5 Size-and-shape space

5.1 Introduction

In the previous chapters we have primarily considered the geometric invariance of objects under translation, rotation and scale. In some applications the joint study of size and shape is appropriate, and we use the term size-and-shape in this case. Another common term used for size-and-shape is form, and this is particularly common in biological applications.

In molecule comparisons (see Sections 1.4.4 and 1.4.9) the relative scales of the molecules are known, and so we do not need scale invariance. Hence, molecules are usually compared geometrically in size-and-shape space, rather than shape space.

Another example where size-and-shape is appropriate is the analysis of the microfossil data of Section 1.4.16. It is of interest to examine whether the size is related to shape. The form of the microfossils is represented in the size-and-shape space, which is a product of the positive real line (for size) and shape space.

The definition of size-and-shape of an object was given in Definition 3.9. When carrying out size-and-shape analysis all objects must be recorded on the same scale, which we denote as being commensurate in scale. In many applications we do have scale information so a choice needs to be made as to whether to work in size-and-shape space, or to work in shape space and consider the size variable separately.

5.2 Root mean square deviation measures

One way of measuring size-and-shape differences between sets of points is to ...

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