classification-of-equal-parts

Classification of Curves with Multiple Equal Parts

Summary: A completeness theorem for the chapter (§§361–363): every algebraic curve with two or more similar and equal parts is either (a) a curve with one or more diameters, classified by n-diameters-by-cos-ns, or (b) a curve with several alternately equal parts but no diameter, classified by equal-parts-without-diameter. There is no third case. The proof is a continuity argument: if a putative equal-pair $OAa,OBb$ in figure 75 did not fit either classification, the law of continuity would force infinitely many such equal pairs — which is impossible for an algebraic curve.

Sources: chapter15 (§§361–363); figures 75, 76 in figures72-75 and figures76-80.

Last updated: 2026-05-11.

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The bootstrap by equilateral triangle (§361, figure 75)

Suppose a continuous curve has two similar and equal parts $OAa$ and $OBb$, both emanating from points $A,B$ on the curve. Construct the equilateral triangle $ACB$ with apex at $C$, where $\angle ACB$ equals the angle $\angle AOB$ subtended at the shared endpoint $O$ (or any consistent angular pair the curve provides). Then:

• The parts $CA,CB$ on opposite sides of the triangle are equal and similar to the original $OA,OB$ pair (since the triangle is equilateral and the curve is symmetric under that triangle’s symmetry).
• More generally, draw $CD,CE$ at the same angle $\angle ACB$ from $CB,CD$. These give new equal-and-similar arcs $Dd,Ee,\dots$ alongside the original $Aa,Bb$.

The construction iterates indefinitely. The result is a sequence of equal arcs around the point $C$, separated by the angle $\angle ACB$.

Termination. The bootstrap closes (returns to its starting arc) iff $\angle ACB$ is a rational multiple of $2\pi$ — equivalently, iff $\angle ACB=2\pi /n$ for some integer $n$. Then the curve has exactly $n$ similar equal parts arranged about $C$, fitting into one of the two classifications above. Otherwise the bootstrap generates infinitely many distinct equal arcs — but no algebraic curve can have infinitely many equal parts, since the bootstrap line would meet the curve in infinitely many points, violating the line-curve-intersection-bound.

Hence:

The curve is contained in those [classes] without a diameter which we have investigated. (source: chapter15, §361)

Handling oblique placements (§§362–363, figure 76)

The §361 argument needed the two arcs $aA,bB$ to be related by a rotation about some center — symmetric in the sense that the perpendicular bisector of $AB$ passes through the rotation center. The §362 step handles the case where the arcs are not arranged this way, e.g., when the arcs $aA,bB$ are on opposite sides of two parallel straight lines $AR,BS$.

The construction: draw $AR,BS$ so that the angle $\angle OAR=\angle OBS=\frac{1}{2}\angle AOB$, making $AR\parallel BS$. Then draw $CV$ parallel to $AR,BS$ through the midpoint $C$ of $AB$. The arcs $aA,bB$ are then equal and similar with respect to $CV$ — but unless $AO=BO$ (i.e., unless $AB\perp CV$), $CV$ does not split the original curve into two equal halves: the construction creates arcs $Ee$ on the other side that match $bB$, and arcs to match $Aa$ on the opposite side, etc. The iteration generates infinitely many similar arcs — impossible for an algebraic curve.

The exceptional case $AO=BO$ (§363): then $AB$ is perpendicular to $AR,BS,CV$, and the bisector $CV$ passes through $O$. The point $b$ coincides with $a$ (after reflection), and the curve is symmetric across $CV$ — meaning $CV$ is a diameter of the curve. This case reduces to the §361 diameter classification.

The two classifications are therefore exhaustive:

It follows that absolutely all algebraic curves with two or more equal parts fall under the classification given in this chapter. (source: chapter15, §363)

The dichotomy in summary

For an algebraic curve with two or more similar and equal parts:

Case	Characterization	Where to find the equation
At least one diameter	Reflective symmetry; equation involves only $\cos ns$ in polar form	n-diameters-by-cos-ns
Center but no diameter	Rotational symmetry only; equation involves both $\cos ns$ and $\sin ns$	equal-parts-without-diameter

The chapter has thus achieved a complete invariant-theoretic description of all algebraic curves with multiple-part symmetry. The structural ingredients — equal angles $\pi /n$ between diameters, $\cos ns$ as the invariant under reflection, $\sin ns$ as the rotational supplement — anticipate the modern theory of dihedral and cyclic group actions on the plane.

Comparison with the diameter / center theory of conics

Chapter 5 derived “diameter” and “center” purely from algebra — sum-of-roots on the quadratic-in-$y$ form of the conic. Chapter 15 derives the same notions from symmetry of the equation, generalized to all algebraic orders. The two routes converge for conics:

• Conic with one diameter (parabola) ↔ §338 form $Z=Z(x,y^2)$.
• Conic with two perpendicular diameters and a center (ellipse, hyperbola) ↔ §342 form $Z=Z(x^2,y^2)$.

Chapter 15 generalizes this to ”$n$ diameters and a center” for all $n$, and adds the ”$n$ equal parts and a center but no diameter” family. For $n\ge 3$, the curves are of order $\ge 3$, and the conic theory does not see them.

Figures

Figures 72–75

Figures 76–80

Related pages

• chapter-15-on-curves-with-one-or-several-diameters
• n-diameters-by-cos-ns — the diameter side of the dichotomy.
• equal-parts-without-diameter — the no-diameter side of the dichotomy.
• concurrence-of-diameters — the $\pi /n$ angle quantization, of which this is the equal-parts counterpart.
• line-curve-intersection-bound — the algebraic-curve constraint that rules out infinitely many equal parts.
• continuous-and-discontinuous-curves — the “law of continuity” Euler invokes in §361.