Five tangents name the same conic

Five marks and the sixth established the fit this collection has used ever since: a conic has six coefficients up to scale, five points give five linear conditions, and the sixth mark is a prediction rather than a fit. Every recovery of a circle’s image on this site rests on it.

Its dual is one sentence away. A line is tangent to the conic C exactly when lᵀ C* l = 0, where C* = adj(C) is the conic of tangent lines — which is linear in the six coefficients of C*. So five tangent lines determine a conic by the same algebra, with the design matrix built from line coefficients instead of point coordinates and the answer taken as an adjugate at the end.

That is a theorem. What it is worth on a drawing is a measurement, and this essay is mostly the measurement.

The fit #

Five tangents of a photographed circle, no points marked at all. The conic that comes back passes through seven sampled points of the true one to 2.6 × 10⁻¹¹ pixels, which is the arithmetic floor.

Nothing about the points of contact is given to the fit. They are drawn in the figure only so a reader can see where the lines touch; the algebra is handed five triples of line coefficients and nothing else.

The one thing that has to be watched #

A line through the origin of the picture has its third coefficient zero, and the dual chart divides by it — the same running-out-of-chart that a point and a line are one object meets when a Desargues configuration is centred on the origin and three of its dual’s points go to infinity.

So the fit refuses such a tangent rather than returning a large number, and tells the caller to move the figure. That is a refusal about the coordinates and not about the geometry: the same five tangents, translated, fit perfectly. Stating it as a refusal is the alternative to a routine that quietly loses a digit whenever a figure happens to sit near the middle of the page.

The asymmetry duality does not remove #

A real conic is rank three and its dual is rank three, which is the case every theorem lives in. A pair of lines is rank two, and its dual is rank one — the single point where the two lines cross, counted twice. A repeated line is rank one and its dual is exactly zero.

So the two descriptions are not interchangeable at the degenerate end. Five points lying on a pair of lines still say which two lines they are; the tangent description of the same object says only where they crossed and cannot be inverted at all. The point description degrades gracefully into a rank-two object; the line description collapses.

This matters wherever a fit might land on something degenerate, which is wherever the evidence is thin — and it is the reason a tangent fit needs the eigenvalue gap watched as carefully as a point fit does, not less.

Which is the better evidence #

Duality says the two fits are the same theorem. It says nothing about which to run on a drawing, and the obvious guess is that tangents win: a tangent is what a straightedge produces and a point on a smooth curve is a guess.

Measured, that guess is wrong, and the reason it is wrong is worth the essay.

The reason can be stated before the measurement, and it is about how many numbers each datum carries.

A point on the curve is one measurement: a position, read to whatever a mark can be read to.

A tangent is two: a position and a direction. And the direction is not observed — it is differentiated, estimated from a short piece of the curve, so a baseline $a$ of curve read to $\varepsilon$ gives a direction good to $\varepsilon/a$. Carried out to the far side of a conic of size $R$, that direction error becomes a line-position error of order $\varepsilon R/a$.

So the tangent datum is worse than the point datum by roughly $R/a$ — the conic’s own size over the baseline each tangent was estimated from — and that ratio is large whenever the tangents are read locally, which is what happens when they are taken off a drawn curve. Differentiation costs, and duality does not make it free.

Which is why the rule is about where the data comes from #

The useful form of the finding is not “point fits are better”. It is:

Use the description the data arrives in.

Marks on a curve are points, and fitting points to them uses each mark once. Straight lines drawn as straight lines are tangents, and fitting tangents to them uses each line once. The moment either is converted into the other, something is differentiated or something is intersected, and the conversion costs the ratio above.

That makes the tangent fit the right tool in exactly the places where lines are primary rather than derived. The sides of a circumscribing polygon — the rhombus in the four-centre construction is four of them. The supporting lines of a silhouette, which are what a visual hull is built from and which are measured as lines with no curve to differentiate. The straight edges of a drawn frame touching an inscribed ellipse. In each of those the tangent is the observation and the contact point is what would have to be computed.

