Machine Kinematics
solved, then drawn
Concept

Pole tangent — where it appears

The direction the instantaneous centre is itself travelling in, which is the common tangent to the two centrodes at their point of contact. It is the direction that orients everything else in the field: the inflection circle's diameter lies along its normal, and the Bobillier construction is stated relative to it.

Named by 7 essays across one field — each of them below, with the objects they name alongside it.

The coupler's motion at 66°. Every point drawn as a stub is a point of the coupler's own plane, and the stub is that point's velocity — solved, not sketched. They all point different ways and they are all consistent with one statement: at this instant the whole plane is turning about a single point, the pole, marked with a cross. It is off this frame at 1.5 coupler lengths from the crank pin, which happens whenever the coupler is close to translating. The arrow through it is the pole's own velocity, which is a quantity about the motion rather than about any point of it, and half of everything in this field follows from its direction and its size. positioned by solving, not by drawing.

The mechanism drops out

Every other field here is about a machine. This one is about the motion a machine makes — a plane sliding over a plane — and near any instant that motion is a handful of numbers with no linkage in them. Two mechanisms that agree on those numbers make the same motion, and one of them can always be thrown away.

curvature · Curvature 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 circle of points going straight, at 66°. Every point on this circle is, at this instant, travelling in a straight line: its path has zero curvature there. The circle passes through the pole — where the point is not moving at all — and its diameter is 47.64, which is the pole's own speed divided by the plane's angular rate. Nothing here was assumed to be a circle. The locus is the zero set of a quadratic whose |w|² coefficient is φ′³, a real number with no cross term and no difference between its two square terms, and a general conic fitted to the sampled locus returns those coefficients at 8.1e-15 and 6.4e-15. At this position the coupler is close to translating, the pole has run off the canvas and the circle with it — the figure is the size of the mechanism, and δ here is 13.6 coupler lengths. positioned by solving, not by drawing.

The circle of points going straight

At any instant some points of a moving plane are travelling in a straight line. They form a circle — not nearly a circle, a circle — and the reason is one real coefficient in a quadratic. A coupler curve has an inflection exactly when that circle sweeps over the tracing point, which turns out to be rare.

curvature · Curvature Two poles at 66°, and only one of them is still. The stubs here are accelerations, not velocities, and they vanish at a different point from the one where the velocities vanish. The cross is the velocity pole — the point the plane is turning about — and the square is the acceleration pole, the point that is momentarily not accelerating. Over a full turn of this crank the two are never nearer than 1.49 coupler lengths and get as far apart as 22.2. The instantaneous centre is about velocities and about nothing else: the point that is not moving is being accelerated, usually hard, and a body pivoting about it in the ordinary sense would not be. positioned by solving, not by drawing.

The other pole

The instantaneous centre is the point of a moving plane that is not moving. There is a second point that is not accelerating, it is somewhere else entirely, and over a full turn of one four-bar the two are never closer than one and a half coupler lengths and get as far apart as twenty-two.

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 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 Holding the coupler still: the frame's inflection circle is the mirror of the coupler's. The four-bar with ground 4, crank 1, coupler 3.5 and rocker 3 at a crank angle of 225°, redrawn in the coupler's own frame, so the coupler is the horizontal bar and the ground link is what moves. The circle on one side of the pole is the ordinary inflection circle — coupler points travelling straight — carried into this frame. The circle on the other side is the inverse motion's: frame points travelling straight when the coupler is held. It is computed by differentiating the swap and, separately, by solving the coupler-held four-bar, and both routes put it at the first circle reflected through the pole, to 7e-15 on a radius of 4.45. The line through the pole is the common tangent. Dragging the crank angle moves both circles and they stay mirror images.

The frame seen from the coupler

Hold a four-bar's coupler still and let its frame move, and every construction of the curvature field has a counterpart. The frame's points that travel straight lie on the ordinary inflection circle reflected through the pole — found by differentiating the swap and, independently, by solving the coupler-held four-bar, both to 10⁻¹⁴ — and a point and the centre of curvature of its path trade places exactly. The inverse motion of one four-bar is the ordinary motion of another.

curvature · Curvature

Named alongside it

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

Path curvatureEuler savaryInflection circleInstantaneous centreCentrodeConjugate pointMoving planeClosed formCollineation axisInstant centreOsculating circleSecond-order

All concepts