Walking on Kendall\u2019s Shape Space: Understanding Shape Spaces and Their Coordinate Systems

Christian Peter Klingenberg

doi:10.1007/s11692-020-09513-x

Abstract

More and more analyses of biological shapes are using the techniques of geometric morphometrics based on configurations of landmarks in two or three dimensions. A fundamental concept at the core of these analyses is Kendall’s shape space and local approximations to it by shape tangent spaces. Kendall’s shape space is complex because it is a curved surface and, for configurations with more than three landmarks, multidimensional. This paper uses the shape space for triangles, which is the surface of a sphere, to explore and visualize some properties of shape spaces and the respective tangent spaces. Considerations about the dimensionality of shape spaces are an important step in understanding them, and can offer a coordinate system that can translate between positions in the shape space and the corresponding landmark configurations and vice versa. By simulation studies “walking” along that are great circles around the shape space, each of them corresponding to the repeated application of a particular shape change, it is possible to grasp intuitively why shape spaces are curved and closed surfaces. From these considerations and the available information on shape spaces for configurations with more than three landmarks, the conclusion emerges that the approach using a tangent space approximation in general is valid for biological datasets. The quality of approximation depends on the scale of variation in the data, but existing analyses suggest this should be satisfactory to excellent in most empirical datasets.

Highlights

Geometric morphometrics is widely used to analyze biological shapes in a variety of research contexts, including systematics and evolutionary biology, anthropology, the biomedical sciences, and increasingly developmental biology and genetics (Klingenberg 2010; Zelditch et al 2012)
Because statistical maneuvers like these are fundamental to the biological applications of geometric morphometrics, an understanding of shape spaces is critically important for biological shape analysis
This paper has explored Kendall’s shape space for triangles by travelling around it on paths that are great circles on its spherical surface

Summary

Introduction

Geometric morphometrics is widely used to analyze biological shapes in a variety of research contexts, including systematics and evolutionary biology, anthropology, the biomedical sciences, and increasingly developmental biology and genetics (Klingenberg 2010; Zelditch et al 2012). Shape spaces are specific to particular classes of shapes, for instance, all possible shapes for a given number of landmarks in two or three dimensions (Small 1996; Kendall et al 1999; Dryden and Mardia 2016; Srivastava and Klassen 2016). The concept of shape spaces gives a firm mathematical basis to statistical maneuvers such as estimating average shapes or characterizing variation of shapes around those averages, which are otherwise not well understood (Small 1996; Kendall et al 1999; Dryden and Mardia 2016; Srivastava and Klassen 2016). Because statistical maneuvers like these are fundamental to the biological applications of geometric morphometrics, an understanding of shape spaces is critically important for biological shape analysis

Objectives

Conclusion