And it makes the point fit right for a photograph of a curve, where the marks are what the picture offers and every tangent would be an estimate.

The degeneracy, dualised #

The point fit’s degeneracy is three collinear; the tangent fit’s is its dual, three concurrent. Five tangents of which three pass through one point determine nothing, and — as with the point case — they do not fail loudly: the fit returns a degenerate dual conic, which is a point-pair rather than a curve, and the algebraic residual is small because a degenerate object satisfies the five conditions exactly.

Three nearly-concurrent tangents are the dangerous version, and they are easy to produce by accident: five tangents taken from a short arc of a curve are nearly concurrent, since a short arc’s tangents all pass near one another. So the tangent fit’s bad case is a short arc, exactly as the point fit’s is, and the two descriptions fail on the same drawings for dual reasons.

The comparison has to be made at equal marking precision, which means saying what a hand actually does to each kind of mark. A point is displaced by whatever the hand can see — call it n pixels. A tangent is laid along a rule of length L using marks with the same error, so its angle is wrong by about n/L and its position is wrong by about n. Rotating a tangent by a fixed angle instead would be comparing a hand with a protractor.

At n = 0.4 pixels and a 200-pixel rule, the point fit is out by 6.64 pixels at its worst and the tangent fit by 7.75 — a ratio of 1.17. Neither wins.

Where the tangent fit’s error actually is #

The third curve in that figure is the one that explains the first two. Anchor each tangent exactly at its point of contact and leave only the angular error, and the fitted conic is out by 0.005 pixels at a 200-pixel rule — three orders of magnitude better — and it improves as one over the rule length, reaching the arithmetic floor by 1,200 pixels.

The drawn tangent does not improve at all: 8.04 pixels at a 40-pixel rule, 7.70 at 1,200, an improvement of four per cent across a thirty-fold change. And the point fit is flat too, because a point has no length for a longer rule to buy anything with.

So the whole of a drawn tangent’s disadvantage is the positional error of the two marks that set it. A tangent looks like better evidence than a point and is not, because it is made of points.

Why this is the right null result to have #

A measurement that came out the other way would have been more quotable and less trustworthy, so it is worth saying what would have made it come out the other way.

If a tangent could be laid without marking its point of contact — along a family of marks spread over a long arc, say, rather than through two marks near where it touches — then its positional error would fall too, and the fit would inherit the improvement. That is exactly what a plumb-line calibration does with a lens: fitting a lens from straightness alone uses many points along each line, and its accuracy is bought with line length in precisely the way an anchored tangent’s is here.

So the result is not “tangents are no better than points”. It is narrower and more useful: a tangent set by two nearby marks is no better than one of those marks, and buying the improvement means spreading the evidence, which is a statement about the drawing rather than about the algebra.

The conditioning number both fits report #

Both routes return the eigenvalue ratio of their design matrix — how far the smallest singular value is from the next, which is what says the five conditions determine one conic rather than a pencil of them.

At exact marks both gaps are at the floor. Under error they stay there, which is the honest and slightly surprising part: the fits go wrong without the spectrum saying so. Five points in general position determine a conic no matter how badly they are marked; the answer is a well-determined conic through five wrong points.

That is the distinction an ambiguity is not an uncertainty is written about, met here for the first time in this round. A small eigenvalue gap says the evidence does not determine the answer. A large gap says nothing at all about whether the answer is right.

What the dual fit is actually good for #

Two arrangements, and both are real.

A curve whose points cannot be marked and whose tangents can. A silhouette against a bright background, a shadow’s edge, the outline of a turned part on a shadowgraph: the boundary is a gradient rather than a set of marks, and a straightedge laid along it is a genuinely easier measurement than a point on it. The positional error is still there, but it is smaller than the error of picking a point on a fuzzy edge.

