2.2.1 Configuration space

Definition 2.1 The configuration is the set of landmarks on a particular object. The configuration matrix X is the k × m matrix of Cartesian coordinates of the k landmarks in m dimensions. The configuration space is the space of all landmark coordinates.

In our applications we have k ≥ 3 landmarks in m = 2 or m = 3 dimensions and the configuration space is typically the space of real k × m matrices (or equivalently ) with possibly some special cases removed, such as coincident points.

2.2.2 Centroid size

Before defining shape we should define what we mean by size, so that it can be removed from a configuration. Consider X to be a k × m matrix of the Cartesian coordinates of k landmarks in m real dimensions, that is the configuration matrix of the object.

Definition 2.2 A size measure g(X) is any positive real valued function of the configuration matrix such that

(2.1)

for any positive scalar a > 0.

Definition 2.3 The centroid size is given by:

(2.2)

where X_ij is the (i, j)th entry of X, the arithmetic mean in the jth dimension is ,

(2.3)

is the centring matrix,

is the Euclidean norm, I_k is the k × k identity matrix (diagonal matrix with ones on the diagonal), and 1_k is the k × 1 vector of ones.

Obviously S(aX) = aS(X) for a > 0, thus satisfying Equation (2.1). The centroid size S(X) is the square root of the sum of squared Euclidean distances from each landmark to the centroid, namely

where (X)_i is the ith row of X(i = 1, …, k) and is the centroid. This measure of size will be used throughout the book. In fact, the centroid size is the most commonly used size measure in geometrical shape analysis (e.g. Kendall 1984; Bookstein 1986; Goodall 1991; Dryden and Mardia 1992). The centroid size could also be used in a normalized form, for example S(X)/√k or S(X)/(km)^1/2, and this would be particularly appropriate when comparing configurations with different numbers of landmarks. The squared centroid size can also be interpreted as (2k)^{− 1} times the sum of the squared inter-landmark distances since

In Figure 2.2 we display boxplots of the centroid size of the six groups of great apes which were described in Section 1.4.8. The plot is obtained using the R commands:

data(apes)
size<-centroid.size(apes$x)
plot(apes$group,size)

It is clear that there are sex differences in centroid size for gorillas and orangutans (pongo), with the males all being larger than the females. For chimpanzees (pan) there is an overlap in the centroid size distributions of the sexes, but the males are significantly larger in mean centroid.size, for example using a t-test (p-value = 0.0002):

> t.test(size[apes$group=="panm"],size[apes$group=="panf"],
                                    alternative="greater")
      Welch Two Sample t-test
 
data: size[apes$group == "panm"] and size[apes$group == "panf"]
t = 3.8176, df = 50.704, p-value = 0.0001841

Example 2.1 de Souza et al. (2001a,b) analysed the sizes of stereolithography (SLA) skull models which had been constructed by a manufacturer using CT scans from a real head. In a test to assess the accuracy of the SLA models, six heads (cadavers) were CT scanned, and the CT scans were used to build SLA models. After the removal of the soft tissues, 37 specific anatomical landmarks were measured on both the SLA models and the corresponding real skulls. Using a paired t-test on the log of the centroid sizes, it was found that the SLA model was significantly larger by 1.5% than the original real skull. In particular, this size error was taken into account when designing custom implants, because even a small size increase would have severe clinical consequences.   □

2.2.3 Other size measures

An alternative size measure is the baseline size, that is the length between landmarks 1 and 2:

(2.4)

The baseline size was used as early as 1907 by Galton (1907) for normalizing faces and its use came into prominence with Bookstein coordinates, described in Section 2.4. This size variable is also useful in calculating size-and-shape distributions (Dryden and Mardia 1992) which we outline in Section 11.5.

