Machine Kinematics
solved, then drawn
Concept

the Burmester curve — where it appears

The locus of points of a moving body whose prescribed positions lie on one circle, so that each of them could be a crank pin. It is what turns a synthesis problem into a choice from a continuum rather than a single answer.

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

Burmester's curves, contoured rather than drawn. With three poses, every point of the moving body works: three images, one circumcircle. With four, a point's four images are concyclic only if it lies on a particular cubic — the circle-point curve — and the fixed pivots those points want lie on a second cubic, the centre-point curve. Both are drawn here as contours of a measured quantity: at each point of a 150×150 grid, how far the fourth image misses the circle through the other three, contoured at zero. Points refined onto the contour are concyclic to 3.8e-15; points 0.47 away from it miss by at least 2.0e-1. The two curves are keyed in the legend and the four prescribed poses are drawn faintly for scale. Three poses leave a designer the whole plane; four leave a curve. The four prescribed poses are outlined faintly for scale, and both curves are clipped to the frame: the centre-point curve is a cubic with unbounded branches that reach 360 units on a mechanism three units across, and the part worth looking at is the part near the machine.

What the fourth position costs

With three prescribed positions every point of the coupler will do, and a designer is spoilt for choice. Add a fourth and the whole plane collapses to a curve — only points on a particular cubic have four images that lie on a circle, and the cubic is Burmester's.

synthesis · Synthesis 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.

SynthesisBurmester theoryCentre-pointCircle-pointCircle-point curveCircumcentreContact orderCubic of stationary curvatureDyadMarching squaresOsculating circlePolynomial fit

All concepts