Curvilinear Asymptote

Summary: When the highest member of the curve equation factorises as $P=(ay-bx{)}^{k}M$ with multiplicity $k\ge 3$, the asymptote is no longer a straight line nor a parabola but an asymptotic curve of order up to $k$. After rotating axes onto the asymptote direction, the asymptotic equation takes forms like $u^3+A{t}^2/g+Bt/g^2+C/g^3=0$ (cubic, $k=3$) or $u^4+A{u}^3/g+\cdots =0$ (quartic, $k=4$). When all $k$ factors of $P$ coincide, the simplest curvilinear asymptote is $(u-\alpha {)}^k=K/t^{\mu }$ — Euler’s normal form. Branches converging to a common straight line may be hyperbolic, super-hyperbolic ($u=C/t^2$), etc., distinguished only by examining successive members.

Sources: chapter7 (§§185-197)

Last updated: 2026-04-28

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Setup

§185. Suppose the highest member has three or more real linear factors, possibly with repetitions. The simple case — all factors distinct — reduces to rectilinear-asymptote-from-equation applied factor-by-factor: $k$ distinct factors give $k$ straight-line asymptotes and $2k$ branches. The interesting case is when factors coincide.

When $P$ has, say, three coincident real linear factors,

$P=(ay-bx{)}^{3}M,M\text{ homogeneous of degree }n-3,$

the analysis of parabolic-asymptote does not apply directly: the multiplicity is now 3, and the asymptotic curve will be of order 3 or higher in the rotated frame.

Triple coincident factor

§186-187. Adapting the procedure of parabolic-asymptote to the triple factor: choose constants

$A:=\frac{Q}{(ax+by{)}^{2}M}{|}_{y=b,x=a},$

so that $A(ax+by{)}^2$ has the same scaling as $(ay-bx{)}^3$ at infinity. The leading asymptotic equation becomes

$(ay-bx{)}^3+A(ax+by{)}^2=0.$

Rotate to $t=(ax+by)/g$, $u=(ay-bx)/g$ with $g=\sqrt{a^2+b^2}$:

$\boxed{u^3+\frac{A{t}^2}{g}=0.}$

This is a semicubical-parabolic asymptote: a single algebraic curve of order 3 in the $(t,u)$-plane, to which the original curve tends at infinity. The shape (i.e. the actual geometry of the asymptote curve) is studied in chapter 8 — Euler defers it (“In the next chapter we will consider these asymptotic curves”). Two branches of the original curve converge to the two branches of $u^3=-A{t}^2/g$.

§188-189. Variants when $Q$ contains the linear factor $ay-bx$:

• $Q$ divisible by $ay-bx$ but not by $(ay-bx{)}^2$ → equation $u^3+Atu/g+B{u}^2/g^2+C/g^3=0$ or similar; two asymptotes appear, one straight ($u=0$ or $u=-B/(Ag)$) and one parabolic ($u^2+At/g=0$).
• $Q$ divisible by $(ay-bx{)}^2$ → equation $u^3+A{u}^2/g+Bt/g^2=0$ — a third-order asymptotic curve.

Branches converging to a common straight line: distinguishing the rate

§190-192. A subtle case arises when $A=B=C=0$ in the leading expansion. Then at the next non-vanishing member,

$u^3+\frac{D}{g^{4}t}=0\text{or}{u}^3+\frac{E}{g^5{t}^2}=0\text{or}{u}^3+\frac{F}{g^6{t}^3}=0,\dots$

In every such case the line $u=0$ is a straight-line asymptote (set $t=\infty$ in the equation; $u^3$ dominates and the only real cubic root is $u=0$). But the rate of approach differs:

• $u=(-D/g^4{)}^{1/3}{t}^{-1/3}$ — slow.
• $u=(-E/g^5{)}^{1/3}{t}^{-2/3}$ — slower.
• And so on.

§191. Even when the leading equation gives three parallel straight lines (all roots $\alpha ,\beta ,\gamma$ real and distinct), this only specifies the lines, not the manner of convergence. The next member tells whether the convergence is hyperbolic ($u-\alpha =C/t$), squared-hyperbolic ($u-\alpha =C/t^2$), etc.

