diameter-and-center-from-equation

Diameter and Center from the Equation

Summary: Three reflective / point-symmetry conditions read directly off the parity of exponents in the curve equation $Z=0$ (§§337–343). Even powers of $y$ only ($Z=Z(x,y^2)$) → the abscissa axis $AB$ is an orthogonal diameter, dividing the curve into two equal parts (§338). Even powers of $x$ only ($Z=Z(x^2,y)$) → $EF$ is a diameter (§339). Substitution $(x,y)\mapsto (-x,-y)$ leaving the equation invariant → a center $C$, where every line through $C$ is bisected by $C$; the equation is a sum of homogeneous parts all of even degree (curve has diameters) or all of odd degree (curve has only a center) (§§340–341). Even powers of both $x$ and $y$ ($Z=Z(x^2,y^2)$) → two perpendicular diameters $AB\perp EF$, and the curve belongs to an even order (§§342–343).

Sources: chapter15 (§§336–343); figure 68 in figures68-71.

Last updated: 2026-05-11.

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Setup: four quadrants split by two perpendicular lines (§337, figure 68)

Take rectangular coordinates $(x,y)$ with the four quadrants labelled $Q,R,S,T$ — formed by the two perpendicular lines $AB$ (horizontal, the abscissa axis) and $EF$ (vertical), intersecting at $C$. Sign conventions:

Quadrant	$x$	$y$
$Q$	$+$	$+$
$R$	$+$	$-$
$S$	$-$	$+$
$T$	$-$	$-$

The three cases below ask which substitution leaves the equation $Z(x,y)=0$ unchanged.

Diameter $AB$: only even powers of $y$ (§338)

The portions of the curve in quadrants $Q$ and $R$ are equal and similar (reflections across $AB$) iff $Z(x,y)=Z(x,-y)$ — i.e., $Z$ contains no odd powers of $y$. Equivalently $Z$ is a “non-irrational function of $x$ and $y^2$.” Then $AB$ is a diameter: every chord perpendicular to $AB$ is bisected by $AB$. The general equation of this kind is

$0=\alpha +\beta x+\gamma x^2+\delta y^2+ϵx^3+\zeta x{y}^2+\eta x^4+\theta x^2{y}^2+\iota y^4+\cdots .$

Note that the same symmetry makes the $S\leftrightarrow T$ pair also equal — reflection across $AB$ acts simultaneously on both sides of $EF$.

Diameter $EF$: only even powers of $x$ (§339)

Symmetrically, $Q$ and $S$ are equal iff $Z(x,y)=Z(-x,y)$ — i.e., $Z$ is a non-irrational function of $x^2$ and $y$. Then $EF$ is the diameter, and the general equation has the form

$0=\alpha +\beta y+\gamma x^2+\delta y^2+\zeta y^3+\eta x^4+\theta x^2{y}^2+\iota y^4+\cdots .$

Center $C$: invariance under $(x,y)\mapsto (-x,-y)$ (§340)

The portions in opposite quadrants $Q$ and $T$ are equal iff $Z(x,y)=Z(-x,-y)$. Let $Z={\sum }_k{Z}_k$ be the decomposition into homogeneous parts. Substituting $(x,y)\mapsto (-x,-y)$ multiplies each $Z_k$ by $(-1{)}^k$. The invariance condition $Z=Z(-x,-y)$ holds in two distinct ways:

1. All homogeneous parts have even degree. Then every $Z_k$ is unchanged, and $Z=Z(-x,-y)$ identically. The general equation: $0=\alpha +\beta x^2+\gamma xy+\delta y^2+ϵx^4+\zeta x^{3}y+\eta x^2{y}^2+\theta x{y}^3+\iota y^4+\kappa x^6+\cdots .$
2. All homogeneous parts have odd degree. Then each $Z_k\mapsto -Z_k$, so $Z\mapsto -Z$, and the equation $Z=0$ is still satisfied — although $Z$ itself is not invariant, the zero set of $Z$ is. The general equation: $0=\alpha x+\beta y+\gamma x^3+\delta x^{2}y+ϵx{y}^2+\zeta y^3+\eta x^5+\theta x^{4}y+\iota x^3{y}^2+\cdots .$

A mixed-parity $Z$ would not have the symmetry, since the even- and odd-degree pieces transform differently.

Diametrically equal vs. alternately equal (§341)

Euler distinguishes two ways a curve can have two equal parts:

Kind	Geometric description	Symmetry	Read off from
Diametrically equal	parts on opposite sides of a straight line; every orthogonal chord bisected by that line	reflection across a diameter	even-$y$ or even-$x$ exponents (cases above)
Alternately equal	parts in opposite quadrants $Q$/$T$ or $R$/$S$; every straight line through $C$ divides the curve into two parts that are alternately equal	$180°$ rotation about $C$	the §340 form (only odd-degree homogeneous parts)

The first kind is the classical diameter. The second kind is a new feature — the curve has a center $C$, but no axis of reflection. The simplest example is the cubic $y=x^3$: rotating $180°$ about the origin sends the curve to itself, but no line of reflection works.

When all parts in opposite quadrants are equal and the curve is built only of even-degree homogeneous parts, the curve has both a diameter and a center.

Two perpendicular diameters (§§342–343)

If $Z$ has even exponents on both $x$ and $y$ — i.e., $Z=Z(x^2,y^2)$ — then both substitutions $y\mapsto -y$ and $x\mapsto -x$ leave $Z$ unchanged. Both $AB$ and $EF$ are diameters, and they are mutually perpendicular. The general equation has the form

$0=\alpha +\beta x^2+\gamma y^2+\delta x^4+ϵx^2{y}^2+\zeta y^4+\eta x^6+\theta x^4{y}^2+\cdots .$

Even order only. Because every term has even total degree, the equation is of degree $2,4,6,\dots$ — never odd. Consequently no curve of odd order can have two mutually perpendicular diameters.

Center as a consequence. Two perpendicular diameters force a center $C$ at their intersection, since the §342 form is a special case of the §340 (all-even-degree) form. Hence in this case all four quadrants $Q,R,S,T$ carry equal and similar portions of the curve.

Such curves include the ellipse and hyperbola (order 2), most quartics with both reflective symmetries, and the higher-order generalizations developed in n-diameters-by-cos-ns.

Relation to the chapter-13 / chapter-12 picture

The symmetries here are global — they relate the curve to itself across all of the plane, not to a single point on the curve. Compare:

• multiple-points-on-curves: a singular point is a local symmetry (the curve passes through one point in two or more directions).
• center-of-conic: every conic with two diameters automatically has a center. The chapter 5 derivation came from sum-of-roots on the quadratic-in-$y$; the chapter 15 derivation comes from substitution invariance. Both reach the same conclusion.
• diameter-and-center-of-cubic: third-order curves have a “sum-preserving” diameter but only sometimes a center (when an algebraic condition on the coefficients holds). The chapter 15 framework restricts attention to orthogonal diameters — the genuine bisecting kind — and asks when an equation in standard rectangular coordinates already has them.

Figures

Figures 68–71

Related pages

• chapter-15-on-curves-with-one-or-several-diameters
• diameter-of-conic — the §90 prototype: a diameter as bisecting locus of parallel chords.
• center-of-conic — every conic with two diameters has a center.
• concurrence-of-diameters — what happens with more than two diameters.
• n-diameters-by-cos-ns — extending these symmetries to $n$ diameters at angle $\pi /n$.