constant-chord-interval-conchoid

Constant Chord Interval and the Conchoid of Nicomedes

Summary: §§410–418. The headline of chapter 17. Starting from $C{M}^2+C{N}^2=2a^2$ and generalizing to $C{M}^2+nCM\cdot CN+C{N}^2=a^2$, Euler arrives at the case $(CN-CM{)}^2=$ const — equivalently the interval $MN=2c$ is constant — and gives two polar solutions: $z=b\cos \varphi \pm c$ (three varieties: conjugate point at $C$ for $b<c$, cusp for $b=c$, node for $b>c$; figures 83–85) and $z=b/\cos \varphi \pm c$, which is the conchoid of Nicomedes (figure 86). §§417–418 extend the construction to a conchoidal-genus family in which any single-intersection curve serves as directrix (figure 87).

Sources: chapter17 §§410–418; figures 83–85 in figures81-85; figures 86–87 in figures86-89.

Last updated: 2026-05-11.

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$C{M}^2+C{N}^2=2a^2$ (§§410–411)

Since $CM+CN=P$ and $CM\cdot CN=Q$, we have $C{M}^2+C{N}^2=P^2-2Q=2a^2$, so $Q=(P^2-2a^2)/2$. Substituting $P=Mz/L$ and $Q=N{z}^2/L$:

$\frac{2N{z}^2}{L}=\frac{M^2{z}^2}{L^2}-2a^2⟺N=\frac{M^2}{2L}-\frac{a^{2}L}{z^2},$

which after substituting $z^2=x^2+y^2$ and clearing denominators gives

$\boxed{2L^2(x^2+y^2)-2LM(x^2+y^2)+M^2(x^2+y^2)-2a^2{L}^2=0.}$

If $M=0$ this collapses to $x^2+y^2=a^2$ — a circle with $C$ at the center, which trivially satisfies $CM=CN=a$.

§411 lists explicit fourth-order specializations:

• $M=2b$, $L=\alpha x+\beta y$: $(\alpha x+\beta y{)}^2(x^2+y^2-a^2)-2b(\alpha x+\beta y)(x^2+y^2)+2b^2(x^2+y^2)=0$.
• $L=x^2+y^2$, $M=2(\alpha x+\beta y)a$ (divide by $2(x^2+y^2)$): $(x^2+y^2{)}^2-2a(\alpha x+\beta y)(x^2+y^2)+2a^2(\alpha x+\beta y{)}^2-a^2(x^2+y^2)=0$.

Every even order $2n+6$ provides such a family, and a second family for orders $2n+4$ when $L$ is divisible by $x^2+y^2$.

The unified family $C{M}^2+nCM\cdot CN+C{N}^2=a^2$ (§§412–413)

Identity: $C{M}^2+nCM\cdot CN+C{N}^2=P^2+(n-2)Q$. Setting this to $a^2$ gives $Q=(a^2-P^2)/(n-2)$, hence

$\frac{M^2{z}^2}{L^2}+\frac{(n-2)N{z}^2}{L}=a^2⟺N=\frac{a^{2}L}{(n-2)z^2}-\frac{M^2}{(n-2)L},$

giving the general equation

$a^2{L}^2+(x^2+y^2)((n-2)L^2-(n-2)LM-M^2)=0.$

A second general form is obtained by letting $L=(x^2+y^2)N$:

$a^2(x^2+y^2)N^2+(n-2)(x^2+y^2{)}^2{N}^2-(n-2)(x^2+y^2)MN-M^2=0.$

§413 specializes:

• $n=2$: $CM+CN$ constant. Both equations are homogeneous in $x,y$, so they force $L,M$ to share a linear factor — meaning the curve is a pair of lines through $C$, not a genuine curve. Confirms the §405 impossibility.
• $n=-2$: $(CN-CM{)}^2=$ const — i.e., the interval $MN$ is constant. This is the rich case.

