chapter-15-on-curves-with-one-or-several-diameters

Chapter 15: Concerning Curves with One or Several Diameters

Summary: An invariant-theoretic classification of algebraic curves with symmetry (§§336–363). The chapter generalizes the diameter / center theory of chapter-5-on-second-order-lines from conics to all orders. Reflective symmetry — a diameter — corresponds to even powers of one coordinate in the equation $Z(x,y)=0$. Point symmetry — a center $C$ — corresponds to invariance under $(x,y)\mapsto (-x,-y)$, which decomposes into two cases: all homogeneous parts of even degree (curve has diameters too) or all of odd degree (curve has only the center, “alternately equal”). $n$ diameters must concur at a common center, separated by equal angles $\pi /n$, alternating in two kinds — a rational fraction of $\pi$ is mandatory, since otherwise the symmetries multiply infinitely and force the circle. The closed-form description uses polar coordinates $(z,s)$ centered at $C$: a curve with $n$ diameters has $z$ depending on $s$ only through $\cos ns$, equivalently on rectangular coordinates through $x^2+y^2$ and the “Chebyshev real part” $T_n(x,y)=\mathrm{Re}(x+iy{)}^n$. Without reflective symmetry (rotational only), $\sin ns$ also enters — giving the ”$n$ equal parts but no diameter” family. A continuity argument shows these two families exhaust all algebraic curves with two or more similar and equal parts.

Sources: chapter15 (§§336–363). Figures 68–76 (in figures68-71, figures72-75, figures76-80).

Last updated: 2026-05-11.

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Setting (§336)

The chapter opens by recalling the second-order results: every conic has at least one diameter; ellipse and hyperbola have two perpendicular diameters and a center; the circle is special, with innumerably many diameters. The aim of the chapter is to extend this from $n=2$ (conics) to all algebraic orders, and to make precise the distinction between “diametrically equal” (reflective) and “alternately equal” (rotational) pairs of parts.

Strand 1 — diameter and center as parity conditions (§§337–343)

The four-quadrant setup with axes $AB$ and $EF$ meeting at $C$ (figure 68) makes the symmetries visible as substitutions:

Substitution invariance	Geometric meaning	Equation form
$y\mapsto -y$	diameter $AB$	$Z=Z(x,y^2)$, only even powers of $y$
$x\mapsto -x$	diameter $EF$	$Z=Z(x^2,y)$, only even powers of $x$
$(x,y)\mapsto (-x,-y)$ (even hom.)	center $C$ with diameters	sum of even-degree homogeneous parts
$(x,y)\mapsto (-x,-y)$ (odd hom.)	center $C$ without diameters	sum of odd-degree homogeneous parts
both $y\mapsto -y$ and $x\mapsto -x$	two perpendicular diameters	$Z=Z(x^2,y^2)$, only even powers of both

The last entry has a striking consequence: no curve of odd order admits two perpendicular diameters, since the equation $Z=Z(x^2,y^2)$ has only even total degree. §341 names the two types of “equal pairs” — diametrically equal (with a diameter), alternately equal (point-symmetric only). See diameter-and-center-from-equation.

Strand 2 — concurrence and angle quantization (§§344–346)

If a curve has only two diameters they must be perpendicular (§344). With more than two, all diameters meet at one point $C$ (the “center”), and consecutive diameters are separated by equal angles. The bootstrap argument: each diameter is an axis of reflection, so reflecting one diameter across another gives a new diameter. Iterating produces an angular sequence that must close after finitely many steps — hence the angle $\angle ACE$ between adjacent diameters must equal $\pi /n$ for some integer $n$. Otherwise the bootstrap generates infinitely many diameters, forcing the circle. The diameters fall into two alternating kinds — odd-indexed and even-indexed in the cyclic order — playing equivalent roles under the symmetries. Also: no algebraic curve has two parallel diameters, since that would also generate infinitely many. See concurrence-of-diameters.

Strand 3 — the polar Chebyshev derivation (§§347–354)

Working in polar coordinates $(z,s)$ centered at $C$, a curve with $n$ diameters has $z$ invariant under both $s\mapsto -s$ (reflection across diameter $CA$) and $s\mapsto 2\pi /n-s$ (reflection across the next diameter). The simultaneous invariant is $\cos ns$:

$z^n\cos ns=\mathrm{Re}(x+iy{)}^n=x^n-(\genfrac{}{}{0ex}{}{n}{2})x^{n-2}{y}^2+(\genfrac{}{}{0ex}{}{n}{4})x^{n-4}{y}^4-\cdots .$

