General Construction by Parabola and Conic

Summary: A systematic method for constructing any equation of any degree as the intersection of two algebraic curves (§§499–504). Choose the first curve in the form $P(z)+Q(z)y=0$, so that $y=-P/Q$ is single-valued in $z$ and no complex intersections arise. Then the second curve is obtained by substituting this $y$ into the target equation $\Phi (z)=0$ and rearranging. The standard worked case takes $P(z)+Q(z)y=0$ to be the parabola $ay=z^2+bz$; the matched second curve is a general conic, and the discriminant $A^2-4B-4ac$ selects hyperbola, parabola, or ellipse — circle when $b=A/2$ and a further condition holds.

Sources: chapter20 (§§499–504).

Last updated: 2026-05-12.

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The single-valued-first-curve principle (§499)

A real root $z_0$ of the target equation produces a real intersection of the two construction curves $F=0$ and $G=0$ iff the recovered ordinate $y_0$ is real. The cleanest way to guarantee this is to choose the first curve so that $y$ is a single-valued rational function of $z$:

$P(z)+Q(z)y=0⟹y=-\frac{P(z)}{Q(z)}.$

For such a curve, every real $z$ produces a real $y$. So no matter what the second curve looks like — even if it has many branches and complex ordinates of its own — the intersection at every real root of the eliminant is real. The “shadow” of complex intersection in chapter 19 is removed.

Deriving the second curve (§500)

Given the target equation $\Phi (z)=0$ of degree $N$, the recipe is:

1. Pick $P(z),Q(z)$ defining the first curve $P+Qy=0$ (a curve of order $\max (\mathrm{deg}P,\mathrm{deg}Q+1)$ at most).
2. Substitute $y=-P/Q$ into $\Phi (z)=0$ and clear denominators.
3. The resulting polynomial in $z$ and $y$ is the equation of the second curve. Its intersections with the first curve are exactly the roots of $\Phi$.

The order of the second curve is fixed by $\mathrm{deg}\Phi$ and the choice of $P,Q$.

Worked case: quartic via parabola and conic (§500)

Target equation:

$z^4+A{z}^3+B{z}^2+Cz+D=0.$

First curve — parabola $ay=z^2+bz$ (rewritten $z^2=ay-bz$). This is order 2, but $y$ is single-valued in $z$.

Second curve — substitute $z^2=ay-bz$ into the quartic:

• $z^4=(ay-bz{)}^2=a^2{y}^2-2abyz+b^2{z}^2=a^2{y}^2-2abyz+b^2(ay-bz)$
• $A{z}^3=Az\cdot z^2=Az(ay-bz)=Aayz-Ab{z}^2=Aayz-Ab(ay-bz)$
• $B{z}^2=B(ay-bz)$

Substituting and collecting,

$\boxed{a^2{y}^2+a(A-2b)yz+(B-Ab+b^2)z^2+Cz+D=0.}$

This is the general second-order curve in $(y,z)$, save for the missing $y$-term (no $\delta y$). The parameters $a,b$ remain free, giving an infinite family of valid second curves.

Even greater variety (§501)

The first curve can be multiplied by extra terms before substitution. Replace $z^2=ay-bz$ by $c{z}^2=acy-bcz$ (multiplying by $c$) and add this to the quartic before substituting in:

$a^2{y}^2+a(A-2b)yz+(B-Ab+b^2+ac)z^2-a^{2}cy+(C+abc)z+D=0.$

Three free constants $a,b,c$ instead of two; the missing $y$-term reappears.

Selecting the conic type (§502)

The conic-type discriminant of the general second-order curve in $(y,z)$ is the standard $A^2-4B-4ac$ (the cross-term squared minus four times the product of the diagonal coefficients):

• $A^2-4B-4ac>0$: hyperbola
• $A^2-4B-4ac=0$: parabola
• $A^2-4B-4ac<0$: ellipse

(The trichotomy is the §131 [[classification-of-conics|sign of $\gamma$]] applied to the second curve.)

Circle subcase: a conic is a circle iff its $y^2$ and $z^2$ coefficients are equal and the cross-term vanishes. The latter requires $b=A/2$; the former, $a^2=B-Ab+b^2+ac$ — i.e. $c=a+A^2/(4a)-B/a$. With this choice the second curve becomes a circle whose center and radius can be read off:

$(y-\frac{a}{2}-\frac{A^2}{8a}+\frac{B}{2a}{)}^2+(z+\frac{C}{2a^2}+\frac{A}{4}+\frac{A^3}{16a^2}-\frac{AB}{4a^2}{)}^2={\text{(explicit radius}}^2\text{)}.$

This recovers the circle-and-parabola construction of §§493–495 as a special case of the general method.

Higher degrees via higher parabolic lines (§503)

For an equation of degree $N$ where $N$ has only awkward factorizations, take the first curve to be a higher-order parabolic line $y=P(z)$:

$y=P(z)\text{with }P\text{ a polynomial of degree }d.$

This is order $d$ but still single-valued. Substituting into the target equation reduces it to a polynomial in $z$ and $y$ of degree $N/d$ (rounded up), defining the second curve.

Example. $z^{12}-f^{10}{z}^2+f^{9}gz-g^{12}=0$ (degree 12). Take the quartic parabola $z^4=a^{3}y$; then $z^{12}=a^9{y}^3$, and substitution gives the third-order curve

$a^9{y}^3-f^{10}{z}^2+f^{9}gz-g^{12}=0.$

Order $4\cdot 3=12$ matches the target.

Multiplying by powers of $z$ (§504)

If the natural construction (parabola + matching conic) misses the target — e.g., a cubic produces a hyperbola which is less convenient than a circle/parabola — multiply the target by a power of $z$ to add easily-recognized extra roots.

Cubic example. $z^3+A{z}^2+Bz+C=0$.

• Substitution $z^2=ay$ alone gives the hyperbola $azy+Aay+Bz+C=0$. Always a hyperbola — fine but inflexible.
• Multiplying the cubic by $z$ gives the biquadratic $z^4+A{z}^3+B{z}^2+Cz=0$ (with $D=0$). Apply §§493–495 to construct this as a circle-and-parabola intersection. The extra root $z=0$ from the multiplication is the origin and is trivially recognized; the three actual roots are the remaining intersection abscissas.

This is the technique tying the cubic Bäcker’s-rule construction back to a circle-and-parabola figure.

Caveat on real-arc coverage (§505)

Even when the construction has no complex intersections, a real arc of the second curve may not span the full range of abscissas where the eliminant has real roots. Some real roots can then go unrealized — the corresponding intersection lies on a “different branch” of the curve, or at infinity, or on the discarded conjugate of the first curve. Euler dismisses this as more curious than useful: when one chooses the first curve in $P+Qy=0$ form, the worry rarely materializes.

Related pages

• construction-of-equations
• chapter-20-on-the-construction-of-equations
• biquadratic-by-circle-and-parabola
• complex-intersections
• classification-of-conics
• intersection-product-degree-bound
• parabola