two-point-equation-from-fixed-point

Two-Point Equation From a Fixed Point

Summary: §§399–404: when every straight line through $C$ meets the curve in exactly two points $M,N$, the chord-distances $z=CM,CN$ are the roots of $z^2-Pz+Q=0$ with $P$ odd and $Q$ even in $\sin \varphi ,\cos \varphi$. The non-irrational rectangular form forces $P=Mz/L$ and $Q=N(x^2+y^2)/L$, giving the general equation $L-M+N=0$ with $L,M,N$ homogeneous of degrees $n+2,n+1,n$ — an order-$n$ curve with a multiplicity-$(n-2)$ point at $C$.

Sources: chapter17 §§399–404.

Last updated: 2026-05-11.

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The setup (§§399–400)

A line through $C$ at angle $\varphi$ meets the curve in two points $M,N$ iff the quadratic

$z^2-Pz+Q=0$

has two roots, where $P=CM+CN$ and $Q=CM\cdot CN$ are functions of $\varphi$. The 180° rotation $\varphi \mapsto \varphi +\pi$ sends $\sin \varphi ,\cos \varphi \mapsto -\sin \varphi ,-\cos \varphi$ and swaps which point is "$M$" vs "$N$", so:

• $P$ (the sum) must change sign under the flip: $P$ is odd in $\sin \varphi ,\cos \varphi$.
• $Q$ (the product) must be preserved under the flip: $Q$ is even in $\sin \varphi ,\cos \varphi$.

In rectangular coordinates (using $x=z\cos \varphi$, $y=z\sin \varphi$), this forces $P/z$ to be a non-irrational homogeneous function of degree $-1$ and $Q/z^2$ to be non-irrational of degree $-2$ in $x,y$.

The $L-M+N=0$ form (§400)

Let $L$ be a homogeneous polynomial of degree $n+2$, $M$ of degree $n+1$, $N$ of degree $n$, all in $x,y$. Then

$P=\frac{Mz}{L},Q=\frac{N{z}^2}{L}=\frac{N(x^2+y^2)}{L}$

are of the required types. Substituting into $z^2-Pz+Q=0$ and dividing by $z^2$:

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

This is the general equation for an order-$n$ curve cut by every line through $C$ in exactly two points (counting only the points other than $C$).

The conic case (§§401–402)

Letting $n=0$ (so $N=$ const, $M$ linear, $L$ quadratic in $x,y$) recovers

$\alpha x^2+\beta xy+\gamma y^2-\delta x-ϵy+\zeta =0,$

i.e., the general conic, with $C$ taken anywhere — not on the curve. Lines through $C$ meet the conic in two points or zero points; the “in one point” exception is two lines (asymptote directions for a hyperbola, axis direction for a parabola) that meet at infinity. §402 makes the algebraic structure explicit by transforming to polar form: the coefficient of $z^2$ vanishes when

$\alpha +\beta \tan \varphi +\gamma {\tan }^2\varphi =0,$

whose two roots are precisely the asymptote directions of the conic.

The multiplicity rule (§§403–404)

§403 takes $n=1$ in $L-M+N=0$, getting

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

When $x=0$, also $y=0$, so this cubic passes through $C$. (Cubics through $C$ are met by every line through $C$ in 2 additional points.)

For order 4 ($n=2$): the homogeneous-of-degree-4 + degree-3 + degree-2 structure requires $C$ to be a double point. For order 5: $C$ must be a triple point. In general:

A curve of order $n$ is cut by every straight line through $C$ in exactly two points (besides $C$ itself) iff $C$ is a multiplicity-$(n-2)$ point of the curve.

§404 frames this via the line-curve-intersection-bound: a line meets an order-$n$ curve in $n$ points counted with multiplicity, complex intersections, and intersections at infinity included. If $C$ contributes $n-2$ of these (as a multiple point), exactly 2 remain.

Connection to the one-point case

This generalizes curves-from-polar-coordinates one step: $z=P$ (one intersection) was the case where $C$ had multiplicity $n-1$; here $C$ has multiplicity $n-2$, leaving two intersections free. The chapter will later (§§426–427) push to three intersections by taking $C$ as a multiplicity-$n$ point of an order-$(n+3)$ curve, see three-point-equation-from-fixed-point.

Why $L,M,N$ have those particular degrees

The condition that $P/z$ be homogeneous of degree $-1$ picks out the ratio of a polynomial of degree $n+1$ to one of degree $n+2$. Similarly $Q/z^2$ being homogeneous of degree $-2$ requires a degree-$n$ numerator over the same degree-$n+2$ denominator. The shared $L$ is what makes the combined equation $L-M+N=0$ purely polynomial.

Related pages

• chapter-17-on-finding-curves-from-other-properties
• curves-from-polar-coordinates — one-intersection precursor.
• sum-and-product-of-chords-from-point — constant sum / constant product applied here.
• constant-chord-interval-conchoid — uses this framework for $(CN-CM{)}^2$ constant.
• multiple-points-on-curves — Euler’s local theory of multiple points, here used at $C$.
• line-curve-intersection-bound.