$n$	Polynomial $T_n(x,y)$	“Simplest” curve with $n$ diameters	Figure
1	$x$	—	(degenerate)
2	$x^2-y^2$	$x^2-y^2=\alpha (x^2+y^2)+\beta$ (ellipse / hyperbola)	70
3	$x^3-3x{y}^2$	$x^3-3x{y}^2=a(x^2+y^2)+b^3$ (cubic with three asymptotes bounding equilateral triangle)	71
4	$x^4-6x^2{y}^2+y^4$	quartic with four asymptotes bounding square	72
5	$x^5-10x^3{y}^2+5x{y}^4$	quintic with five asymptotes bounding regular pentagon	(not figured)
$n$	$\mathrm{Re}(x+iy{)}^n$	order-$n$ curve with $n$ asymptotes bounding regular $n$-gon	—

These curves all share the structure: equation a non-irrational function of $x^2+y^2$ and $T_n(x,y)$. See n-diameters-by-cos-ns.

Strand 4 — equal parts without a diameter (§§355–360)

A curve with $n$ diameters has $2n$ similar and equal parts. But a curve can also have $n$ equal parts with $n$-fold rotational symmetry yet no reflective symmetry — only “alternately equal” arrangements. In polar coordinates this corresponds to $z$ invariant under $s\mapsto 2\pi /n+s$ (rotation) but not under $s\mapsto -s$ (reflection). The natural invariants are $\sin ns$ and $\cos ns$ together; in rectangular coordinates, both the real and imaginary parts of $(x+iy{)}^n$ appear:

$n$	$\cos ns$-part (real)	$\sin ns$-part (imaginary)
2	$x^2-y^2$	$2xy$ (i.e., the $xy$ term)
3	$x^3-3x{y}^2$	$3x^{2}y-y^3$
4	$x^4-6x^2{y}^2+y^4$	$4x^{3}y-4x{y}^3$
5	$x^5-10x^3{y}^2+5x{y}^4$	$5x^{4}y-10x^2{y}^3+y^5$

Dropping the $\sin ns$ part recovers the $n$-diameter case. The presence of the imaginary part is exactly what breaks the reflective symmetry while preserving the rotational one. See equal-parts-without-diameter.

Strand 5 — completeness (§§361–363)

The chapter closes by proving that the two classifications above — curves with $n$ diameters, and curves with $n$ equal parts but no diameter — exhaust all algebraic curves with two or more similar and equal parts. The argument is a continuity bootstrap: an isolated equal-pair would, by the law of continuity together with the equilateral-triangle (figure 75) or parallel-line (figure 76) construction, generate infinitely many copies of itself unless the angles match the $\pi /n$ quantization. Infinite copies imply infinite intersections with any transverse line, which the line–curve intersection bound forbids for algebraic curves. See classification-of-equal-parts.

Connection to other chapters

• Conics (chapter-5-on-second-order-lines): The §342 case $Z=Z(x^2,y^2)$ recovers the central conics’ two perpendicular diameters; the §338 case $Z=Z(x,y^2)$ recovers the parabola.
• Cubics (chapter-10-on-the-principal-properties-of-third-order-lines): Cubic curves can have a “sum-preserving” diameter or none, depending on coefficients. The chapter 15 framework restricts to orthogonal diameters, which forces $n=3$ and gives the very specific curve $x^3-3x{y}^2=a(x^2+y^2)+b^3$. Chapter 10’s diameter is more general but does not directly bisect.
• Branches at infinity (chapter-7-on-the-investigation-of-branches-which-go-to-infinity): The simplest $n$-diameter curve of order $n$ has $n$ asymptotes through $C$ bounding a regular $n$-gon — a direct application of the branch-at-infinity machinery.
• Modern symmetry theory: Chapter 15 is implicitly classifying curves invariant under the cyclic group $C_n$ (rotation by $2\pi /n$) or the dihedral group $D_n$ (rotation plus reflection). Without naming groups, Euler arrives at the same invariant decomposition into real and imaginary parts of $(x+iy{)}^n$ that the modern theory would call the irreducible $\mathbb{R}[x,y{]}^{C_n}$-module decomposition.

Figures

Figures 68–71

Figures 72–75

Figures 76–80

Related pages

• diameter-and-center-from-equation — §§337–343: parity conditions.
• concurrence-of-diameters — §§344–346: angle $\pi /n$ quantization, alternating kinds, no parallel diameters.
• n-diameters-by-cos-ns — §§347–354: polar Chebyshev derivation; simplest curves bounded by regular polygons.
• equal-parts-without-diameter — §§355–360: alternately equal classification.
• classification-of-equal-parts — §§361–363: completeness theorem.
• chapter-5-on-second-order-lines — the conic prototype.
• diameter-of-conic, center-of-conic — second-order origin of “diameter” and “center.”
• diameter-and-center-of-cubic — third-order diameter / center theory (sum-preserving, not orthogonal).