Double-Factor Asymptote Cases

Summary: When the highest member factors as $P=(ay-bx{)}^{2}M$ — multiplicity 2 — chapter 8 refines the parabolic-asymptote analysis. After axis rotation, two leading-equation alternatives survive at infinity: I. $\alpha u^2+\beta t=0$ (a parabola, figure 38) or II. $\alpha u^2+\beta u+\gamma =0$ (parallel-line family, complex, or double). Case II splits further by discriminant: real distinct roots give two parallel straight asymptotes, each refinable individually; a double root demands $(u-c{)}^2+A/t^k=0$ refinements. The most general form is $z^2-Az/t^p+B/t^q=0$ with $z=u-c$ and three sub-cases on $q$ vs $2p$; the $z^2=C/t^k$ shapes are catalogued by parity of $k$ (figures 39, 40).

Sources: chapter8 (§§204–209), figures35-39 (figures 38, 39), figures40-43 (figure 40).

Last updated: 2026-04-28

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Setup

§204. With $P=(ay-bx{)}^{2}M$, the rotation $t=(ax+by)/g$, $u=(ay-bx)/g$ from curvilinear-asymptote-refinement takes the members to

$P=\alpha t^{n-2}{u}^2+\alpha t^{n-3}{u}^3+\cdots ,$

$Q=\beta t^{n-1}+\beta t^{n-2}u+\beta t^{n-3}{u}^2+\cdots ,$

$R=\gamma t^{n-2}+\gamma t^{n-3}u+\gamma t^{n-4}{u}^2+\cdots ,$

$S=\delta t^{n-3}+\delta t^{n-4}u+\cdots ,$

and so on. (Reused letters denote different coefficients in different members.) The leading $u^2$-coefficient of $P$ is $\alpha t^{n-2}$.

The leading equation at infinity has two possibilities depending on which of $Q,R$ contributes their first term:

Equation I — $Q$‘s leading term $\beta t^{n-1}$ is present ($\beta \ne 0$):

$\alpha t^{n-2}{u}^2+\beta t^{n-1}=0⟹\alpha u^2+\beta t=0.$

Equation II — $\beta =0$ (so $Q$‘s leading term is absent), and $R$‘s leading term $\gamma t^{n-2}$ takes its place; also $Q$ may contribute $\beta t^{n-2}u$ (relabel that coefficient $\beta$):

$\alpha t^{n-2}{u}^2+\beta t^{n-2}u+\gamma t^{n-2}=0⟹\alpha u^2+\beta u+\gamma =0.$

Equation I: parabolic asymptote (§204)

$\alpha u^2+\beta t=0$ is a parabola in $(t,u)$. The two branches of this parabola coincide with two branches of the original curve at infinity, lying in quadrants $P$ and $R$ (figure 38) — for $\beta <0$ and $\alpha >0$ this is $u=\pm \sqrt{-\beta t/\alpha }$, two branches in the right half plane $t>0$.

This case reproduces the parabolic asymptote of parabolic-asymptote and is the canonical “asymptote without straight line” — the first species of hyperbolic-and-parabolic-branches parabolic family.

Equation II: parallel-line family (§§205–207)

$\alpha u^2+\beta u+\gamma =0$ has discriminant ${\beta }^2-4\alpha \gamma$. Three sub-cases:

Distinct real roots: two parallel straight asymptotes (§205)

Roots $u=c$ and $u=d$ with $c\ne d$. Each root gives a straight-line asymptote parallel to the new axis $AX$. To refine the asymptote $u=c$, factor

$\alpha u^2+\beta u+\gamma =(u-c)(u-d)\cdot \alpha ,$

then in the full equation substitute $u=c$ everywhere except in the factor $u-c$:

$\alpha (c-d)(u-c)t^{n-2}+t^{n-3}(\alpha c^3+\beta c^2+\gamma c+\delta )+t^{n-4}(\alpha c^4+\beta c^3+\gamma c^2+\delta c+ϵ)+\cdots =0.$

Dividing by the leading factor produces the same refinement chain as the simple-factor case in curvilinear-asymptote-refinement: $(u-c)+A/t^k=0$ for the first non-vanishing $A$. The refinement of $u=d$ proceeds analogously with $c$ replaced by $d$.

Complex roots: no real branches (§205)

If ${\beta }^2<4\alpha \gamma$, the equation $\alpha u^2+\beta u+\gamma =0$ has no real roots; no branch goes to infinity in this direction.

Double root: refinement to $(u-c{)}^2+A/t^k=0$ (§206)

If ${\beta }^2=4\alpha \gamma$, the equation reduces to $\alpha (u-c{)}^2=0$. Substituting $u=c$ everywhere except the factor $(u-c{)}^2$:

$\alpha t^{n-2}(u-c{)}^2+t^{n-3}(\alpha c^3+\beta c^2+\gamma c+\delta )+t^{n-4}(\cdots )+\cdots =0.$

Dividing by $\alpha t^{n-2}$:

$(u-c{)}^2+\frac{A_1}{t}+\frac{A_2}{t^2}+\cdots +\frac{A_{n-2}}{t^{n-2}}=0.$

The refinement is $(u-c{)}^2+A/t^k=0$ for the smallest $k$ with $A_k\ne 0$. If all $A_k$ vanish, the curve contains $u=c$ as a complex factor (see complex-curves).

