The polar with a straightedge

There is a small family of constructions on this site that use no measurement at all. They draw lines through pairs of points and mark where lines cross, and nothing else. That restriction is not asceticism: it is the exact condition for a construction to survive a projection, and therefore the exact condition for it to be carried out on a photograph of a thing rather than on the thing.

The polar of a point with respect to a conic is the most useful member of that family, and this essay is about doing it with a ruler.

The construction #

Take a conic and a point $P$ not on it.

Draw any two lines through $P$ that cut the conic. The first meets it at $A$ and $B$; the second at $C$ and $D$.

Those four points form a complete quadrangle. A complete quadrangle is four points and the six lines joining them in pairs; the six lines meet in three diagonal points, and one of them is $P$ itself, being $AB \cap CD$.

The other two diagonal points are $AC \cap BD$ and $AD \cap BC$. Mark them.

The line through those two is the polar of $P$.

That is all of it. Six lines drawn, three crossings marked, one line drawn through two of them. A child with a ruler can do it and it is exact.

Why it is worth having #

The algebraic definition is shorter still: writing the conic as a symmetric matrix $C$, the polar of $P$ is the line $C\,P$. One matrix product.

So why construct it?

Because the matrix product needs the conic’s coefficients in some coordinate system, and the construction needs nothing at all. Handed a photograph with a conic in it and a point marked, the construction gives the polar of that point in the picture — and because every step is an incidence, and incidences survive projection, that line is the image of the polar in the world.

Which means the construction transports a fact about the world through a picture whose camera is unknown, unrecorded and unrecoverable. That is the property the whole foundations field is built to collect.

Checking it, and checking that the check means something #

The construction and the matrix product are two independent routes to one line, so they can be compared. Normalised so that the coefficient pair $(a,b)$ has unit length — a line being defined only up to scale and sign — they agree to about $4\times10^{-13}$.

That number on its own is worth very little. Two lines built from four points on one conic will tend to be near each other whatever the method, and a construction that quietly returned some fixed line would agree with itself perfectly.

So the test has a second half. The secants are varied. Three unrelated pairs of chords through the same point, three complete quadrangles with no vertices in common, three constructed lines — and all three land on the algebraic polar to the same tolerance.

And a third half, which is the control. Move the point and the line has to move. Shifting $P$ by a few dozen pixels moves the polar by tens of pixels on the same normalised scale — fifteen orders of magnitude more than the disagreement between the two methods. Without that, a routine insensitive to its input would pass everything above.

That is the pattern every claim here is asked to have: the thing that should agree, the freedom it should be indifferent to, and the parameter it must not be indifferent to.

What the polar of a point is #

The construction is exact before it is interpreted, but the interpretations are what make it useful, and there are four worth carrying.

It is the harmonic partner. Take any line through $P$ cutting the conic at $X$ and $Y$. It meets the polar at one further point $Q$, and $(X, Y; P, Q)$ is harmonic — cross-ratio exactly $-1$. That holds for every line through $P$, which is the strongest way to state what the polar is, and it makes the connection to the diagonal-point constructions exact rather than analogical.

It is the chord of contact. If $P$ is outside the conic, the two tangents from it touch at two points, and the polar is the line through them. So the construction hands over the tangents for free — draw the polar, mark where it cuts the curve, join those to $P$.

It is a duality. The map “point to its polar” is a bijection between points and lines, and it reverses incidence: if $P$ is on the polar of $Q$, then $Q$ is on the polar of $P$. That single sentence is what makes conics the natural home of duality on this site, and it is what the essay on four lines and their cross-ratio is quietly using.

And it is a fixed structure. The map that sends every point to the harmonic conjugate of itself across the conic — the polarity — is an involution, and it is the fixed structure that distinguishes conic-based constructions from the homologies and elations the plane-map census sorts.

Where the construction stops #

A construction is only as trustworthy as its refusals, and this one has three that are genuinely different in kind. Each is paired below with a case one step away that must still work, because a routine that refused everything would satisfy every refusal test ever written.

The centre. The polar of a conic’s centre is the line at infinity. That is not a failure of the construction — the quadrangle still builds, and the two diagonal points still exist — but the line through them is at infinity, which no normalisation can express as a finite line. The machinery here refuses rather than returning a very large number, and a point a thousandth of a unit off the centre has a perfectly good polar, which is asserted beside it.

Inside the conic. A point inside has a polar; it simply misses the curve, so there are no real tangents to hand over. The construction still works — the quadrangle needs the secants to cut, and every line through an interior point does — and what fails is the chord-of-contact reading. This is a refusal from the geometry, not a large residual, and it is the third time this site has had to record that distinction.

A secant that misses. From a point far outside the conic, most lines miss it entirely. Then there are no four points, no quadrangle, and nothing to construct. The failure is complete rather than degraded, which is the good kind: a solver that returned two complex intersections and carried on would produce a confident line from a construction that never happened.

