Machine Kinematics
solved, then drawn
Concept

Euler savary — where it appears

The relation fixing every point's path curvature from its position relative to the pole, through one length and one angle. It is the field's workhorse relation, and this site checks it against derivatives of the loop equation rather than deriving it — the two agree to fourteen places over dozens of points.

Named by 8 essays across 3 fields — 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 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 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 A cam profile is an envelope, and its curvature obeys the same law. The dashed curve is the pitch curve — where the roller's centre travels — and the solid one is the surface that has to be cut, which is the pitch curve offset inward by the roller radius. That offset is the conjugate law of this field with one centre of curvature sent to infinity, and it says ρ_cut = ρ_pitch − r. Measured off the drawn polyline at six angles, the worst departure is 4.8e-7: at 95° the pitch curve has radius 29.52 and the cut profile 21.52, against 21.52 predicted. It is also why undercutting is a curvature condition rather than an accident: where ρ_pitch falls below the roller radius the offset turns itself inside out.

A profile is an envelope

A cam's surface is not a curve somebody drew. It is the envelope of the roller as the roller runs round the pitch curve, and its curvature is the pitch curve's less the roller radius — a law that this site's cam library was breaking for six months, in the one curve that gets manufactured and the only one no check looked at.

cams · Cam The string is the radius. An involute is generated by unwinding a taut string from the base circle, and the taut string is the flank's normal — so the point where it leaves the base circle is the centre of curvature and the string's length is the radius. At a flank radius r that length is √(r² − r_b²), and the dashed curve is that expression. The dots are the curvature of the polyline this site actually draws, measured by circumcircle through consecutive points, over 324 of them. Worst relative disagreement 1.9e-6. At the base circle the radius of curvature is zero, which is why a flank cut below it is not an involute and why undercutting removes exactly that part.

Two flanks, one law

At a gear mesh the two tooth flanks are conjugate profiles, the pitch point is the pole of their relative motion, and their radii of curvature are tied to each other rather than free. Each flank's radius varies fourfold across the mesh and the sum of the two does not vary at all.

gears · Tooth 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.

A curvature is a size with a minus sign

Everything in the curvature field is a similarity invariant in shape and a reciprocal length in value. Scale a moving plane and its inflection circle scales, its cubic of stationary curvature scales, and every curvature it computes is divided by the factor — so a bigger machine traces gentler paths than its drawing suggests.

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 curvatureInflection circleConjugate pointPole tangentOsculating circleCentrodeInstant centreInstantaneous centreCollineation axisCoupler curveEnvelopeUndercut

All concepts