Other alternative size measures include the square root of the area of the convex hull for planar configurations or the cube root of the volume of the convex hull for configurations in three dimensions which intuitively describes size, but these measures have not become popular. For triangles with coordinates (X₁, Y₁), (X₂, Y₂) and (X₃, Y₃) the area is simply given by:

Note that the volume of a parallelepiped in three dimensions formed by v₁, v₂, v₃ is given by the absolute value of the determinant of the matrix with columns v₁, v₂, v₃. Other size measures for triangles include the radius of the inscribed circle and the circumradius (e.g. Miles 1970), the latter being equal to S(X)/√3.

In general the choice of size will affect reported conclusions. In particular, if object A has twice the centroid size of object B it does not necessarily follow that object A is twice the size of object B when using a different measure. When reporting size differences one needs to state which size measure was used. The exception is when both objects have the same shape, in which case the ratio of sizes will be the same regardless of the choice of size variable. If comparing sets of collinear points, then very different conclusions about size are reached, with these pathological configurations all having zero area but non-zero centroid size for non-coincident collinear points.

Example 2.2 Consider the triangles in Figure 2.3. The centroid sizes of the triangles to three decimal places are:

The baseline sizes are:

The square root of area sizes to three decimal places are:

The relative ranking of triangles in terms of size differs depending on the choice of size measure. In particular, in terms of area A is larger than B, but in terms of centroid size B is larger than A. This example demonstrates that different choices of size measure can lead to different conclusions about size.   □

The following R commands give the calculations in the example:

> library(shapes)
> x1<-matrix(c(-1.5,-0.5,-1,1,1,sqrt(3)/2+1),3,2)
> x2<-matrix(c(0.5,2,1.25,1,1,1+sqrt(3)/4),3,2)
> x3<-matrix(c(-1.5,-0.5,-1.5,-1,-1,-1+1/sqrt(3)),3,2)
> x4<-matrix(c(0.5,2,1.25,-1,-1,0.75-1),3,2)
> centroid.size(x1)
[1] 1
> centroid.size(x2)
[1] 1.118034
> centroid.size(x3)
[1] 0.942809
> centroid.size(x4)
[1] 1.224745
> #baseline size
> sqrt( (x1[2,1]-x1[1,1])**2+(x1[2,2]-x1[1,2])**2 )
[1] 1
> sqrt( (x2[2,1]-x2[1,1])**2+(x2[2,2]-x2[1,2])**2 )
[1] 1.5
> sqrt( (x3[2,1]-x3[1,1])**2+(x3[2,2]-x3[1,2])**2 )
[1] 1
> sqrt( (x4[2,1]-x4[1,1])**2+(x4[2,2]-x4[1,2])**2 )
[1] 1.5
> #areas
> sqrt(abs((x1[3,2]-x1[1,2])*(x1[2,1]-x1[1,1])-
                  (x1[3,1]-x1[1,1])*(x1[2,2]-x1[1,2]))/2)
[1] 0.658037
> sqrt(abs((x2[3,2]-x2[1,2])*(x2[2,1]-x2[1,1])-
                  (x2[3,1]-x2[1,1])*(x2[2,2]-x2[1,2]))/2)
[1] 0.5698768
> sqrt(abs((x3[3,2]-x3[1,2])*(x3[2,1]-x3[1,1])-
                  (x3[3,1]-x3[1,1])*(x3[2,2]-x3[1,2]))/2)
[1] 0.537285
> sqrt(abs((x4[3,2]-x4[1,2])*(x4[2,1]-x4[1,1])-
                  (x4[3,1]-x4[1,1])*(x4[2,2]-x4[1,2]))/2)
[1] 0.75

Mosimann (1970) considers a definition of size based on length measurements which satisfies Equation (2.1), but with vectors of positive measurements (length, width, etc.) in place of the configuration matrix X.

In order to describe an object’s shape it is useful to specify a coordinate system. We initially consider some of the most straightforward coordinate systems, which helps to provide an elementary introduction to aspects of shape. A suitable choice of coordinate system for shape is invariant under translation, scaling and rotation of the configuration. Further coordinate systems are discussed later in Section 4.4.

