Which of the triangles in Figure E.1 have the same size and the same shape (assume by ‘shape’ we mean labelled shape)?

Which triangles have the same shape but different sizes?

Which triangles have different shapes but the same ‘reflection shape’ (i.e. a translation, rotation, scale and a reflection will match them exactly)?

Rank the triangles in Figure E.1 in order of increasing centroid size. Repeat the rankings using baseline size (with baseline 1, 2) and square root of area.

Informally rank the triangles in Figure E.1 in order of ‘closeness’ to the anti-clockwise labelled equilateral triangle.

In a forensic study photographs of the front view of the faces of alleged criminals are taken. Some landmarks are to be located on the face. For each of the following landmarks decide whether it is a Bookstein’s Type I, II or III landmark. Also, state whether they are anatomical, mathematical or pseudo-landmarks.

1. Pupil of the eye
2. Tip of the nose
3. Corners of the mouth
4. Point on the cheek halfway between the pupil of the eye and corner of the mouth.
5. Centre of the forehead.
6. Lowest point on the chin.

Which landmarks are easy to locate and which are difficult?

Prove the result that

Consider the Helmert submatrix H. Write down H for k = 3, 4, 5 and verify that H^TH = C and HH^T = I.

Prove the general results for k ≥ 2 that H^TH = C and HH^T = I.

Rank the triangles in Figure E.1 in terms of Riemannian distance from the anti-clockwise labelled equilateral triangle.

1. For the male gorilla data (see Section 1.4.8) find the centroid sizes of the gorilla landmarks.
2. Obtain Bookstein’s shape variables for the specimens and provide a scatter plot of the data, registered on the baseline 1, 2.
3. Give the Bookstein’s shape variables for the largest specimen using landmarks 1, 2 (pr, l) as the baseline.

4. Provide a scatter plot of the data using landmarks 1, 6 (pr, na) as the baseline, and comment on the scatter compared with that using baseline 1, 2.

   Hint: to permute the rows 1, 2, 3, 4 to 1, 4, 2, 3 for an array x with 4 rows, one would use the command x <- x[c(1, 4, 2, 3),,].

5. Obtain the mean shape from Bookstein’s shape variables (using the arithmetic mean) using both baselines. Are they the same shape?
6. Find the specimen which is the furthest away from the mean in terms of Riemannian distance.

Prove the linear relationship between Bookstein coordinates U^B and Kendall coordinates U^K.

Consider a pre-shape z = x + iy (k − 1 −vector). Obtain the pre-shapes which are the furthest possible Riemannian distance from this pre-shape.

On Kendall’s spherical shape space where are the right-angled triangles located (in terms of the spherical coordinates θ and ϕ)?

Obtain an expression for the full Procrustes distance d_F in terms of the partial Procrustes distance d_P. Obtain a Taylor series expansion of d_F in terms of d_P giving the first two non-zero terms explicitly.

Prove that the Riemannian distance ρ is a distance, that is show that: (i) ρ(x, y) = 0 if and only if x = y; (ii) ρ(x, y) = ρ(y, x) for all x, y; and (iii) ρ(x, y) ≤ ρ(x, z) + ρ(z, y) for all x, y, z.

Plot the full and partial Procrustes and Riemannian distances versus ρ over the range of ρ. Discuss the relative values of the distances over the range of ρ.

For a triangle with Bookstein’s shape variables (U^B, V^B) find an expression for the Riemannian distance ρ to the (a) closest equilateral shape and (b) closest collinear shape.

Consider two centred configurations z₁, z₂ of k points in two dimensions, which are not necessarily of unit size. Find the closest Euclidean distance between z₁ and z₂ by rotating (but not scaling) z₂ to be as close as possible to z₁ (complex notation should help). Express the distance in terms of the centroid sizes of z₁ and z₂ and the Riemannian shape distance ρ(z₁, z₂).

Find the same expression for k points in m dimensions.

Consider two configurations of k points in m dimensions with pre-shapes Z₁, Z₂. Derive an expression for the partial Procrustes distance d_R between pre-shapes Z₁, Z₂ where the minimization is over the orthogonal group O(m) rather than SO(m).

Consider the female gorilla data. Find the partial Procrustes tangent coordinates v for the largest female gorilla skull, using the full Procrustes mean as the pole of the projection.

By using the inverse projection from the tangent plane to the figure space obtain a suitable icon for the largest gorilla, which is Procrustes rotated to be as close as possible to the full Procrustes mean female gorilla shape.

Provide a plot of all the female gorilla icons, Procrustes rotated to the full Procrustes mean (i.e. for each skull find the tangent coordinates and project back to an icon, using the full Procrustes mean as the pole).

Provide a pairwise plot of the coordinates of the icons and comment on any structure present in the plots.

Find the full Procrustes mean shapes of the the male and female gorillas separately by using the routine procGPA with options eigen=TRUE and eigen=FALSE. Check whether they give the same result (up to an arbitrary scale and rotation). Compare the percentages of variability explained by the first three PCs using the routines with either option.

What percentage of variability is explained by the first three PCs for the three groups of mouse T2 vertebrae (Small, Large, Control). Describe the geometrical features highlighted by each PC.

Give an educated guess as to what the full Procrustes mean shape would be from a dataset of two equilateral triangles – one clockwise and the other anti-clockwise labelled. Find the full Procrustes mean shape using the routine procGPA with options eigen=TRUE and eigen=FALSE, and comment on any differences or similarities in the results.

Provide plots of the thin-plate spline transformation from the orangutan female to male mean shapes. Test whether the mean shapes are significantly different or not.

Carry out relative warps analysis of the orangutan data, with respect to bending energy and inverse bending energy, and comment on the differences.

Plot the partial warp scores for the female and male orangutan data, and comment.

Investigate allometry in the 3D macaque data.