right hand and a left hand. In some cases, both full and partial PS may also include reflection. Reflection allows, for instance, a successful (possibly perfect) superimposition of a right hand to a left hand. Thus, partial PS with reflection enabled preserves size but allows translation, rotation and reflection, while full PS with reflection enabled allows translation, rotation, scaling and reflection. Optimal translation and scaling are determined with much simpler operations (see below). Ordinary Procrustes analysis Here we just consider objects made up from a finite number k of points in n dimensions. Often, these points are selected on the continuous surface of complex objects, such as a human bone, and in this case they are called landmark points. The shape of an object can be considered as a member of an equivalence class formed by removing the translational, rotational and uniform scaling components. Translation For example, translational components can be removed from an object by translating the object so that the mean of all the object's points (i.e. its centroid) lies at the origin. Mathematically: take points in two dimensions, say . The mean of these points is where Now translate these points so that their mean is translated to the origin , giving the point . Uniform scaling Likewise, the scale component can be removed by scaling the object so that the root mean square distance (RMSD) from the points to the translated origin is 1. This RMSD is a statistical measure of the object's scale or size: The scale becomes 1 when the point coordinates are divided by the object's initial scale: . Notice that other methods for defining and removing the scale are sometimes used in the literature. Rotation Removing the rotational component is more complex, as a standard reference orientation is not always available. Consider two objects composed of the same number of points with scale and translation removed. Let the points of these be , . One of these objects can be used to provide a reference orientation. Fix the reference object and rotate the other around the origin, until you find an optimum angle of rotation such that the sum of the squared distances (SSD) between the corresponding points is minimised (an example of least squares technique). A rotation by angle gives . where (u,v) are the coordinates of a rotated point. Taking the derivative of with respect to and solving for when