A null space of fifteen is not noise

A sextic that comes apart divided a parallelogram chain’s coupler sextic by a circle and found the remainder was rounding. The quotient was a quartic, and each factor turned out to be one of the two things the machine can do: with the crank and rocker parallel the coupler point runs round the circle, and with them crossed it runs round the quartic. The division was exact because the elimination was exact. Nothing in it was fitted.

It left a question about the other route. The equation a four-bar satisfies found a coupler curve’s sextic from traced points alone, by asking which combination of twenty-eight monomials vanishes on them, and it decided the answer by a gap in the singular values. On a parallelogram’s circle motion that question has more than one answer. Every sextic that is the circle times some quartic vanishes on every point of the circle, and there are fifteen independent quartics. A fit to that motion should report fifteen, not one.

It does, behind a drop of more than thirteen decades. And a fit to half an oval of an ordinary crank-rocker also reports more than one, behind a drop of about two. The two counts look alike in a table and are different kinds of fact, and this essay measures what separates them.

What a fit answers with #

A degree-six implicit fit writes each traced point as a row of twenty-eight numbers, the values at that point of twenty-eight basis polynomials, and looks for coefficient vectors the whole matrix sends to nought. The singular values of that matrix measure how nearly it sends each direction to nought. A polynomial that vanishes on every point is a direction with singular value zero; in floating point it comes out at rounding, about 10⁻¹⁵ of the largest.

The basis is products of Chebyshev polynomials on the box the points fill rather than bare powers of x and y, so that high-degree columns are not orders of magnitude apart before any curve is involved. For the parallelogram all three fits share one box, the one both motions fill together, so that a polynomial means the same coefficient vector in each.

The figure orders the singular values from largest to smallest and scales them by the largest. Read left to right, each curve stays within four decades of the top for a while and then falls off a cliff. Where the cliff falls is the count: the number of values beneath it is the dimension of the null space, the number of independent polynomials that vanish on the points. The height of the cliff is the evidence for that count.

This is a different reading from the one the four-bar fit used. That fit compared the smallest singular value with the second smallest, which is the right test when the answer is known to be one polynomial or none. Here the answer is not known to be one, so the fit takes the largest ratio between any two consecutive values and reports how many lie below it. On the circle motion that ratio is 3.7 × 10¹³, between the thirteenth value at 0.23 and the fourteenth at 6.2 × 10⁻¹⁵. Fifteen values lie below it. Comparing only the last two would have found a ratio near one and concluded that nothing decides.

Fifteen is the number of quartics #

The fifteen are not a property of the points. They are what a circle is.

Write the circle’s equation as C(x, y) = 0. Any sextic of the form C·Q, with Q a polynomial of degree four or less, vanishes wherever C does, and the polynomials of degree at most four in two variables form a space of dimension fifteen: one constant, two linear terms, three quadratic, four cubic, five quartic. So the degree-six polynomials that vanish on the circle form a space of exactly fifteen dimensions, and the fit to the circle motion can find no more and no fewer, whatever the number of points, provided they are points of the circle.

The same arithmetic predicts the other two. The quartic motion’s points lie on the quartic factor, and a sextic vanishing on a quartic curve is the quartic times a polynomial of degree two or less, of which there are six. The fit reports six, behind 3.0 × 10¹⁰. Both motions together lie on both factors, and a sextic vanishing on both is a multiple of their product, which is already a sextic, so the multiplier is a constant. The fit reports one, behind 2.5 × 10¹².

The table asks each factor at its own degree as well. The circle motion’s points at degree two give a null space of one behind 1.2 × 10¹⁴: the circle itself, and nothing else of degree two. The quartic motion’s points at degree four give one behind 1.9 × 10¹². Each factor, asked at its own degree, names itself, and asked at a higher degree names its multiples.

Degree five is the control. No quintic vanishes on both motions, since the smallest curve containing both is their product, a sextic. The fit to both motions at degree five has a largest consecutive ratio of 3.44 and places it two values from the end, so as a count it says two. The drop says there is nothing: 3.44 is the kind of step singular values take among themselves when no polynomial is present. A count reported without its drop would have been a false two.

The fifteen, one at a time #

A dimension is a count, and the claim is sharper than a count: the fifteen are the circle’s multiples, not fifteen other polynomials that happen to be small on the traced points.

That can be checked polynomial by polynomial. Take the circle’s exact equation, multiply it by a quartic with random coefficients, and write the product in the fit’s Chebyshev basis. If the fifteen-dimensional null space is the space of circle multiples, the product’s coefficient vector lies inside it, and its distance from the null space, as a fraction of its own length, is rounding.

The four random products sit at 3.0 × 10⁻¹⁵, 4.7 × 10⁻¹⁵, 3.3 × 10⁻¹⁵ and 3.5 × 10⁻¹⁵. The parallelogram’s own sextic, from the elimination, sits at 5.4 × 10⁻¹⁵, as it must, since it is the circle times its quartic factor. The standard crank-rocker’s sextic, which has no circle for a factor, sits at 0.46: nearly half of it is outside.

