three-point-equation-from-fixed-point

Three-Point Equation From a Fixed Point

Summary: §§426–434: the chapter-17 framework promoted to three intersections per line through $C$. With $z^3-P{z}^2+Qz-R=0$ ($P,R$ odd and $Q$ even in $\sin \varphi ,\cos \varphi$), the rectangular form is $K-L+M-N=0$ with $K,L,M,N$ homogeneous of degrees $n+3,n+2,n+1,n$, and $C$ is a multiplicity-$n$ point of the order-$(n+3)$ curve. Symmetric-function constraints (constant $pq+pr+qr$, $p^2+q^2+r^2$, $p^4+q^4+r^4$) are solved as in ordinate-powers-from-fixed-point; §434 closes the chapter by noting the four-intersection problem requires nothing new.

Sources: chapter17 §§426–434.

Last updated: 2026-05-11.

---

The setup (§§426–427)

A line through $C$ meets the curve in three points $M,N,P$ iff their chord-distances $z_1,z_2,z_3$ are the roots of

$z^3-P{z}^2+Qz-R=0,$

where $P=p+q+r$, $Q=pq+pr+qr$, $R=pqr$ are functions of the angle $\varphi$. Parity:

• $P=$ sum, odd in $\sin \varphi ,\cos \varphi$.
• $Q=$ pairwise-product sum, even in $\sin \varphi ,\cos \varphi$.
• $R=$ triple product, odd in $\sin \varphi ,\cos \varphi$.

In rectangular coordinates, write

$P=\frac{Lz}{K},Q=\frac{M{z}^2}{K},R=\frac{N{z}^3}{K},$

with $K,L,M,N$ homogeneous polynomials in $x,y$ of degrees $n+3,n+2,n+1,n$. Substituting into the cubic and dividing by $z^3$:

$1-\frac{P}{z}+\frac{Q}{z^2}-\frac{R}{z^3}=0⟺\boxed{K-L+M-N=0.}$

This is the general equation of an order-$(n+3)$ curve cut by every straight line through $C$ in three points (besides $C$).

The multiplicity rule (§§426–427)

§427 observes:

• All third-order lines satisfy the equation when $C$ is distinct from the curve.
• All fourth-order lines satisfy it when $C$ lies on the curve.
• All fifth-order lines with a double point satisfy it when $C$ is the double point.
• In general, an order-$(n+3)$ curve satisfies the condition iff $C$ is a multiplicity-$n$ point.

This continues the multiplicity pattern from earlier sections:

Intersections	Order	Multiplicity of $C$
1	$n$	$n-1$
2	$n$	$n-2$
3	$n+3$	$n$

So all curves of order at most three automatically satisfy the three-intersection property (subject to $C$ being placed correctly), since the multiplicity requirement is satisfied trivially.

Symmetric-function constraints (§§428–433)

The same Vieta machinery as in ordinate-powers-from-fixed-point applies. Substituting $P=Lz/K$, etc., into a constant symmetric-function condition yields a constraint among $K,L,M,N$.

$p+q+r$ or $pqr$ constant: no solution (§428)

Both $P$ and $R$ are odd, so equating them to a constant means dividing by $z$ or $z^3$ — irrational. §428 notes: “there are no curves of the kind we are considering in which $p+q+r$ or $pqr$ are constant,” for the same reason §405 ruled out the analogous $CM+CN=$ const.

