Quaternions and spatial rotation the rotation. As the rotation angles become larger, the slice moves in the negative "z" direction, and the circles become larger until the equator of the sphere is reached, which will correspond to a rotation angle of 180 degrees. Continuing southward, the radii of the circles now become smaller (corresponding to the absolute value of the angle of the rotation considered as a negative number). Finally, as the south pole is reached, the circles shrink once more to the identity rotation, which is also specified as the point at the south pole. Notice that a number of characteristics of such
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" −