curves-from-polar-coordinates

Curves from Polar Coordinates at a Fixed Point

Summary: Euler’s polar setup for chapter 17: at a fixed point $C$, write $z=CM$ and $\varphi =\angle ACM$ (figure 81); convert to rectangular coordinates via $y/z=\sin \varphi$, $x/z=\cos \varphi$, $z=\sqrt{x^2+y^2}$. Curves cut by every line through $C$ in exactly one point are characterized by $z=P$ with $P$ odd in $\sin \varphi$ and $\cos \varphi$ — equivalently, an order-$n$ curve with a multiplicity-$(n-1)$ point at $C$.

Sources: chapter17 §§391–398; figures 81–82 in figures81-85.

Last updated: 2026-05-11.

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The polar frame (§§391–392)

The chapter-16 setup expressed a curve through its parallel ordinates $PM,PN$ at varying abscissa $z=CP$. That framework cannot directly state properties of straight lines through a fixed point $C$, such as ”$CM\cdot CN$ is constant” — those properties are radial, not vertical. So Euler reorients:

• Pick a fixed point $C$ and a fixed direction $CA$ as polar axis (figure 82).
• For a point $M$ on the curve, set $z=CM$ (the chord-distance) and $\varphi =\angle ACM$ (the polar angle).
• Every angle $\varphi$ picks out a chord $CM$, and the equation $z=$ (function of $\varphi$) determines the curve.

Conversion to rectangular coordinates, with $CP$ the projection of $CM$ onto $CA$ and $PM$ the perpendicular: $CP=z\cos \varphi =x$, $PM=z\sin \varphi =y$, so

$\frac{x}{z}=\cos \varphi ,\frac{y}{z}=\sin \varphi ,z=\sqrt{x^2+y^2}.$

The one-intersection parity rule (§§393–394)

If $z=P$ where $P$ depends only on $\sin \varphi$, the same straight line through $C$ corresponds to two angles, $\varphi$ and $\varphi +\pi$, since rotating by two right angles puts the line in the same position (only the direction flips). Under that flip $\sin \varphi \mapsto -\sin \varphi$. If $P(\sin \varphi )=Q$, then $P(-\sin \varphi )=Q^{\prime }$ and $Cm=Q^{\prime }$ — a second intersection of the curve with the same line, unless $Q^{\prime }=-Q$, i.e., unless $P$ is odd in $\sin \varphi$.

The same flip sends $\cos \varphi \mapsto -\cos \varphi$. So if $P$ depends on both, the single-intersection condition is that $P$ be odd in both arguments simultaneously: $P(-\sin \varphi ,-\cos \varphi )=-P(\sin \varphi ,\cos \varphi )$.

Rectangular form: $P$ odd in $x/z$ and $y/z$ (§§395–397)

Because $x/z$ and $y/z$ are odd in $\varphi \mapsto \varphi +\pi$ (both flip sign), $P$ being odd in $\sin \varphi ,\cos \varphi$ is the same as $P$ being odd in $x/z,y/z$. Euler tabulates the lowest cases:

$z=\frac{\alpha x}{z}+\frac{\beta y}{z}+\frac{\gamma z}{x}+\frac{\delta z}{y}$

and proceeding to higher powers,

$z=\frac{\alpha x}{z}+\frac{\beta y}{z}+\frac{\gamma z}{x}+\frac{\delta z}{y}+\frac{ϵx^3}{z^3}+\frac{\zeta x^{2}y}{z^3}+\frac{\eta x{y}^2}{z^3}+\frac{\theta y^3}{z^3}+\frac{\iota x^2}{yz}+\frac{\kappa y^2}{xz}+\frac{\lambda yz}{x^2}+\cdots .$

§396 observes: dividing by $z$ leaves only even powers of $z$, which $z^2=x^2+y^2$ then eliminates. So every such equation reduces to a non-irrational rectangular equation of degree $-1$ in $x,y$.

§397 packages this into the order-$n$ master form

$\alpha x^{n+1}+\beta x^{n}y+\gamma x^{n-1}{y}^2+\delta x^{n-2}{y}^3+ϵx^{n-3}{y}^4+\cdots =c(A{x}^n+B{x}^{n-1}y+C{x}^{n-2}{y}^2+D{x}^{n-3}{y}^3+\cdots ).$

The left side is homogeneous of degree $n+1$; the right side is $c$ times a homogeneous form of degree $n$.

The order-by-order catalogue (§§397–398)

Order	Equation	Geometry
I	$\alpha x+\beta y=c$	straight line
II	$\alpha x^2+\beta xy+\gamma y^2=c(Ax+By)$	conic through $C$
III	$\alpha x^3+\beta x^{2}y+\gamma x{y}^2+\delta y^3=c(A{x}^2+Bxy+C{y}^2)$	cubic with double point at $C$
IV	$\alpha x^4+\beta x^{3}y+\cdots +ϵy^4=c(A{x}^3+B{x}^{2}y+Cx{y}^2+D{y}^3)$	quartic with triple point at $C$

Reading the pattern: the homogeneous-of-degree-$n+1$ left side and homogeneous-of-degree-$n$ right side meet at $C=(0,0)$ to order $n$, so $C$ is a multiplicity-$(n-1)$ point of the order-$n$ curve. By the line-curve-intersection-bound, a line through a $(n-1)$-fold point of an order-$n$ curve meets the rest of the curve in exactly $1$ further point — recovering the single-intersection property.

Reading: conic example

For order II, fix $C$ on the conic. Every line through $C$ meets the conic at $C$ itself and at one other point — exactly the lemma underlying many of Newton’s Principia constructions. The right side $c(Ax+By)$ is the tangent direction of the conic at $C$ generalized to a variable scale.

Figures

Figures 81–85

Related pages

• chapter-17-on-finding-curves-from-other-properties
• two-point-equation-from-fixed-point — the immediate generalization to two-intersection lines.
• multiple-points-on-curves — Euler’s local theory of singular points, here applied at $C$.
• line-curve-intersection-bound — the multiplicity-$(n-1)$ result rests on this.
• chord-rectangle-property — the conic precursor in chapter 5.