Machine Kinematics
solved, then drawn
Concept

Ball point — where it appears

The one point of a moving plane whose path has four-point contact with its own tangent line, lying on both the inflection circle and the cubic. It is where a linkage draws its straightest line, and it is generally not where a designer put the tracing point — the site measures how much is left on the table.

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

Ball's point at 66°. Two curves and one point. The circle is the locus of points whose path is momentarily straight; the cubic is the locus of points whose path curvature is momentarily not changing. A point on both has a path that is straight and staying straight — four-point contact with its own tangent line, one order better than anything else in the plane. That is Ball's point. Walking the circle and looking for a zero of κ′ turned up 1 sign changes here and 1 of them survived being checked for continuity — the others are the asymptote the pole itself produces. Its measured order of contact with the line is 3.95. positioned by solving, not by drawing.

The straightest point there is

Where the circle of points going straight meets the cubic of points whose curvature is standing still, there is one point whose path is straight and staying straight. Neither Watt nor Chebyshev put their tracing point there, and moving Watt's to it more than halves the error over the whole stroke.

curvature · Curvature Three points, three orders of contact. Replace the coupler point by a crank pivoting at the centre of its path's osculating circle, drive the linkage away from the instant, and measure how far the point gets from that circle. The distance grows like a power of the step, and the power is the number of derivatives that agreed. An ordinary point gives 3.00; a point of the cubic of stationary curvature gives 3.94; a Burmester point, where the curvature is stationary and its rate of change is too, gives 4.97. Nothing in the measurement knows which kind of point it was handed. This is what the two special curves are for: they are where a single pivot can replace a whole linkage for longest. positioned by solving, not by drawing.

How long a pivot stands in for a linkage

Throw the four-bar away and replace a coupler point by a crank pivoting at the centre of its path's curvature. The two separate like the cube of the crank step for an ordinary point, the fourth power on the cubic, and the fifth at a Burmester point — three exponents that are the whole field, measured with no derivative anywhere in the measurement.

curvature · Curvature

Named alongside it

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

Contact orderCubic of stationary curvaturePath curvatureStructural errorBurmester pointInflection circleOsculating circleParameterisationPrecision-pointStraight line mechanism

All concepts