Concept

Signed area — where it appears

The area inside a closed curve, counted positive when the curve runs anticlockwise and negative when it runs clockwise. It is half the integral of position crossed with its own increment, so it has a formula, and the two lobes of a figure-eight cancel in it rather than adding.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

One crank, four machines, one area. Four crank-rockers sharing only a crank of length 1, with grounds, couplers and rockers all different, each tracing the point 30% of the way from the crank pin to the rocker pin. The curves have different shapes and sizes and different places in the plane, and the area each one encloses is 2.199115 = (1 − 0.3)·π·1² — on both assemblies of every one, with a worst difference of 3.2 × 10⁻¹⁴.

The area a coupler point encloses

Trace a point on the line through a crank-rocker's two moving pins and the region its curve encloses has area (1 − u)πa²: the crank pin's own circle, scaled by how far along the line the point sits. No ground, coupler or rocker length appears in it. Four machines that share only a crank enclose 2.199115 each, to 3 × 10⁻¹⁴.

curves · Coupler

Three machines drawing one oval, and where each keeps its area. The crank-rocker with ground 4, crank 1, coupler 3.5 and rocker 3, its tracing point at (0.45, 0.5) of the coupler, and the two other four-bars Roberts's construction gives for the same curve, each drawn holding the same point of one oval and each with its input pin's path dashed in the input colour and its output pin's in the output colour. The shaded oval encloses 2.116354 for all three. In the crank-rocker the crank pin goes round and carries 1.727876 of it; in the double rocker neither pin goes round and the coupler's turn carries 1.727876; in the rocker-crank the output pin goes round and carries 1.727876. The remaining 0.388478 is the same in all three.

Where three machines keep one area

Roberts's theorem gives every four-bar two others that draw the same coupler curve, and so enclose the same areas. Measured, they do, to 10⁻¹² — but each keeps the area in a different place: the crank-rocker in its crank pin's circle, the double rocker in its coupler's turn, the third machine in its output pin's circle. What is left over has no closed form, and it is one number in all three.

curves · Coupler

What region each four-bar's two cognates land in. One row per region of length space. Against each, the regions of the two four-bars Roberts's construction gives for the same coupler curve, and how many of the 24000 chains in the census landed in that region. Every row has one entry: across the whole census, and at each of 5 tracing points, the original's region decides its cognates' regions with nothing left over. The four Grashof regions are above the rule and the four triple rockers below it, and no row crosses it — a crank-rocker has a double rocker and a rocker-crank, a double crank has a double crank and a double crank, a rocker-crank has a rocker-crank and a double rocker, a double rocker has a crank-rocker and a crank-rocker, a 0–π rocker has a 0–π rocker and a 0–π rocker, a π–π rocker has a π–π rocker and a π–0 rocker, a π–0 rocker has a 0–0 rocker and a 0–0 rocker, a 0–0 rocker has a π–0 rocker and a π–π rocker.

The kind is decided before the lengths are

Roberts's construction hands a four-bar two others that draw its curve, and which of the eight kinds those two are is settled by the kind of the first — not by its lengths within that kind, and not by where the tracing point sits. Twenty-four thousand chains at five tracing points produce no exception, and the reason is one line: the tracing point enters the construction only as a scale, and a region is scale-blind.

curves · Coupler

Named alongside it

The objects these essays reach for when they reach for this one.

Coupler curve Crank-rocker Grashof's condition Cognate linkage Double rocker Holditch theorem the Roberts–Chebyshev theorem Assembly branch Classification Coupler point Enumeration Scale invariance

All concepts