And a curve given as tangents in the first place. A drawing office’s french curve, a template, an ellipse guide and a set square all produce tangents by construction, and the ellipse the drawing office draws is a curve built entirely out of tangency conditions. Asking whether such a curve is a conic is therefore a question about its tangents, and it has a straightedge answer — which is what six tangents and the point nobody drew does with it.

Why the adjugate rather than a second nullspace #

A detail of how the fit is done, because it is the step where a reader’s own implementation is most likely to differ and get a different answer.

The five tangents give a nullspace problem in the six coefficients of the dual conic C*. What comes out is therefore C*, and the conic itself is C = adj(C*) — the adjugate again, because for a nondegenerate three-by-three the adjugate of the adjugate is the original up to scale.

The alternative anybody would try first is to solve directly for C by writing the tangency condition as a discriminant, and it does not work: tangency written in terms of C is quadratic in the coefficients, not linear, so five conditions do not give a linear system and the problem stops being a nullspace at all. Going through the dual is what makes it linear, and that is the whole reason the dual conic is worth having as an object rather than as a definition.

The price is one place where the answer can be lost, and it is the rank collapse above: if the fitted C* comes back rank two — five tangents through a common point, say — the adjugate is rank one and there is no conic to hand back. The routine refuses there rather than returning a curve, for the same reason the polar with a straightedge refuses a secant that misses.

Five tangents that determine nothing #

The degenerate arrangement of five points is well known here — four of them collinear, or all five on a conic that is really two lines — and it has a partner.

Five tangents through one common point are five members of a pencil, and every conic inscribed in that pencil satisfies them; the dual design matrix loses a rank and the answer is a family. Five tangents in two families of parallels do the same at infinity. Neither is exotic: a drawing whose edges are two sets of parallels is an ordinary orthographic drawing, and asking for the conic inscribed in its outline is a question with no single answer.

That is the same shape as the degeneracy the two-view field is about this round — a flat scene fixes no second eye is a rank collapse in a design matrix too, and the diagnostic is the same one: not the smallest singular value, which always vanishes, but the second smallest.

The two fits on one photograph #

Run both on the same imaged circle and they land on the same conic, which is the theorem; run both under the same hand and they land in the same place, which is not.

The practical advice that falls out is short. Use whichever marks the drawing actually supplies. If it supplies both, the point fit is marginally better and considerably simpler, because it needs no adjugate and no chart. If it supplies neither cleanly, spreading the evidence buys more than choosing between the two descriptions does.

What a sixth tangent is for #

The point fit’s discipline is that the sixth mark is withheld: fit on five, predict the sixth, measure the prediction. That discipline transfers exactly, and it is worth transferring because it is the only thing separating a fit from a description.

Six tangents give one more condition than the fit needs, and the residual is a distance rather than a number with arbitrary units — how far the sixth line is from touching the conic the other five named. On an exact photograph it is at the floor. Under a hand it is the same order as the fit’s own error, which is what says the sixth tangent is a genuine test and not an easier version of the fit.

And six tangents give something the point route does not, which is the next essay’s subject: they can be tested against each other with no conic fitted at all. Six tangents and the point nobody drew is Brianchon’s theorem used as an instrument, and it is a straightedge construction with a yes-or-no answer where this essay’s fit is arithmetic with a residual.

What is measured here #

Four numbers.

The tangent fit lands on the conic to 2.6 × 10⁻¹¹ pixels from five lines alone. At 0.4 pixels of marking error and a 200-pixel rule the two fits are 6.64 and 7.75 pixels out, a ratio of 1.17. With the tangents anchored and only their angles in error the tangent fit is 0.005 pixels out and falls as one over the rule. And a thirty-fold longer rule improves the drawn tangent fit by four per cent and the point fit by nothing.

Together those say that duality’s promise holds exactly and that the practical difference people expect from it does not exist — for a reason that is about hands and not about conics.