2.3.1 Angles

For k = 3 points in m = 2 dimensions two internal angles are an obvious choice of coordinates that are invariant under the similarity transformations. For example, x₁ and x₂ could measure the shape of the triangle in Figure 2.4, where x₁ + x₂ + x₃ = 180°.

However, it soon becomes apparent that using angles to describe shape can be problematic. For cases such as very flat triangles (three points in a straight line) there are many different arrangements of three points. For example, see Figure 2.5, where the triangles all have x₁ = 0°, x₃ = 0°, x₂ = 180°, and yet the configurations have different shapes.

For larger numbers of points (k > 3) one could subdivide the configuration into triangles and so 2k − 4 angles would be needed. For the m = 3 dimensional case angles could also be used, but again these suffer from problems in pathological cases. Also, probability distributions of the angles themselves are not easy to work with (see Mardia et al. 1977). If the angles of the triangle are x₁, x₂ and x₃, then the use of log (x₁/x₃) and log (x₂/x₃) (where x₁ + x₂ + x₃ = 180°) has some potential for analysing triangle shape using compositional data analysis (Aitchison 1986; Pukkila and Rao 1988) and the approach can be adapted to higher dimensions. Mardia et al. (1996b) used angles from triangulations of landmark data on photographs of faces, and Mardia et al. (1977) used the angles between towns in central place data. In such cases, analysis of shape can proceed as in directional statistics (Mardia and Jupp 2000).

A Ramachandran plot (Ramachandran et al. 1963) is a method for visualizing backbone dihedral torsion angles ψ versus ϕ of amino acid residues in protein secondary structures. The two angles describe the torsion angles either side of each α-carbon atom, and provide very useful shape information. This plot contains important angular shape information about the protein backbone and has proved very popular in the proteomics literature. The torsion angles are helpful in the prediction of a protein’s 3D folding. Note that each angle is on the circle, S¹, and so the plots are on a torus S¹ × S¹. The plot is also known as a [ϕ, ψ] plot or a Ramachandran diagram. Another geometrical description of a protein backbone was given by Zacharias and Knapp (2013) called a (d,θ) plot.

2.3.2 Ratios of lengths

Another traditional approach to representing shape is through ratios of length measurements, or lengths divided by size. A considerable amount of work has been carried out in multivariate morphometrics using distances, ratios, angles, and so on and it is still very commonly used in the biological literature (Reyment et al. 1984). There are many situations, such as classification problems, where the techniques can be very powerful. However, sometimes the interpretation of the important linear combinations of ratios of lengths and angles can be difficult. It is often easier to interpret pictures in the original space of the specimens than in some derived multivariate space, so we consider methods where the geometry of the object is retained.

Typical applications of the multivariate morphometrics approach include the classification of species (taxonomy) or the sexing of skulls using lengths, angles or ratios of lengths between landmarks. A very commonly used method is to perform a principal component analysis (PCA) of the multivariate measurements and to interpret each component as a measure of some aspect of size or shape (e.g. Jolicoeur and Mosimann 1960). Quite often the first principal component has approximately equal loadings on each variable and so can be interpreted as an overall measure of size.

Mosimann (1970) gives a rigorous treatment of this subject and provides a theoretical framework for the study of the size and shape of positive length measurements. Mosimann (1970) defines the shape vector to be l/g(l), where g(l) is a size measure. He provides theorems for the independence of population size and shape, including characterizations of various distributions. Further details of the approach can be found in Mosimann (1975a,b, 1988) and Darroch and Mosimann (1985) and the references therein.

2.3.3 Penrose coefficent

Given a set of measurements d₁, …, d_m the Penrose (1952) size coefficient is:

and the Penrose shape coefficient is:

The coefficients are related to a coefficient of racial likeness (Penrose, 1952; after Pearson, 1926) given by:

These very traditional shape measures usually only capture part of the shape or size-and-shape information, so we prefer a newer approach where the geometry of the objects under study is retained.

