equation derived above with $n=2$, $M=\gamma$, gives directly the parabolic shape $u^2+At/g=0$. So the parabolic asymptote of chapter 7 is the higher-degree generalisation of the conic parabola: any algebraic curve of any order with a double real linear factor in its highest member behaves at infinity like a parabola in the relevant direction.

Related pages

• branches-at-infinity
• rectilinear-asymptote-from-equation
• curvilinear-asymptote
• chapter-7-on-the-investigation-of-branches-which-go-to-infinity
• parabola
• classification-of-conics