Machine Kinematics
solved, then drawn
Concept

Circle-point curve — where it appears

The locus of points of a moving body whose four prescribed positions lie on one circle, so that each of them could serve as a crank pin. It is the finite-position version of the cubic, and as the four positions are brought together it tends to it as the square of their spread.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

Five positions, and what is left of the curve. Five prescribed poses of a moving body. With four of them, every point of the pale curve is a usable fixed pivot — a one-parameter family. The fifth pose is one more equation, and it leaves 4 points. Bézout's number for the system is 16; 4 paths arrive; 4 of those are real. Every pair of the 4 is a four-bar, so there are 6 candidate linkages and 2 of them reach all five poses in one piece and in order. 1 of the 4 pivots is too far away to draw in frame and is marked at the edge with its true distance — which is why some of the linkages have a bar twenty times the size of the body.

Five positions, and what is left

Three prescribed poses leave a whole plane of choices. Four leave a curve. Five leave four points, and finding them is the first thing in this site's synthesis field that a compass cannot do — it needs two cubics intersected, which is algebra rather than construction. Four points give six four-bars, and two of them can be built.

synthesis · Synthesis Where the curvature is standing still, at 66°. The cubic of stationary curvature: every point on it traces a path whose curvature has stopped changing at this instant, so its osculating circle fits for one order longer than an ordinary point's. It passes through the pole — twice, with a double point there — and through both moving pins, which is the plainest case there is, since a pin traces an exact circle and a constant curvature is certainly a stationary one. The expression whose zero set this is looks like a quartic and its fourth-degree terms cancel to 1.2e-14 of the third-degree ones. positioned by solving, not by drawing.

Where the curvature stands still

One derivative further on there is a second locus: the points whose path curvature has momentarily stopped changing. The expression whose zero set it is looks like a quartic, its fourth-degree terms cancel, and the cancellation is measured at ten to the minus fourteen rather than assumed.

curvature · Curvature Four positions brought together. Burmester's construction for four prescribed positions gives a curve of points that can be fixed pivots. Bring the four positions together and it has to become the cubic of stationary curvature, because a circle through four coalescing positions is a circle of four-point contact. Measured along one ray from the pole: the finite curve crosses at one radius and the cubic at another, and the gap between them falls as the square of the spread — fitted exponent 2.024, and 7.9e-5 coupler lengths at a spread of 0.01 radians. The finite construction is the same concyclic test the site's four-position synthesis is built on, so this compares that machinery against the infinitesimal one rather than against a re-implementation of either.

Four positions brought together

Burmester's construction finds the points of a moving plane whose four prescribed positions lie on a circle. Bring the four positions together and the curve it draws has to become the cubic of stationary curvature — and the gap between the two closes as the square of the spread, with the exponent measured rather than assumed.

curvature · Curvature

Named alongside it

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

Contact orderCubic of stationary curvatureOsculating circlePolynomial fitBezoutBranch defectthe Burmester curveBurmester pointBurmester theoryCentre-point curveConic fitDyad

All concepts