Euler’s normal form $(u-\alpha {)}^k=K/t^{\mu }$

§192-193. The general curvilinear asymptote — applicable to any multiplicity $k$ when more than one (and perhaps all) of the factors coincide — has the form

$\boxed{(u-\alpha {)}^k=\frac{K}{t^{\mu }}}$

where $\alpha$ is the residual root (so the asymptote line is $u=\alpha$), $k$ is the number of coincident roots in the asymptotic equation (so $k\in \{2,3,\dots \}$), and $\mu$ is the leading non-vanishing power of $1/t$ from the lower members. The special cases that have appeared:

• $(u-\alpha )=K/t$ — single coincident pair, hyperbolic.
• $(u-\alpha {)}^2=K/t^k$ — double coincident pair, parabolic family.
• $(u-\alpha {)}^3=K/t^k$ — triple coincident, cubic family.

This is the unified pattern. Each higher multiplicity merely raises the exponent on the left.

§192. When two of the three roots are equal (say $\beta =\alpha$), the two corresponding asymptotes coalesce into a single double asymptote with $(u-\alpha {)}^2=K/[(\alpha -\gamma )t^k]$. When all three are equal, $(u-\alpha {)}^3=K/t^k$.

Quadruple factor: $k=4$

§193-196. For $P=(ay-bx{)}^{4}M$, the asymptotic equations available are

$u^4+\frac{A{t}^3}{g}=0(\text{Q not divisible by }ay-bx),$

$u^4+\frac{A{t}^{2}u}{g}+\frac{B{t}^2}{g^2}=0(\text{Q divisible by }ay-bx\text{ but not }(ay-bx{)}^2),$

$u^4+\frac{A{u}^2}{g}+\frac{But}{g^2}+\frac{Ct}{g^3}=0(\text{Q divisible by }(ay-bx{)}^2),$

$u^4+\frac{A{u}^3}{g}+\frac{B{u}^2}{g^2}+\frac{Cu}{g^3}+\frac{D}{g^4}=0(\text{Q divisible by }(ay-bx{)}^3).$

The first is a quartic asymptotic curve. The second contains both a straight-line component $u+B/(gA)=0$ and a cubic asymptote $u^3+Bt/g^2=0$. The third gives a third-order asymptotic curve. The fourth, for real and distinct roots, gives four parallel straight lines as asymptotes; for two equal roots, three parallel lines (one doubled); for two pairs of complex conjugate roots, no real branch; etc. — the same trichotomy on signs and discriminants seen in parabolic-asymptote and curvilinear-asymptote.

§196. One quartic case worth noting: the equation $u^4+B{t}^2/g^2=0$. If $B>0$ there is no real solution (no branch); if $B<0$, the equation factors as two parabolas with a common vertex but opening in opposite directions — a bicuspid-like asymptotic figure.

General multiplicity

§197. For $P=(ay-bx{)}^{k}M$ with multiplicity $k$, the asymptotic equation is of order $k$ in $(t,u)$, and the asymptote is an algebraic curve of order at most $k$ in the rotated frame. Each non-coincident factor of $P$ contributes its own straight-line asymptote (§175 / rectilinear-asymptote-from-equation); the coincident factors collectively contribute one curvilinear asymptote of order equal to the total multiplicity. Hence the maximal asymptotic order of an algebraic curve of order $n$ is $n$ itself, attained when $P$ collapses to a single $n$-fold real linear factor.

Hand-off to chapter 8

§§186-187 explicitly defer the shape of these curvilinear asymptotic curves to the next chapter (“if the shape of the curve $u^3+A{t}^2/g=0$ is known, then we will also know the shape at infinity of $P+Q+R+S+\cdots =0$”). Chapter 7 has identified which asymptotic curves arise; chapter 8 will draw and analyse them.

Related pages

• branches-at-infinity
• rectilinear-asymptote-from-equation
• parabolic-asymptote
• chapter-7-on-the-investigation-of-branches-which-go-to-infinity
• asymptotes-of-hyperbola
• order-of-an-algebraic-curve