n-diameters-by-cos-ns

Equations for Curves with $n$ Diameters via $\cos ns$

Summary: A curve has $n$ diameters meeting at $C$ with angle $\pi /n$ between adjacent diameters iff, in polar coordinates $(z,s)$ centered at $C$, the equation $Z=0$ depends on $s$ only through $\cos ns$ (§§347–354). Converting back to rectangular coordinates via $z^2=x^2+y^2$ and $z^n\cos ns=\mathrm{Re}(x+iy{)}^n$ — the Chebyshev expansion — the equation becomes a non-irrational function of $x^2+y^2$ and the “Chebyshev real part” $T_n(x,y)$: $x^2-y^2$ for $n=2$, $x^3-3x{y}^2$ for $n=3$, $x^4-6x^2{y}^2+y^4$ for $n=4$, $x^5-10x^3{y}^2+5x{y}^4$ for $n=5$, etc. The “simplest” curve in each family — order $n$ with $n$ asymptotes bounding a regular $n$-gon centered at $C$ — falls out as $T_n(x,y)=a(x^2+y^2{)}^{(n-2)/2}+\cdots +c$ (§351 for $n=3$, §353 for $n=5$).

Sources: chapter15 (§§347–354); figures 70 ($n=2$), 71 ($n=3$), 72 ($n=4$) in figures68-71 and figures72-75.

Last updated: 2026-05-11.

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The polar reformulation (§347)

Take rectangular coordinates $(x,y)$ centered at $C$, and let

$z=CM=\sqrt{x^2+y^2},s=\angle ACM,\cos s=x/z.$

For a diameter $CA$ to be an axis of reflective symmetry, replacing $s$ by $-s$ should leave the equation unchanged. Hence $z$ as a function of $s$ must depend on $s$ only through $\cos s$ (the simplest function invariant under $s\mapsto -s$). The equation $Z(z,s)=0$ becomes a non-irrational function of $z$ and $\cos s$ — equivalently, since $\cos s=x/z$, of $z$ and $x/z$, equivalently of $z^2=x^2+y^2$ and $x$ (§348). The single-diameter case is recovered.

Two perpendicular diameters: $\cos 2s$ (§349)

If the curve has a second diameter $CB$ perpendicular to $CA$, then $s\mapsto \pi -s$ must also leave $z$ unchanged. The simultaneous invariants under $s\mapsto -s$ and $s\mapsto \pi -s$ are functions of $\cos 2s$, since $\cos 2s=\cos (-2s)=\cos (2\pi -2s).$ (Note that $\sin s$ is invariant under $s\mapsto \pi -s$ but not under $s\mapsto -s$, so it doesn’t work.) Using $\cos 2s=\frac{x^2-y^2}{z^2}=\frac{x^2-y^2}{x^2+y^2},$ the equation becomes a non-irrational function of $z^2=x^2+y^2$ and $x^2-y^2$ — matching the §342 form (function of $x^2$ and $y^2$). The two routes agree.

Three diameters at $60°$: $\cos 3s$ (§§350–351, figure 71)

Three diameters $AB,EF,GH$ at $60°$ means $\angle ACE=\angle ECG=\angle GCB=\pi /3$. The simultaneous invariants under $s\mapsto -s$ and $s\mapsto \frac{2\pi }{3}-s$ are functions of $\cos 3s$: $\cos 3s=\cos (-3s)=\cos (2\pi -3s).$ From the triple-angle formula, $\cos 3s=\frac{x^3-3x{y}^2}{z^3}.$ Hence the equation is a non-irrational function of $x^2+y^2$ and $x^3-3x{y}^2$. Letting $t=x^2+y^2$ and $u=x^3-3x{y}^2$, the general equation is

$0=\alpha +\beta t+\gamma u+\delta t^2+ϵtu+\zeta u^2+\eta t^3+\cdots .$

The simplest three-diameter curve. A third-order line with three diameters: $x^3-3x{y}^2=a{x}^2+a{y}^2+b^3⟺x^3-3x{y}^2=a(x^2+y^2)+b^3.$ This curve has three asymptotes bounding an equilateral triangle centered at $C$. Each asymptote is of the species $u=A/t^2$ (cf. curvilinear-asymptote), so the curve belongs to the fifth species in the chapter-9 cubic enumeration (cf. cubic-species-classification).

