''A collection of corresponding border points.'' Though the above captures the characteristics of the term shape fairly well; this thesis will adapt the definition by D.G. Kendall  and define shape as:
''The characteristic surface configuration of a thing;
an outline or a contour.'' 
''Something distinguished from its surroundings by its outline.'' 
The term shape is - in other words - invariant to Euclidean transformations. This is reflected in figure 4.1. The next question that naturally arises is: How should one describe a shape? In everyday conversation, unknown shapes are often described as references to known shapes - e.g. "Italy has the shape of a boot". Such descriptions can obviously not easily be utilized in an algorithmic framework.
Dryden & Mardia further more discriminates landmarks into three subgroups :
If k denotes the Euclidean dimensions and n denotes the number of landmarks, the dimension of the shape space, follows from the above definition:
ProofInitially we have kn dimensions. The translation removes k dimensions, the uniform scaling one dimension and the rotation dimensions.If a relationship between the distance in shape space and Euclidean distance in the original plane can be established, the set of shapes actually forms a Riemannian manifold containing the object class in question. This is also denoted as the Kendall shape space . This relationship is called a shape metric. Often used shape metrics include the Hausdorff distance , the strain energy  and the Procrustes distance [21,20,6,14]. Where the two former compare shapes with unequal amount of points, the latter requiring corresponding point sets. In the following, the Procrustes distance is used.
In the following the Frobenius norm is used as a shape size metric: