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).
Zero degrees of freedom concentrates all probability at zero. All of this leaves open the question of what happens with zero degrees of freedom when the noncentrality parameter is not zero. The noncentral chi-squared distribution with zero degrees of freedom and with noncentrality parameter "μ" is the distribution of This concentrates probability "e" at zero; thus it is a mixture of discrete and continuous distributions Zero degrees of freedom In statistics, the non-central chi-squared distribution with zero degrees of freedom can be used in testing the null hypothesis that a sample is from a uniform distribution on the interval (0, 1). This distribution was