Four diameters at $45°$: $\cos 4s$ (§352, figure 72)

For four diameters at $\pi /4$, invariants are functions of $\cos 4s$: $\cos 4s=\frac{x^4-6x^2{y}^2+y^4}{z^4}.$ The equation is a non-irrational function of $x^2+y^2$ and $x^4-6x^2{y}^2+y^4$. Equivalently — and Euler points this out — it can be written as a function of $x^2+y^2$ and $x^2{y}^2$, since $x^4-6x^2{y}^2+y^4=(x^2+y^2{)}^2-8x^2{y}^2.$

Five diameters at $36°$: $\cos 5s$ (§353)

The Chebyshev real part of $(x+iy{)}^5$ is $\cos 5s=\frac{x^5-10x^3{y}^2+5x{y}^4}{z^5}.$ The simplest fifth-order curve with five diameters: $x^5-10x^3{y}^2+5x{y}^4=a(x^2+y^2{)}^2+b(x^2+y^2)+c.$ Five asymptotes bound a regular pentagon centered at $C$. The pentagonal pattern echoes the equilateral triangle of the cubic case.

General $n$ diameters (§354)

The pattern continues. For $n$ diameters at angle $\pi /n$, the equation is a non-irrational function of $z^2=x^2+y^2$ and the Chebyshev real part $z^n\cos ns=x^n-\frac{n(n-1)}{1\cdot 2}{x}^{n-2}{y}^2+\frac{n(n-1)(n-2)(n-3)}{1\cdot 2\cdot 3\cdot 4}{x}^{n-4}{y}^4-\cdots .$ This is the real part of $(x+iy{)}^n$, alternating signs on the even powers of $y$. With $t=x^2+y^2$ and $u$ this polynomial, the general $n$-diameter curve has the form $0=\alpha +\beta t+\gamma u+\delta t^2+ϵtu+\zeta u^2+\eta t^3+\theta t^{2}u+\cdots .$

It follows that we can always find a curve with any desired number of diameters which meet in a single point $C$ with equal angles. Conversely, all algebraic curves with a given number of diameters are obtained in this way. (source: chapter15, §354)

This is a complete classification: every algebraic curve with exactly $n$ orthogonal diameters arises from this construction, and every such construction gives an $n$-diameter curve.

The circle absorbed into every family

The equation $0=\alpha +\beta x^2+\beta y^2$ is a circle (provided $\alpha ,\beta$ have opposite signs). Setting $\gamma ,\delta ,\dots =0$ in any of the general equations above recovers it — consistent with the fact that the circle has infinitely many diameters (and so satisfies the $n$-diameter equation for every $n$ simultaneously). §351 makes the point explicitly.

Connections

• Chebyshev polynomials. The polynomial $T_n(x,y):=z^n\cos ns=\mathrm{Re}(x+iy{)}^n$ is the bivariate homogenization of the Chebyshev polynomial of the first kind: $T_n(\cos s,\sin s)=\cos ns$. Euler arrives at it via the trigonometric multiple-angle identities of Book I (Chapter 8) without naming it.
• Regular polygon of asymptotes. Each “simplest” curve of order $n$ has $n$ rectilinear asymptotes through $C$ at the same angular spacing $\pi /n$ as the diameters, but offset — bounding the regular $n$-gon centered at $C$. The diameters bisect the sides (or pass through the vertices, depending on parity).
• Equal parts. A curve with $n$ diameters has $2n$ equal and similar parts of the curve — each diameter divides into two equal halves, and the $n$-fold rotational symmetry multiplies this further. See chapter-15-on-curves-with-one-or-several-diameters §355.

Figures

Figures 68–71

Figures 72–75

Related pages

• chapter-15-on-curves-with-one-or-several-diameters
• diameter-and-center-from-equation — the $n=1,2$ predecessor.
• concurrence-of-diameters — why all $n$ diameters must meet at a common $C$ with equal angles.
• equal-parts-without-diameter — the parallel theory using $\sin ns$ together with $\cos ns$ for curves with $n$ equal parts but no diameter.
• cubic-species-classification — the fifth species, in which the simplest $n=3$ curve $x^3-3x{y}^2=a(x^2+y^2)+b^3$ lives.
• curvilinear-asymptote — the $u=A/t^2$ asymptote shape that bounds the regular triangle / pentagon.