Freedom and Constraints: Bearings

Abstract

A point has three degrees of freedom in space: to define where it is, we need three coordinates x, y and z, or r, θ and z. It can make independent movements in three directions mutually at right angles. If we connect it by a link of fixed length to a fixed point with ball joints at each end, then we remove one of its degrees of freedom. If, for example, the fixed point is the origin and the link is of length L, then the point must lie somewhere on the surface of the sphere which is of radius L and has its centre at the origin, and it has only two degrees of freedom which can be expressed as the latitude and longitude on that sphere. The co-ordinates x, y, z must always represent a point on that sphere, which will be the case if

$$ {x^2} + {y^2} + {z^2} = {L^2} $$

((4.1))

One equation relating the co-ordinates is equivalent to the loss of one degree of freedom.