Machine Kinematics
solved, then drawn
Concept

Collineation axis — where it appears

The line from the pole through the meeting of two conjugate pairs' joins, which is isogonal to the pole tangent in the pair's two rays. It belongs to a pair of conjugate pairs rather than to the motion, so ten pairs on one linkage give ten different axes — and the relation between them is what Bobillier's theorem asserts.

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

A point, its pole, and the centre it is turning about. The tracing point is on the coupler at (0.45, 0.5) of its length. The cross is the pole, the faint curve is the path the point traces over a whole turn, and the circle is the one that path is momentarily on — centre marked, radius 0.267. The point, the pole and the centre are collinear, which is not an accident of this position: a point's centre of curvature always lies on its own ray from the pole, and Euler and Savary's relation says where on it. Here that relation puts the centre 5.6e-16 of a unit from where differentiating the loop equation three times puts it. positioned by solving, not by drawing.

Every point has a centre

A point of a moving plane traces a curve, and near an instant that curve is a circle. Which circle is decided by one relation with two numbers in it — the same relation for every point of the plane at once, and it was written down in 1830 with no derivatives visible anywhere in it.

curvature · Curvature The collineation axis at 66°. The coupler line extended and the frame line extended meet at Q, and the line from the pole through Q is the collineation axis. Bobillier's theorem is that the axis and the pole tangent make equal angles with the two rays PA and PB, in opposite senses — so having the axis gives the pole tangent, which is otherwise the one quantity here that needs the motion differentiated. Measured over 50 pairs of conjugate points the relation holds to 2.5e-14 radians. positioned by solving, not by drawing.

A construction with no arithmetic in it

Everything in this field so far has needed the motion differentiated. Bobillier's theorem gets the pole tangent — the one quantity that otherwise needs a derivative — out of two lines that are already drawn on the mechanism, and the inflection circle follows from three points and a pair of compasses.

curvature · Curvature

Named alongside it

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

Conjugate pointEuler savaryInflection circleInstantaneous centrePole tangentClosed formCouplerDesign ruleOsculating circlePath curvature

All concepts