Constant interval $MN=2c$: the four-line family (§§413–414)

Setting $n=-2$ gives $(CN-CM{)}^2=a^2=4c^2$, hence

$a^2{L}^2=(x^2+y^2)(2L-M{)}^2,a^2(x^2+y^2)N^2=(2(x^2+y^2)N-M{)}^2.$

The simplest case in the first equation: $N=1$, $M=2bx$:

$a^2(x^2+y^2)=4(x^2+y^2-bx{)}^2.$

With $a^2=8c^2$ this rearranges to

$(x^2+y^2{)}^2=2(c^2+bx)(x^2+y^2)-b^2{x}^2,x^2+y^2=c^2+bx\pm c\sqrt{c^2+2bx}.$

Transforming to polar coordinates with $a=2c$ recovers the clean form

$\boxed{z=b\cos \varphi \pm c.}$

The simplest case in the second equation gives the other clean polar form

$\boxed{z=\frac{b}{\cos \varphi }\pm c.}$

Three varieties from $z=b\cos \varphi \pm c$ (§415, figures 83–85)

The geometric construction: draw line $ABC$ through $C$ with $CD=b$; from $D$ measure $c$ in both directions on $ABC$ to find $A,B$ (so $DA=DB=c$). For any line $NCM$ through $C$ at angle $\varphi$, drop perpendicular $DL$ from $D$; on $NCM$ from $L$ measure $LM=LN=c$ in both directions. The points $M,N$ are on the curve, with $MN=2c$ always. The diameter is $ACB$, and the normal line $ECF$ through $D$ has length $2c$.

Three sub-cases:

Case	Geometry at $C$	Figure
$b<c$	$C$ is a conjugate point (isolated point off the rest of the curve)	83
$b=c$	$C$ is a cusp; the interval $AC$ vanishes	84
$b>c$	$C$ is a node (double point); $A$ lies between $C$ and $B$	85

These are exactly the three local types from multiple-points-on-curves and first-species-cubic-configurations, here all instantiated in the same one-parameter family.

The Conchoid of Nicomedes from $z=b/\cos \varphi \pm c$ (§416, figure 86)

Construction: principal line $CAB$ through $C$ with $CD=b$ and $DA=DB=c$, so $A,B$ are on the curve. Draw the normal line $EDF$ through $D$. For any line through $C$ meeting $EDF$ at $L$ with $\angle DCL=\varphi$, $CL=b/\cos \varphi$. Then $LM=LN=c$ in both directions on $CL$ gives points $M,N$ on the curve.

This is the classical conchoid of Nicomedes, with $C$ as pole and $EF$ as the directrix (asymptote). The curve has four branches converging to $EF$ at infinity; the upper part $hBh$ is the exterior of the conchoid, the lower part $gAg$ is the interior; and there is a conjugate point at $C$.

The conchoid was used by Nicomedes (c. 200 BCE) for trisecting angles and duplicating the cube; Euler’s contribution is to derive it inversely as the unique constant-chord-interval companion to the line-directrix.

The conchoidal genus (§§417–418, figure 87)

§417 generalizes: replace the polar-line $z=b/\cos \varphi$ with any single-intersection curve $z=bP$ (where $P$ is odd in $\sin \varphi ,\cos \varphi$, as in curves-from-polar-coordinates). Then $z=bP\pm c$ has the constant-interval property too.

§418 illustrates with figure 87: $CEDLF$ is an arbitrary single-intersection curve; on each line $CL$ through $C$, measure $LM=LN=c$ on either side of $L$ on that line; the locus of all such $M,N$ is the conchoidal-companion curve $AMPCQBNRC$, which still has $MN=2c$.

When the directrix is a circle through $C$, the resulting curve is the same fourth-order one from §414 (with $z=b\cos \varphi \pm c$). This furnishes one of the earliest unifications of the conchoid and the limaçon of Pascal as cases of the same conchoidal-genus construction.

Figures

Figures 81–85

Figures 86–89

Related pages

• chapter-17-on-finding-curves-from-other-properties
• two-point-equation-from-fixed-point — provides the $P,Q$ framework used here.
• sum-and-product-of-chords-from-point — the basic-sum / basic-product cases.
• ordinate-powers-from-fixed-point — pushes to higher symmetric functions.
• curves-from-polar-coordinates — directrix family for the conchoidal-genus generalization.
• multiple-points-on-curves — node/cusp/conjugate-point trichotomy at $C$.
• first-species-cubic-configurations — same trichotomy in the cubic context.