2.4.1 Planar landmarks

Let (x_j, y_j), j = 1, …, k, be k ≥ 3 landmarks in a plane (m = 2 dimensions). Bookstein (1986, 1984) suggests removing the similarity transformations by translating, rotating and rescaling such that landmarks 1 and 2 are sent to a fixed position. If landmark 1 is sent to (0, 0) and landmark 2 is sent to (1, 0), then suitable shape variables are the coordinates of the remaining k − 2 coordinates after these operations. To preserve symmetry, we consider the coordinate system where the baseline landmarks are sent to and .

Definition 2.4 Bookstein coordinates (u^B_j, v_j^B)^T, j = 3, …, k, are the remaining coordinates of an object after translating, rotating and rescaling the baseline to and so that

(2.5a)

(2.5b)

where j = 3, …, k, D²₁₂ = (x₂ − x₁)² + (y₂ − y₁)² > 0 and − ∞ < u^B_j, v_j^B < ∞.

A geometrical illustration of these transformations is given in Figure 2.6. If the baseline is taken as (0, 0) and (1, 0), then there is no in the equation for u^B_j, as originally proposed by Bookstein (1986). We subtract the in u^B_j to also simplify the transformation to Kendall coordinates in Section 2.5, although precisely where we send the baseline is an arbitrary choice. These coordinates have been used widely in shape analysis for planar data. Bookstein coordinates are the most straightforward to use for a newcomer to shape analysis. However, because of the lack of symmetry in choosing a particular baseline and the fact that correlations are induced into the coordinates, many practitioners often prefer to use the Procrustes tangent coordinates (see Section 4.4).

Galton (1907) defined precisely the same coordinates [with baseline (0,0), (1,0)] at the beginning of the last century, but the statistical details of using this approach needed to be worked out.

The construction of Bookstein coordinates is particularly simple if using complex arithmetic. Bookstein coordinates are obtained from the original complex coordinates z^o₁, …, z_k^o, where :

(2.6)

where . Consider how the formulae in Equation (2.5a) are obtained (Mardia 1991). To find (u^B_j, v_j^B) for a fixed j = 3, …, k, we have to find the scale c > 0, the rotation A, and the translation b = (b₁, b₂)^T such that

(2.7)

where X = (x_j, y_j)^T and U = (u^B_j, v_j^B)^T are the coordinates of the jth point before and after the transformation, j = 3, …, k, and A is a 2 × 2 rotation matrix, namely

where we rotate clockwise by θ radians. Applying the transformation to landmarks 1 and 2 gives four equations in four unknowns (c, θ, b₁, b₂)

Now, we can solve these equations and see that the translation is

the rotation (in the appropriate quadrant) is

and the rescaling is c = {(x₂ − x₁)² + (y₂ − y₁)²}^{− 1/2}. So,

(2.8)

Substituting X = (x_j, y_j)^T(j = 3, …, k) and c, A, b into Equation (2.7) we obtain the shape variables of Equation (2.5a). This solution can also be seen from Figure 2.6 using geometry.

Example 2.3 In Figure 2.7 we see scatter plots of the Bookstein coordinates for the T2 Small vertebrae of Section 1.4.1. We take landmarks 1 and 2 as the baseline. Note that the marginal scatter plots at landmarks 4 and possibly 3 and 5 are elliptical in nature. The plot can be obtained using:

data(mice)
qset2<-mice$x[,,mice$group=="s"]
u<-bookstein2d(qset2)$bshpv
plotshapes(u,symbol=as.character(1:6))

For example, the first mouse has Bookstein shape variables given by the last four rows of:

> u[,,1]
, , 1

         [,1] [,2]
[1,] -0.50000000 0.0000000
[2,] 0.50000000 0.0000000
[3,] 0.09121051 0.2826022
[4,] 0.04065577 0.5089238
[5,] -0.07367575 0.2735386
[6,] -0.02672182 -0.3054334

and this could also be obtained directly using the command:

bookstein.shpv(qset2[,,1])