Mixed forms when the next coefficient also vanishes on $u-c$ (§207)

If the coefficient of $t^{n-3}$ above happens to be divisible by $u-c$, then setting $u=c$ kills it as well, and we must keep the next term in the $t^{n-3}$ position with its own $u-c$. This produces equations of the form

$(u-c{)}^2+\frac{A(u-c)}{t^p}+\frac{B}{t^q}=0,p<q,q\le n-2.$

If the third term is also divisible by $u-c$, push to the fourth, etc. The general refined equation has up to three powers of $u-c$:

$(u-c{)}^2+\frac{A(u-c)}{t^p}+\frac{B}{t^q}=0.$

The general form $z^2-Az/t^p+B/t^q=0$ (§208)

Substituting $z=u-c$:

$z^2-\frac{Az}{t^p}+\frac{B}{t^q}=0.$

The behavior at infinity depends on how $q$ compares to $2p$:

Case 1: $q>2p$ — two distinct curvilinear asymptotes, four branches

The two roots of the quadratic in $z$ are

$z=\frac{A}{2t^p}\pm \sqrt{\frac{A^2}{4t^{2p}}-\frac{B}{t^q}}.$

When $q>2p$, the term $B/t^q$ is much smaller than $A^2/(4t^{2p})$ at infinity, so the two roots separate into

$z\approx \frac{A}{t^p}\text{and}z\approx \frac{B}{A{t}^{q-p}}.$

Both equations $z=A/t^p$ and $Az=B/t^{q-p}$ hold at $t=\infty$. So the original curve has two distinct curvilinear asymptotes — each a $z=C/t^k$ form — making four branches approach the straight line $u=c$ at different rates.

Case 2: $q=2p$ — discriminant condition

$z^2-Az/t^p+B/t^{2p}=0$ becomes $(z{t}^p{)}^2-A(z{t}^p)+B=0$, a true quadratic in $z{t}^p$. Roots:

• If $A^2<4B$: complex roots, no real asymptote.
• If $A^2>4B$: two real distinct roots $C_1,C_2$, giving two similar asymptotes $z=C_1/t^p$ and $z=C_2/t^p$ (same exponent, different constants).
• If $A^2=4B$: a double root, leading to higher-order refinement.

Case 3: $q<2p$ — middle term negligible

When $q<2p$, the term $Az/t^p$ vanishes faster than the others at infinity (since $z\to 0$ as well). The leading equation reduces to

$z^2+\frac{B}{t^q}=0,$

i.e. an asymptote of pure $z^2=C/t^q$ form.

Shape catalogue: $z^2=C/t^k$ (§209)

§209. Take the straight-line asymptote $u=c$ as the axis $XY$, set $z=u-c$ as the new ordinate, and consider the curvilinear asymptote $z^2=C/t^k$ for an integer $k$ less than $n-1$. The four quadrants from $XY$ and the perpendicular through $A$ are labelled $P$ (upper-right), $Q$ (upper-left), $R$ (lower-right), $S$ (lower-left) as in figure 39.

Odd $k$: two branches in opposite quadrants

For $z^2=C/t$, only $t>0$ gives real $z$ (assuming $C>0$); $z=\pm \sqrt{C/t}$ gives one branch $EX$ above and one branch $FX$ below the asymptote, both for $t>0$. So branches lie in quadrants $P$ and $R$ — figure 39. Same pattern for any odd $k$.

Even $k$ with $C>0$: four branches in all four quadrants

For $z^2=C/t^2$, both signs of $t$ give real $z$: $z=\pm \sqrt{C}/|t|$. Four branches: $EX$ (quadrant $P$), $FX$ (quadrant $R$) for $t>0$, and $GY$ (quadrant $Q$), $HY$ (quadrant $S$) for $t<0$ — figure 40.

Even $k$ with $C<0$: no real branches

$z^2=C/t^k$ with $C<0$ and $k$ even has no real solutions; no branch goes to infinity in this direction.

Summary of cases

Leading equation	Multiplicity profile	Asymptote	Branches	Figure
$\alpha u^2+\beta t=0$ (eq I)	parabolic	parabola $u^2=-\beta t/\alpha$	2 in $P,R$	38
$\alpha u^2+\beta u+\gamma =0$ distinct real roots	two parallel lines	each line refinable	2 + 2	—
$\alpha u^2+\beta u+\gamma =0$ complex roots	no asymptote	—	0	—
$\alpha u^2+\beta u+\gamma =0$ double root	one line, refined as $(u-c{)}^2+A/t^k$	$z^2=C/t^k$	per parity below	—
$z^2=C/t^k$, $k$ odd	—	—	2 in $P,R$	39
$z^2=C/t^k$, $k$ even, $C>0$	—	—	4 in all quadrants	40
$z^2=C/t^k$, $k$ even, $C<0$	—	—	0	—

Figures

Figures 35–39

Figures 40–43

Related pages

• chapter-8-concerning-asymptotes
• parabolic-asymptote
• curvilinear-asymptote-refinement
• triple-factor-asymptote-cases
• hyperbolic-and-parabolic-branches
• curvilinear-asymptote
• branches-at-infinity
• complex-curves