The quintic's cheap pivots lie off the line

A compiled machine draws a curve with the tip of a two-link arm, and the price is on the equation. Every curve is a sum of cosines of whole-number combinations of the arm’s two angles, and each cosine is a gadget the machine, compiled has to build. Four terms of eighteen draw the quintic found one catalogue curve that is nearly free to draw approximately. The quintic $x^5 + 0.4x^2y - y = 0$ costs eighteen cosines exactly, and four of them, a line through the pivot bent by the square of the radius, come within a few per cent of the arm’s reach.

The quintic’s cheapness runs along a line then moved the arm’s pivot and priced the same arc of the quintic from each new place. Along the line $x = 0$ the four-term price stayed below 7.5% to four tenths of a link. Across it the price passed 18% within a tenth. That left the region between two lines unmeasured, and a mounting error is almost never purely along one or purely across. The question was the shape of the region of pivots from which four cosines stay cheap. If it is a strip whose width is set by the across tolerance, a mounting can be toleranced one direction at a time. If it narrows or turns, the two directions interact.

It turns. The cheap pivots lie in a valley ten degrees off the line, and the line itself runs out of the valley within a fifth of a link.

Pricing a grid of pivots #

The arc is the earlier essay’s: the part of the quintic within 1.4 of the curve’s own point, carried with the curve, which every pivot considered here can reach. The price is two-sided. It is the worst distance from the arc to the cheap curve and from the cheap curve back to the arc, so a close fit that grows a stray branch elsewhere in the reach is charged for it. Prices are quoted as shares of the arm’s reach of two links.

Finding the nearest four-term curve at one pivot means trying every set of four cosines, a census that takes seconds. A grid of a thousand pivots would take most of an hour. But along the line $x = 0$ the census had always picked the same four: the line through the pivot, $x$ and $y$, and the same line bent by the square of the radius, $r^2x$ and $r^2y$. Fitting that one set at a pivot is a single least-squares solve. Since the census takes the least price over every set, the bent line’s price is an upper bound on the nearest four-term curve’s: wherever it is low, four cosines are certainly enough.

So the bent line was priced at pivots a two-hundredth of a link apart across and a twentieth along, out to 0.16 across and 0.45 along. To check the bound where it matters most, the full census, every set of four and every set of six, was run at two pivots where the map is cheapest.

The valley #

The cheap region is not a strip around $x = 0$. At each height along, the price across is lowest somewhere off the line, and the lowest points lie on a straight line through the catalogue’s pivot. Its slope is $dx/dy = -0.172$, a tilt of 9.8° from $x = 0$. Moving the pivot up, towards positive $y$, the cheapest place moves to negative $x$, and moving it down moves it to positive $x$, by the same amount: the valley is odd in the height, to the grid’s step.

Along the valley the four-term price stays between 1.4% and 2.7% of the reach at every height from −0.45 to 0.45. It is lowest a tenth of a link from the catalogue’s pivot, 1.4% at $(0.015, -0.1)$ and 1.5% at $(-0.015, 0.1)$. That is lower than at the catalogue’s pivot itself, where this arc prices at 2.5%. The catalogue placed the pivot where the quintic’s exact equation is shortest, as moving the pivot only adds terms found, and the cheapest place for the approximation is a little away from it.

On the line $x = 0$ the price rises quickly away from the catalogue’s pivot: 2.3% at a tenth of a link, 4.7% at 0.15, 6.0% at 0.2 and 7.3% at 0.3. Past that it eases back slightly, to 6.6% at 0.45. These are the numbers the earlier essay reported as the cheapness running along the line. They were true, and they were the price on the valley’s slope rather than on its floor. A pivot moved 0.3 along and 0.055 across, against the tilt, pays 2.1%, a third of what the same move along pays with no move across.

Where four cosines miss on the line #

The price is two-sided, and splitting it says what goes wrong on the line. From a pivot 0.3 along on $x = 0$, the bent line fits the priced arc itself to 3.1% of the reach. It is close everywhere along the part of the quintic it was asked to follow. The 7.3% comes from the other direction. Beyond the arc’s ends the bent line carries on, and its own points stray from the quintic. On the steep branch at the upper right the fitted curve leans away, and its farthest point inside the reach is 7.3% from the nearest point of the quintic. The figure at the top shows it: the heavy arc and the thin curve agree, and the thin curve’s tail parts company with the quintic’s thin continuation.