For practical data analysis it is sensible to choose the baseline as landmarks that are not too close together, as in this example. As often happens with using Bookstein coordinates, the variability in the points away from the origin appears larger than the points nearer to the origin. This is an artifact of this coordinate system, and will be explored further in Sections 9.4 and 11.1.4. The centroid sizes of the bones are obtained using:

sz<-centroid.size(qset2)

A pairwise scatter plot of Bookstein’s shape variance and centroid size is obtained using:

x<-rbind(u[3:6,1,],u[3:6,2,])
x<-rbind(matrix(sz,1,23),x)
pairs(t(x),labels=c("s","u3","u4","u5","u6","v3","v4","v5","v6"))

In Figure 2.8 we see pairwise scatter plots of Bookstein coordinates. There are strong positive correlations in the v^B₃, v₄^B and v^B₅ coordinates. There are also positive correlations between v^B₃ and v^B₆, between v^B₅ and v^B₆, and between u^B₃ and u^B₄. However, it is also an artifact of the coordinate system that correlations are induced into the shape variables, even when the landmark coordinates are uncorrelated (see Section 11.1.4), and so correlations can be difficult to interpret.

We can also examine the joint relationship between size and shape, which will be considered in more detail in Chapter 5. Scatter plots of the centroid size S versus each of the shape coordinates are also given in Figure 2.8. We see that there are quite strong positive correlations between S and each of v^B₃, v₄^B and v^B₅.   □

Throughout the text we shall often refer to the real (2k − 4)-vector of Bookstein coordinates u^B = (u^B₃, …, u_k^B, v^B₃, …, v_k^B)^T, stacking the coordinates in this particular order.

One approach to shape analysis is to use standard multivariate analysis of the Bookstein coordinates, ignoring the non-Euclidean nature of the space. Provided variations in the data are small, then the method is adequate for mean estimation and hypothesis testing. For example, we could obtain an estimate of the mean shape of the configuration by taking the arithmetic average of the Bookstein coordinates (the Bookstein mean shape). The Bookstein mean shape for the T2 Small mouse vertebrae of Example NaN is given by the last four rows of:

> bookstein2d(mice$x[,,mice$group=="s"])$mshape
        [,1] [,2]
[1,] -0.50000000 0.000000e+00
[2,] 0.50000000 -6.634366e-21
[3,] 0.08469746 2.933430e-01
[4,] 0.01215768 5.613175e-01
[5,] -0.06874750 2.991278e-01
[6,] -0.02502185 -3.041418e-01

which when rounded to three decimal places is:

The variability of the shape variables is less straightforward to interpret. Transforming the objects (or registering) to a given edge induces correlations into the shape variables in general and this can lead to spurious correlations (Kent 1994), see Sections 9.4 and 11.1.4. So the method should not be used to interpret the structure of shape variability unless there is a good reason to believe that two landmarks are essentially fixed (see Section 9.4).

2.4.2 Bookstein-type coordinates for 3D data

Registration on a base to filter out the similarity transformations can also be carried out for 3D data, in a similar way to edge registration in Bookstein coordinates for 2D data. Consider k landmarks , in three dimensions. The number of shape variables is 3k − 7 since we have 3k landmark coordinates but must remove 3 location, 1 scale and 3 rotation parameters. The Bookstein coordinates can be taken as

(2.9)

where A is a 3 × 3 rotation matrix (a function of (X₁, X₂, X₃)) and

where u₂₃ ≥ 0, u₃₃ = 0 and X_j → u_j for j = 4, …, k. We drop the superscript B denoting Bookstein coordinates in this section. Geometrically the figure has been translated, rotated and rescaled so that the baseline (landmarks 1 and 2) is of unit length, has midpoint at the origin and is rotated to lie along the x-axis. The figure is further rotated so that landmark 3 lies in the plane of the first two dimensions (X, Y), with Y ≥ 0. The remaining non-fixed coordinates are the shape variables (cf. Mardia and Dryden 1997). The details are as follows: first translate the figure,

We then rescale by dividing by the length of the baseline between points 1 and 2: D₁₂ = 2(w²₁₂ + w²₂₂ + w₃₂²)^1/2, and rotate through clockwise angles of θ about the z-axis, ω about the y-axis and ϕ about the x-axis where

So, using matrix notation if

then the 3D Bookstein coordinates are:

An explicit expression for the shape coordinates is:

where a = 2(w²₁₂ + w₂₂² + w²₃₂),  j = 4, …, k, and

As in the m = 2 dimensional case, care must be taken when interpreting correlations between the shape variables, because spurious correlations can occur with edge registration methods such as this (see Sections 9.4 and 11.1.4).