$pq+pr+qr=a^2$ constant (§428, end)

Since $Q=M{z}^2/K$ is even, setting $Q=a^2$ gives $M{z}^2/K=a^2$, i.e. $M(x^2+y^2)=a^{2}K$. Combined with $K-L+M-N=0$:

$M(x^2+y^2)-a^{2}L+a^{2}M-a^{2}N=0,$

or eliminating $M$,

$(x^2+y^2)K-(x^2+y^2)L+a^{2}K-(x^2+y^2)N=0.$

§430 specializes to the simplest such curve. With $n=0$ (so $N=b^3$ const, $K$ cubic, $L=0$, $M=K/(x^2+y^2)\cdot a^2/...$): take $K=x(x^2+y^2)$, $L=0$:

$2(x^2+y^2{)}^2{x}^2-a^2{x}^2(x^2+y^2{)}^2-2b^{3}x(x^2+y^2{)}^2=0,$

which divides by $2x(x^2+y^2{)}^2$ to give the third-order curve

$x(x^2+y^2)-\frac{1}{2}{a}^{2}x-b^3=0.$

A fourth-order alternative obtained from $L=2c(x^2+y^2)$:

$x^2(x^2+y^2)+(2c-x{)}^2(x^2+y^2)=a^2{x}^2+2b^{3}x.$

$p^2+q^2+r^2=a^2$ (§429)

Identity: $p^2+q^2+r^2=P^2-2Q$. Substituting and using $K-L+M-N=0$ to eliminate $M$:

$2(x^2+y^2)K^2-2(x^2+y^2)KL+(x^2+y^2)L^2-a^2{K}^2-2(x^2+y^2)KN=0.$

The highest-degree term is $2(x^2+y^2)K^2$ (degree $2n+8$); the lowest is $2(x^2+y^2)KN$ (degree $2n+5$).

$p^4+q^4+r^4=c^4$ (§§431–432)

Identity: $p^4+q^4+r^4=P^4-4P^{2}Q+2Q^2+4PR$. Substituting and solving for $N$:

$4K^{3}LN{z}^4=c^4{K}^4-z^4(L^4-4K{L}^{2}M+2K^2{M}^2),$

then $K-L+M-N=0$ closes the system. §432 handles the simultaneous conditions $p^2+q^2+r^2=a^2$ and $p^4+q^4+r^4=c^4$ via a coupled system, leading to a degree-eight relation that simplifies to

$8K^{3}L{z}^4-8K^3{L}^2{z}^4+4K{L}^3{z}^4-4a^2{K}^{3}L{z}^2-2c^4{K}^4-L^4{z}^4+2a^2{K}^2{L}^2{z}^2+a^4{K}^4=0.$

§433’s simplest example takes $K=z^2=x^2+y^2$ and $L=bx$, giving a seventh-order curve where $C$ is a quadruple point.

§434 — the four-intersection question

“We could proceed to the equation for curves which are intersected in four points by straight lines through the point $C$, and then to find particular such curves which satisfy certain conditions. However, if careful attention has been paid to the preceding arguments, there are no further difficulties to be encountered, and everything of this sort which could be desired can be found with very little labor. Except for the question about the existence of genuine solutions, the answers are immediate.”

The framework generalizes verbatim: $z^4-P{z}^3+Q{z}^2-Rz+S=0$ with the alternating parities, then a relation $K-L+M-N+O=0$ of homogeneous forms of degrees $n+4,\dots ,n$. The multiplicity at $C$ rises to $n$, and the order of the curve is $n+4$. Symmetric-function constraints work as before.

Three lineages converging

The chapter’s three multiplicity rules — $(n-1)$ for one intersection, $(n-2)$ for two, $n$ for three — fit a uniform pattern: a line through a $k$-fold point of an order-$N$ curve meets it in $N-k$ further points, so $N-k$ intersections requires $k=N-\text{(intersections)}$. The single-intersection case (curves-from-polar-coordinates) is the “maximal singularity” case, the conic-through-$C$ case (two-point-equation-from-fixed-point) is one less, and so on.

Related pages

• chapter-17-on-finding-curves-from-other-properties
• curves-from-polar-coordinates — one-intersection case.
• two-point-equation-from-fixed-point — two-intersection case.
• ordinate-powers-from-fixed-point — Vieta machinery shared.
• sum-of-ordinate-powers-curves — the §372 result reused.
• three-ordinate-curves — chapter 16 ordinate analogue of three-root Vieta.
• multiple-points-on-curves — multiplicity-$n$ point classification.