Machine Kinematics
solved, then drawn
Concept

Polynomial fit — where it appears

Fitting a polynomial of stated degree to a sampled expression, so that the size of coefficients that should vanish can be reported. It is how this site measures that a curve is of a stated degree, since a coefficient that should be zero coming back at the fourteenth decimal place is evidence and an assertion is not.

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

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 Three circles at one point, at 50°. The osculating circle of the fixed centrode at the pole, the osculating circle of the moving centrode there, and the inflection circle. All three pass through the pole and all three have their centres on the common normal — which is what makes the relation between them a relation between three numbers on a line. The two centrode radii are 16.180 and 9.730, signed along that normal, and the inflection circle's diameter is 24.398 against the 24.409 the other two give. Nothing in that arithmetic knows there is a linkage. The window is 30.25 wide and the larger osculating circle is 32.36 across, so it is cut at the edge: what is drawn is the neighbourhood of the pole, where the three curvatures are the same three numbers however far the circles carrying them reach.

The two numbers are the curves' own

Every essay in this field runs on the inflection circle and the pole tangent, and every one has taken them from the motion's derivatives. They are the two centrodes' radii of curvature at the pole, combined harmonically — measurable off the curves with no mechanism in the arithmetic, and carrying a sign that changes eight times in a turn.

curvature · Curvature

Named alongside it

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

Osculating circleCircle-point curveContact orderCubic of stationary curvatureInflection circlePath curvaturethe Burmester curveBurmester theoryCentrodeConditioningConic fitEuler savary

All concepts