Example 2.4 In Figure 2.9(a) we see a male macaque skull from the dataset described in Section 1.4.3, with 24 landmarks located in three dimensions and a wire box regular grid drawn over the skull. Let us consider the page to be the x–y plane and the z-axis is perpendicular to the page. The 3D Bookstein coordinates involve selecting two landmarks (e.g. landmarks 1 and 6 in Figure 1.11) to translate, rotate and rescale to lie horizontal in the plane of the page at points and , and a third landmark (e.g. landmark 10) is chosen to lie in the x–y plane, as displayed in Figure 2.9(b).

Important point: For affine shape (where invariance is under affine transformations rather than Euclidean similarity transformations – see Section 12.2), there are 12 rather than 7 constraints. Hence all the coordinates of the first four landmarks are sufficient to form a base in the affine case and the expressions are simpler than for similarity shape.

2.6.1 Bookstein coordinates for triangles

For the case of a triangle of landmarks we have two shape coordinates.

Example 2.5 In Figure 2.3 we see the triangles with internal angles at points 1, 2, 3 given by:

A: 60°, 60°, 60° (equilateral),

B: 30°, 30°, 120° (isosceles) ,

C: 90°, 30°, 60° (right-angled),

D: 45°, 45°, 90° (right-angled and isosceles).

Bookstein coordinates for these three triangles with baseline points 1, 2 are:

A: ,

B: ,

C: ,

D: .    

A plot of the shape space of triangles for Bookstein coordinates is given in Figure 2.10. Each triangle shape is drawn with its centroid in the position of the Bookstein coordinates (U^B, V^B) corresponding to its shape in the shape space. For example, the equilateral triangles are at (0, √3/2) and (0, −√3/2) (marked E and F ). The right-angled triangles are located on the lines U^B = −0.5, U^B = 0.5 and on the circle (U^B)² + (V^B)² = 0.5; the isosceles triangles are located on the line U^B = 0 and on the circles (U^B + 0.5)² + (V^B)² = 1 and (U^B − 0.5)² + (V^B)² = 1; and the flat triangles (3 points in a straight line) are located on V^B = 0.

The plane can be partitioned into two half-planes – all triangles below V^B = 0 can be reflected to lie above V^B = 0. In addition, each half-plane can be partitioned into six regions where the triangles have their labels permuted. If a triangle has baseline 1, 2 and apex 3 and side lengths d₁₂, d₁₃ and d₂₃, then the six regions are:

Thus, if invariance under relabelling and reflection of the landmarks was required, then we would be restricted to one of the 12 regions, for example the region AOE, bounded by the arc of isosceles triangles AE, the line of isosceles triangles OE and the line of flat triangles AO in Figure 2.10.

It is quite apparent from Figure 2.10 that a non-Euclidean distance in (U^B, V^B) is most appropriate for the shape space. For example, two triangles near the origin that are a Euclidean distance of 1 apart are very different in shape, but two triangles away from the origin that are a Euclidean distance of 1 apart are quite similar in shape. Non-Euclidean shape metrics are described in Section 4.1.1. Bookstein (1986) also suggested using the hyperbolic Poincaré metric for triangle shapes, which is described in Section 12.2.4.

2.6.2 Kendall’s spherical coordinates for triangles

For k = 3 we will see in Section 4.3.4 that the shape space is a sphere with radius . A mapping from Kendall coordinates to the sphere of radius is:

(2.13)

and r² = (u^K₃)² + (v₃^K)², so that , where u^K₃ and v^K₃ are Kendall coordinates of Section 2.5.

