Polar coordinates and conic sections

A point in the plane can be named by a pair $(r,\phi )$ instead of a Cartesian pair $(x,y)$. This math handbook measures $\phi$ in radians, with $0\le \phi <2\pi$. Cartesian axes sit with lines in the plane.

Polar coordinates

Distance from a chosen origin is $r$. Direction from the positive $x$-axis is $\phi$. That is all this board is:

$x=r\cos \phi ,y=r\sin \phi ,0\le \phi <2\pi ,$

and the other way,

$r=\sqrt{x^2+y^2},\tan \phi =\frac{y}{x}.$

Drag $P$. The labeled segment is $r$; the marked angle is $\phi$. The next boards reuse this pair — they only move the origin onto a focus.

One family on a cone

An ellipse, a parabola, a hyperbola, and a circle look like four topics. They are four ways a plane can cut a double cone (two nappes meeting at a tip).

• Regular conic sections: circle, ellipse, hyperbola, parabola.
• Degenerate conic sections: two lines, one line, or a point — the plane through the apex.

Turn the figure. This plane meets the upper nappe in a closed curve: an ellipse. The other regular cuts are a different tilt of the same plane. Cartesian equations of those four live on circles in the plane, ellipses in the plane, hyperbolas in the plane, and parabolas in the plane. In the plane they share one rule, next.

The ratio $r/d=\epsilon$

Fix a point $B$ (a focal point) and a line $L$ (the directrix). A point $P$ lies on a regular conic when the ratio of distances

$\frac{r}{d}=\epsilon$

is constant. Here $r$ is the distance $BP$, and $d$ is the distance from $P$ to $L$. The constant $\epsilon$ is the numerical eccentricity. It is the shape:

• $\epsilon >1$ — hyperbola
• $\epsilon =1$ — parabola
• $0\le \epsilon <1$ — ellipse
• $\epsilon =0$ — circle (the directrix has gone to infinity, so $d=\infty$)

The four boards below are that one rule, not four new topics. $r$ and $\phi$ are the polar pair from the first board, with the origin now at a focus. Drag $P$ on each: $r$ and $d$ both change, but their ratio does not. That frozen ratio is $\epsilon$. On the circle there is no finite $L$. Apollonius of Perga (c. 260–190 BCE) named the three non-circular cases from those inequalities: hyperbola “excess,” parabola “equality,” ellipse “deficiency.”

Sit at the focus

Sit at $B$ and reuse the first board's language. Then the $r$ in $r/d=\epsilon$ is exactly the polar $r$, and $\phi$ is the angle at $B$ — the same marks already on the four boards above. For a directrix perpendicular to the axis, $d$ is linear in $r\cos \phi$. The constant ratio rearranges to one equation for all four shapes:

$r=\frac{p}{1-\epsilon \cos \phi }.$

The half-parameter $p$ is a length (size). $\epsilon$ is the same eccentricity as in the table (shape). Choose $\phi$; the formula returns the $r$ that lands on the conic.

The board starts at $p=2$, $\epsilon =0.6$: an ellipse. $B$ is a focus, so it is not the center — the oval sits to the right of the origin. $L$ is the same kind of directrix as above, now at $x=-p/\epsilon$, so $d$ is still visible. Drag $P$ to walk one pair $(r,\phi )$. Drag $\epsilon$: the same focus and the same formula pass through the four cases you just dragged.

The half-parameter is also a radius

The $p$ in that formula is not only size. At a vertex $S$ one can draw a circle that meets the conic there and shares the tangent: the vertical circle (the table calls it an inscribed circle). Its radius $R$ equals $p$. The same circle exists at an arbitrary point $P=(x_0,y_0)$, with radius $R_0$. Curvature at $P$ is

$K=\frac{1}{R_0}.$

This is how fast the curve you already have turns, not a new definition of the conic. The three boards are the ellipse, hyperbola, and parabola from the table. Drag $P$; the smaller circle stays at $S$.