So the null space is the circle’s multiples, identified member by member, and the fit found it from traced points without being told the circle existed.

This has a consequence for what a fit on one motion can be used for. The usual fit reports the smallest singular vector as “the curve”. On the circle motion that vector is some member of a fifteen-dimensional family, chosen by the rounding in the last digits of fifteen equal singular values, and it is almost never the parallelogram’s sextic. It is the circle times a quartic nobody chose. One motion of a reducible machine does not determine the machine’s curve, and the fit says so in the only way it can, by leaving fifteen directions at rounding instead of one. A reader who took the last vector and plotted its zero set would see the circle, correctly, and a stray quartic curve that belongs to no motion of any machine.

Above its degree every curve has multiples #

Reducibility is one way to make a fit’s null space larger than one. Asking above the curve’s degree is another, and it applies to every curve.

The standard crank-rocker, ground 4, crank 1, coupler 3.5, rocker 3, with its coupler point at 0.45 along the coupler and 0.5 to the side, draws an irreducible sextic. Its real points form two ovals, one for each assembly. A fit of degree seven to those points has a null space of three, because the sextic times x, times y or times a constant is a septic vanishing on them. At degree eight it has six, one for each polynomial of degree two or less. In general a fit of degree d has a null space of (d − 5)(d − 4)/2.

On both ovals, 5,760 points, the fits find exactly that: one at degree six behind 2.0 × 10¹¹, three at seven behind 2.5 × 10¹⁰, and six at eight behind 1.8 × 10⁹. The drops shrink by about a decade per degree as the columns multiply, and every one still reaches rounding.

This explains a number that the curve nobody eliminates reported and passed over. Its fit to the four-bar’s coupler curve decided degree six by a ratio of 4.3 × 10¹⁰ between the two smallest singular values, and at degree seven the ratio fell back to 1.12. That is not the fit losing its grip. Fitted again on the same 800 traced points, in the basis used here, degree seven leaves a null space of three behind a drop of 2.0 × 10¹⁰, and the ratio of the last two values is 1.89. Three singular values are at rounding, and the ratio between two of them says nothing. The null space was there, three strong, two steps to the left of where that instrument looked.

On one oval alone, 2,880 points, the picture is different. At degree six it finds the right count, one, but behind a drop of 8.9 × 10³ rather than 10¹¹. At degree seven it finds four behind 113, and at degree eight fourteen behind 57. The sextic’s multiples number three and six. One oval is still a curve with infinitely many points on an irreducible sextic, and in exact arithmetic it would determine the sextic as completely as both ovals do. In double precision it leaves the fit with near-dependencies that have nothing to do with the sextic, and above degree six they outnumber the multiples.

An arc slopes instead of dropping #

So the question becomes how much of a curve a fit needs, and the answer can be measured by shortening the arc.

The crank-rocker’s first oval was traced at 2,880 points, and arcs of it were cut from three different starting places, a third of the oval apart: the whole oval, three quarters, a half, three eighths, a quarter, an eighth and a sixteenth. Each arc was fitted at degree six on its own box, which is the best treatment a short arc can get, since a box fitted to the arc keeps its Chebyshev columns as far from dependent as they can be.

The whole oval reports one from every start, behind drops of 8.9 × 10³. Three quarters also reports one from every start, behind drops between 2.6 × 10³ and 6.1 × 10³. At half an oval the count is three, five or two depending on where the arc begins, behind drops of 115, 134 and 54. At a quarter it is seven from every start, behind drops near a hundred. At an eighth it is sixteen, fifteen or thirteen.

Two things in the picture are the argument. The first is that the count depends on the arc. A null space that changes from three to five when the same length of the same curve is taken from a different place is not a property of the curve, and nothing about the sextic has a dimension of five.

The second is the height of the dots. The parallelogram’s exact null spaces sit between 3.0 × 10¹⁰ and 3.7 × 10¹³. The highest drop any arc of the oval reaches is under 10⁴, which is more than five decades below the lowest of them. And the sixteenth of an oval, 22.5° of crank, reports thirteen from one start behind a drop of 2.05 × 10³, within a factor of 1.3 of the smallest drop behind which three quarters of the oval reported the right answer. A threshold that accepted the right count of one at three quarters would accept a wrong count of thirteen at a sixteenth.

An arc is a spectrum in that plot and a stretch of curve on the machine, and the two are easy to hold apart until they are put side by side. What the fit is given is not a sample of some abstract object: it is the piece of its own path the coupler point has actually covered, ending where the crank has got to.

Shortening the arc walks the machine back along its own path, and the verdict changes under it. At the whole oval and at three quarters the fit says one, which is the right answer. At a half it says three and the arc still looks like most of a coupler curve. At a sixteenth the machine has barely left where it started, the thick stretch is a short bend that could belong to almost anything, and the fit says thirteen — with a confidence, measured as the drop below the count, that a threshold cannot tell from the right answer’s.

That is what makes this a measurement problem rather than a numerical one. Nothing has gone wrong in the arithmetic: a short arc genuinely is fitted by a thirteen-dimensional family of sextics, all of which agree with it and disagree everywhere else. The curve has one sextic; the arc has thirteen.