Definition 2.7 Kendall’s spherical coordinates (θ, ϕ) are given by the polar coordinates

(2.14)

where 0 ≤ θ ≤ π is the angle of latitude and 0 ≤ ϕ < 2π is the angle of longitude.

The relationship between (u^K₃, v₃^K) and the spherical shape variables (Mardia 1989b) is given by:

(2.15)

The sphere can be partitioned into 6 lunes and 12 half-lunes. In order to make the terminology clear, let us note that one example full-lune is 0 ≤ ϕ ≤ π/3, 0 ≤ θ ≤ π and one example half-lune is 0 ≤ ϕ ≤ π/3, 0 ≤ θ ≤ π/2.

In Figure 2.11 we see triangle shapes located on the spherical shape space. The equilateral triangle with anti-clockwise labelling corresponds to the ‘North pole’ (θ = 0) and the reflected equilateral triangle (with clockwise labelling) is at the ‘South pole’ (θ = π). The flat triangles (three collinear points) lie around the equator (θ = π/2). The isosceles triangles lie on the meridians ϕ = 0, π/3, 2π/3, π, 4π/3, 5π/3. The right-angled triangles lie on three small circles given by:

and we see the arc of unlabelled right-angled triangles on the front half-lune in Figure 2.11.

Reflections of triangles in the upper hemisphere at (θ, ϕ) are located in the lower hemisphere at (π − θ, ϕ). In addition, permuting the triangle labels gives rise to points in each of the six equal half-lunes in each hemisphere. Thus, if invariance under labelling and reflection was required, then we would be restricted to one of these half-lunes, for example the sphere surface defined by 0 ≤ ϕ ≤ π/3, 0 ≤ θ ≤ π/2. Consider a triangle with labels A, B and C, and edge lengths AB, BC and AC. If the labelling and reflection of the points was unimportant, then we could relabel each triangle so that, for example, AB ≥ AC ≥ BC and point C is above the baseline AB.

If we have a triangle in three dimensions, then we see that we can translate, rotate and rescale so that

where u₂₃ > 0. Hence, the triangle has two shape coordinates (u₁₃, u₂₃)^T which lie in a half-plane. We could transfer to Kendall’s spherical shape coordinates of Equation (2.14), although the range of the latitude angle θ will be , as u₂₃ > 0. Hence, the shape space for triangles in three dimensions is the hemisphere with radius half, that is , as we will see in Section 4.3.4. This is also the case for triangles in more than three dimensions.

2.6.3 Spherical projections

For practical analysis and the presentation of data it is often desirable to use a suitable projection of the sphere for triangle shapes. Kendall (1983) defined an equal area projection of one of the half-lunes of the shape sphere to display unlabelled triangle shapes. The projected lune is bell-shaped and this graphical tool is also known as ‘Kendall’s Bell’ or the spherical blackboard (an example is given later in Figure 14.2).

An alternative equal-area projection is the Schmidt net (Mardia 1989b) otherwise known as the Lambert projection given by:

In Figure 2.12 we see a plot of one of the half-lunes on the upper hemisphere of shape space projected onto the Schmidt net. Example triangles are drawn with their centroids at polar coordinates (ξ, ψ) in the Schmidt net.

2.6.4 Watson’s triangle coordinates

Watson (1986) considers a coordinate system for triangle shape. Let be the complex landmark coordinates. Let ω = exp(iπ/3) and u = (1, ω, ω²)^T, So, 1₃, u and are orthogonal and have length √3. One can express the triplet of points z as:

The vector 1₃ represents the degenerate triangle shape (all points coincident), u is an equilateral triangle and is its reflection. Since z has the same shape as cz + d1₃, where , the shape of the triangle can be obtained from . The Watson shape coordinates are given by the complex number b. All unlabelled triangles can be represented in the sector of the unit disc, with |b| ≤ 1 and 0 < Arg(b) < π/3. Watson (1986) constructs sequences of triangle shapes from products of circulant matrices.