The practical consequence for a reader working on a photograph is the third one: aim the secants at the curve, rather than drawing them in fixed directions. From an exterior point a randomly chosen direction usually misses, and aiming at a point known to be on the curve cannot.

The centre of a circle, from its photograph #

The polarity is what settles a question this field has already asked twice, and the answer is worth having in one line.

The image of a circle’s centre is not the centre of the image. What is it? It is the pole of the horizon.

The circle’s centre is the pole of the plane’s line at infinity with respect to the circle. A projection carries poles to poles, so the image of the centre is the pole of the image of that line — which is the horizon. So: find the horizon, construct its pole with respect to the drawn conic, and that point is where the circle’s centre went.

And the pole of a line is constructed with the same straightedge, dually: take two points on the line, build each one’s polar, and the two polars cross at the pole.

The identity that fell out of a tilted plane #

There is a place on this site where a pole–polar relation turns up in a field with no conics drawn in it, and it is worth naming here because this essay is the machinery it was using.

Tilt a camera’s picture plane and two things move: the horizon drops from the principal point by $f\tan\theta$, and the vertical vanishing point arrives from infinity to $f/\tan\theta$. Their product is $f^2$, at every tilt.

That is a polar relation. The horizon is the polar of the vertical direction’s vanishing point with respect to the absolute conic, and “the product of the two offsets is $f^2$” is what a polar relation looks like when both points are on the principal axis. It is also a one-line calibration from two things a straightedge finds in a photograph — which is this essay’s whole argument arriving somewhere it was not expected.

The dual construction, for a line #

Duality means every statement here has a partner, and the partner is the one that finds a pole from a line — which is what the circle-centre problem actually needs.

Given a line $\ell$, pick two points on it. Build each point’s polar with the quadrangle construction above. The two polars cross at one point, and that point is the pole of $\ell$.

The proof is the incidence reversal in one line: if $X$ is on $\ell$, then the pole of $\ell$ is on the polar of $X$; that is true for both chosen points, so the pole is on both polars, so it is their crossing.

Which makes the whole apparatus symmetric and equally cheap in either direction. Points to lines, lines to points, six drawn lines each time, and nothing measured.

The economy of incidence #

It is worth being explicit about what class of statement this construction belongs to, because that class is small and everything in it is unusually durable.

A construction made only of joins (the line through two points) and meets (the point where two lines cross) is a projective construction. It commutes with every projectivity, so it can be done before or after a projection with the same result. There is no approximation involved: the result is not nearly the same, it is the same.

Everything else fails. A midpoint is not projective, so bisecting is out. An angle is not projective, so a perpendicular is out. A circle is not projective, so a compass is out. What survives is a ruler, and the surprising thing is how much can be done with one — the harmonic conjugate, the polar, the tangents, Desargues’ configuration, the repeated bay in a perspective pavement, the fourth point of a harmonic range.

That list is the reason the classical treatments spend so long on the complete quadrangle. It is not a curiosity: it is the only tool that works on a picture.

Why the quadrangle produces the polar #

It is a short argument and it is worth having, because it shows that the construction is the harmonic property rather than a trick that happens to agree with it.

Let the two secants through $P$ meet the conic at $A, B$ and $C, D$, and let $E = AC \cap BD$ and $F = AD \cap BC$. Consider the line $EF$ and where it meets the secant $AB$ — call that point $Q$.

The four points $A$, $B$, $P$, $Q$ lie on one line, and $E$, $F$ are two diagonal points of the quadrangle $ACBD$ with $P$ the third. The classical harmonic property of a complete quadrangle says exactly this: on any line through one diagonal point, the two vertices it passes through and the two points where the other diagonal meets it form a harmonic set. So $(A, B; P, Q) = -1$.

The same argument on the other secant gives $(C, D; P, R) = -1$, where $R$ is where $EF$ meets $CD$. So $EF$ contains the harmonic conjugate of $P$ on both secants — and since the polar is defined as the locus of those conjugates, and two points determine a line, $EF$ is the polar.

Two things follow that the drawing makes vivid. The construction is indifferent to which secants were used because the harmonic conjugate on each is determined by the conic and $P$ alone. And the whole argument is a statement about cross-ratios, which is why the projection that made the picture leaves it alone.

What to do with it #

Three uses, in increasing order of how much they buy.

Find a tangent that is not drawn. The tangent to a photographed curve at a marked point is the polar of that point, and it is built with the straightedge — useful where the curve is drawn faintly or is partly hidden, since only five other marks on it are needed.

Find the centre of a photographed circle. Construct the horizon from any two families of parallels in its plane, then take its pole. The result is the image of the centre, which is the point a compass would have been placed at.

And find where the camera was. The polarity of a photographed circle relates the picture’s own geometry to the absolute conic, which is the object a focal length is read from — so a circle, a horizon and a ruler are between them a calibration. It takes two more rungs of this ladder to say exactly how, and this construction is the tool used at every one of them.