The spectra show why no threshold of that size can work.

The circle motion’s values run within a decade of the top to the thirteenth and then drop more than thirteen decades in one step. The arcs’ values do not step. The whole oval slides down twelve decades across its first twenty-seven values, never more than two at a time, and then falls four at the last, which is the sextic; three quarters of the oval does the same with a last step of 3.4. Half an oval reaches rounding at its twenty-seventh value, and the largest single step on the way is 2.1 decades. The quarter oval begins falling at the ninth value, is below 10⁻¹⁵ by the twenty-fourth, and never steps by more than 1.9.

A slope has a steepest step, and the fit reports the count below it. But on a slope that step is decided by small differences between neighbouring values, and moving the arc’s starting place changes them. The count follows. On a cliff the count is fixed by the gap; on a slope the count is fixed by whichever step happened to be steepest.

The slope has a plain cause. Over a short stretch, the curve is close to a polynomial in the distance along it of low degree, a parabola to begin with and nearly a cubic a little further on. Twenty-eight Chebyshev products evaluated along such a stretch are twenty-eight polynomials in that one parameter, and polynomials of one variable on a short interval become nearly dependent in a graded way, a little more so for each degree. The singular values record that grading. None of it involves the sextic, and all of it shrinks as the arc grows, which is why the whole oval is mostly clear of it and why two ovals are clearer still.

A count and its drop are one measurement #

Put together, the fits divide into two kinds, and the two kinds are separated by more than five decades.

In the first kind the null space is algebra. The parallelogram’s circle motion has fifteen because quartics have fifteen coefficients; the quartic motion has six because conics have six; the crank-rocker’s whole curve has three at degree seven because a sextic times a linear polynomial has three free coefficients. Every one of those null spaces sits behind a drop to rounding, from 1.8 × 10⁹ to 1.2 × 10¹⁴, and every count agrees with a formula written before the fit was run.

In the second kind the null space is arithmetic. The arcs’ counts of two to nineteen, and the single oval’s counts of four and fourteen above degree six, sit behind drops of 10⁴ or less, change when the arc is moved, and agree with no formula.

The count alone cannot tell the two apart. Fifteen from a circle and fourteen from an oval are both integers beneath a largest ratio. What tells them apart is the ratio, and the discipline that decided a mechanism’s mobility by a gap rather than a residual applies without change: a rank is reported with its gap, and a rank with a gap of two decades is not a rank.

The same discipline answers the question the parallelogram raised. A fit to a reducible machine’s motion does not fail. It reports, exactly and with overwhelming evidence, that the motion is a smaller curve than the machine’s sextic, and it reports how much smaller: fifteen multiples means the curve is a conic, six means a quartic. Run backwards, the count of multiples reads a factor’s degree off traced points with no elimination at all. Three linkages, one equation used a fit to show that three cognates draw one curve; this is the complementary use, a fit showing that one machine draws two.

Where the drop has not been tested #

No machine a micron away from reducible is fitted. A parallelogram a micron wrong measured what a small length error does to the motion. What it does to the fit, whether the fifteen survive behind a smaller drop or collapse to one, is not measured here.

The arc experiment is one oval of one curve at one degree. The counts on shorter arcs will differ on other curves, and nothing here predicts them. The claim is only that they are not properties of the curve, and one curve is enough to show that.

The Chebyshev basis is one basis. A basis orthogonalised along the arc itself would flatten some of the slope. It would move the steepest step and not create a cliff, because the near-dependence is in the geometry of a short arc rather than in the columns chosen to describe it; that reasoning is stated, and the alternative basis is not run.

Drops are compared across fits of different sizes. A fit with 5,760 rows and one with 180 are not conditioned alike, and the five decades between the exact null spaces and the arcs are wide enough that the comparison does not depend on it.

What comes next: the micron-wrong parallelogram, fitted #

The micron-wrong parallelogram, fitted. A coupler 10⁻⁶ too long makes the sextic irreducible. Its fit on the motion that used to be the circle must eventually report one curve, and on the way the fifteen singular values at rounding must separate. How far they separate as a function of the error, and whether the drop that decides fifteen shrinks as the square root of the error, like the stall angle in the micron-wrong parallelogram, or linearly, would say how close to reducible a built machine can be and still read as reducible from its traced points.

Reading a factor’s degree from a fit. The formula (d − k + 1)(d − k + 2)/2 for a factor of degree k inverts: a null space of fifteen at degree six says the traced curve has degree two. A six-bar whose first loop is a parallelogram has motions that split with the parallelogram’s, and a point on its arm draws a curve that comes apart in pieces nobody has eliminated. The count of multiples on each motion’s traced points would give each piece’s degree, where the sextic essay’s division is not available to check against.

How many points of an arc a fit needs, as a law. The arcs here were cut at seven lengths. A finer sweep, with the drop at the right count plotted against arc length on log axes, would show whether the drop grows as a power of the arc and which power, and so give the length of arc below which no double-precision fit can decide a sextic.