That is a stray-branch charge of exactly the kind the two-sided price was designed to make. A compiled machine draws the whole of its curve, not the part a designer cares about. The tail is what the arm’s tip traces if the machine is driven past the arc, and a curve that is close on the arc but wrong beyond it is a machine that has to be stopped at the arc’s ends.

In the valley the two directions agree. At 0.3 along and 0.055 across, the arc-to-curve distance is 2.10% and the curve-to-quintic distance 2.12%. At every height measured the pattern is the same. On the line the arc fits to 2.1–3.3% while the curve strays by 6.0–7.3%. In the valley the two distances are equal to within a few hundredths of a per cent. So part of what the move across buys is a fitted curve whose tails lie on the quintic beyond the arc as well as on it. The fit on the arc improves too, from 3.1% to 2.1% at 0.3 along, but the larger gain is the tail.

A V at every height #

Read across at a fixed height, the price is a V. At the catalogue’s height the V is broad and flat-bottomed, under 3.3% from 0.06 one way to 0.06 the other, and its five per cent band is an eighth of a link wide. That broad bottom is why the earlier essay found the move across costing little at first and then a great deal: 3.2% at 0.05, then 18% at 0.10.

At every other height the V is narrower and its bottom has moved. At 0.15 along, the least is 1.6% at $x = -0.025$ and the band within five per cent runs from −0.045 to 0. At 0.3 along, the least is 2.1% at $x = -0.055$, the band from −0.085 to −0.02, and the line $x = 0$ lies outside it altogether. At −0.3 along everything is mirrored. The band’s width away from the catalogue’s height is a twentieth to a fourteenth of a link, 0.045 to 0.075.

So the earlier essay’s guess at a width was about right, a twentieth of a link set by the across tolerance. What it could not see from two lines was that the strip is not centred on $x = 0$. The two directions interact: the move across that keeps four cosines cheap depends on the move along, linearly, at 0.17 of a link across for each link along.

The catalogue’s pivot is on the flat #

The catalogue’s pivot sits where the V is broadest, and that explains both why it looked like the cheapest place and why it is not quite. At its height the price across is 2.5% at $x = 0$, 2.7% at 0.04 one way, 3.3% at 0.04 the other, and under five per cent across a band an eighth of a link wide. It is a plateau, not a point. A pivot moved a little along drops off the plateau into the valley’s floor, which lies lower: 1.4% a tenth of a link down and 1.5% a tenth up.

So the catalogue’s pivot is the best place for the exact equation and a good place for the approximation, robust to small moves in every direction. It is not the cheapest place for four cosines, and the cheapest places are a tenth of a link away, along the valley. For a designer who wants the approximate machine and nothing else, moving the pivot a tenth of a link along and a hundredth and a half across buys the arc at 1.4% instead of 2.5%, nearly halving the error with the same four cosines.

Every set of four, and every set of six #

The map rests on one set of four cosines being the right four. At two pivots in the valley, $(-0.055, 0.3)$ and $(0.015, -0.1)$, the full census tried every set of four and every set of six.

At both, the census picks the bent line, at exactly the map’s price: 2.1% and 1.4%. The bound is the price there. Two further results are worth noting. Six cosines do no better than four at either pivot, so in the valley the bent line is already as close as the arc allows within six terms. And the exact equation costs 30 cosines at both pivots, the most its shape allows. The pivots that are cheapest to approximate are among the most expensive to draw exactly. The two measures of cost, exact and close, point in different directions here as they did for straight-line linkages.

What the amplitudes do not explain #

The earlier essay explained its two lines by the size of the cosines a move adds. Moving the pivot along $y$ adds four new cosines, small and fixed in size. Moving it along $x$ adds twelve, sixty to ninety times larger in total. On that reading the cheap direction is the one that adds least amplitude outside the bent line’s four. It can be computed. To first order a move $(x_0, y_0)$ adds $x_0\mathbf a + y_0\mathbf b$ to the cosines outside the support, and for a given $y_0$ the $x_0$ that adds least is $-(\mathbf a\cdot\mathbf b)/(\mathbf a\cdot\mathbf a)\,y_0$.

