Choice of support points
A kinematic support for a one-dimensional beam requires exactly two support points. Three or more support points will not share the load evenly (unless they are hinged in a non-rigid whiffle tree or similar). The position of those points can be chosen to minimize various forms of gravity deflection. A beam supported at the ends will sag in the middle, resulting in the ends moving closer together and tilting upward. A beam supported only in the middle will sag at the ends, making a similar shape but upside down.Airy points
Supporting a uniform beam at the Airy points produces zero angular deflection of the ends. The Airy points are symmetrically arranged around the centre of the length standard and are separated by a distance equal to : of the length of the rod. "End standards", that is standards whose length is defined as the distance between their flat ends such as long gauge blocks or the , must be supported at the Airy points so that their length is well-defined; if the ends are not parallel, the measurement uncertainty is increased because the length depends on which part of the end is measured. For this reason, the Airy points are commonly identified by inscribed marks or lines. For example, a 1000 mmBessel points
"Line standards" are measured between lines marked on their surfaces. They are much less convenient to use than end standards but, when the marks are placed on the neutral plane of the beam, allow greater accuracy. To support a line standard, one wishes to minimise the ''linear'', rather than angular, motion of the ends. The Bessel points (after Friedrich Wilhelm Bessel) are the points at which the length of the beam is maximized. Because this is a maximum, the effect of a small positioning error is proportional to the square of the error, an even smaller amount. The Bessel points are located 0.5594 of the length of the rod apart, slightly closer than the Airy points. Because line standards invariably extend beyond the lines marked on them, the optimal support points depend on both the overall length and the length to be measured. The latter is the quantity to be maximized, requiring a more complex calculation. For example, the 1927–1960 definition of the metre specified that theOther support points of interest
Other sets of support points, even closer than the Bessel points, which may be wanted in some applications are: * The points for minimum sag, 0.5536 times the length. Minimum sag occurs when the centre of the rod sags the same amount as the end points, which is not quite the same thing as minimum ''horizontal'' motion of the ends. * TheSee also
* History of measurement *References