Charts on SO(3) closed set under composition. This visualization can be extended to a general rotation in 3-dimensional space. The identity rotation is a point, and a small angle of rotation about some axis can be represented as a point on a sphere with a small radius. As the angle of rotation grows, the sphere grows, until the angle of rotation reaches 180 degrees, at which point the sphere begins to shrink, becoming a point as the angle approaches 360 degrees (or zero degrees from the negative direction). This set of expanding and contracting spheres represents a hypersphere in four-dimensional space (a 3-sphere).
Degrees of freedom (statistics) the vectors of residuals are constrained to lie in a space of smaller dimension than the number of components in the vector. That smaller dimension is the number of degrees of freedom for error, also called residual degrees of freedom. Perhaps the simplest example is this. Suppose are random variables each with expected value μ, and let be the "sample mean." Then the quantities are residuals that may be considered estimates of the errors "X" − μ. The sum of the residuals (unlike the sum of the errors) is necessarily 0. If one knows the values of any "n" −