That gives a slope of −0.023, the right sign and a seventh of the size. The valley is tilted seven times more than the amplitudes say. So the amplitude argument does not price the approximation by itself. The fit spends its four cosines on a curve close to the arc, and how close it can get depends on more than how much of the equation lies outside them: on which parts of the arc those outside cosines move, and in which direction. What sets the tilt is not established here. It is measured, it is the same at every height, and it is ten degrees.

The split of the two-sided price above points at where to look. The amplitudes describe the whole equation, every part of the curve the arm can reach, weighted equally. The price is decided at two places: the arc, and the tails just beyond it, which on the line carry most of it. A spectral size of the added cosines weighs a change at the far side of the reach the same as one at the arc’s end. The fit, and the price, care only about the second.

Pivot by pivot #

The ledger is the practical form of the map. A designer who has to mount the arm away from the catalogue’s pivot, because the frame is in the way or the work is, should move it along the valley, not along $x = 0$. Going 0.2 of a link along means going 0.035 across against it, at 1.8% instead of 6.0%. Going 0.4 along means 0.07 across, at 2.5% instead of 7.1%. In every case the band within five per cent is at least 0.045 wide, so a mounting held to a fiftieth of a link across, either side of the valley, stays within budget.

A worked case shows the size of the difference. Suppose a frame member forces the arm’s pivot 0.25 of a link along from the catalogue’s place. Kept on $x = 0$, the four cosines miss by 7.0%, well outside a five per cent budget. Moved 0.045 across against the tilt, they miss by 1.9%. The across move is smaller than many mounting tolerances, and it is the difference between a machine within budget and one well outside it.

The mistake to avoid is the one two lines invite: holding the across position at its catalogue value while the along position drifts. On the line $x = 0$ a drift of 0.2 along already costs more than half of a five per cent budget, and the valley would have cost a third of it.

The finding also bears on the scale question a compiled machine and its own scale took up: a compiled machine’s bars are its equation’s coefficients, and the coefficients are set by where the pivot is. A designer choosing a pivot for a cheap approximate machine is choosing a point on this map, and the cheapest curves at each degree shows how rare a curve is that has such a valley at all.

The limits of the map #

An upper bound, confirmed at two points. The map prices one set of four cosines, an upper bound on the best four. At two valley pivots the census confirms the bound is the price. Elsewhere, and especially far from the valley, some other set of four may do better, so the regions shaded expensive may be cheaper than shown. The valley itself is exact where it was checked.

One arc. The arc priced is the part of the quintic within 1.4 of its own point. A longer arc asks more of four cosines, and both the valley’s floor and its width would change.

The split of the price, at six heights. That on the line the price is mostly the tail and in the valley the two directions balance was measured at six heights, from −0.4 to 0.4 along, on the line and on the valley’s floor. It held at all six. Between the two, on the valley’s walls, how the price divides was not measured.

A grid, not a contour. The band’s edges are read off a grid a two-hundredth of a link apart, so each edge is known to that step and each width to about a hundredth.

Still open: what tilts the valley #

The valley’s slope is −0.172, the same at every height, and the first-order amplitude of the cosines a move adds outside the bent line’s support predicts −0.023. Something about the fit sets the tilt, and the amplitudes alone do not see it.

The distinct argument would be a second-order account of the bent line’s fit to a moved arc. The fitted curve’s error at each point of the arc is a sum over the outside cosines, weighted by how far each displaces the curve at that point, which depends on the curve’s gradient there. The measurement would compute that weighted amplitude, a geometric rather than a spectral size of what a move adds, find the move across that minimises it at each height, and compare the slope with −0.172. If it matches, the tilt is a fact about where on the arc the quintic’s added cosines bite. If it does not, the tilt belongs to the fit itself, to how the four cosines and their free constant are spent. That would make a compiled machine’s cheapest approximate pivot something only a search can find.