For small areas, distances on the surface of the hypersphere are approximated well by distances on its projection (Figure 14.2) because a sin a for small angles. For what size angle is the distortion a/ sin(a) (i) 1.01, (ii) 1.05 and (iii) 1.1?

x5 projectedH Figure 14.2 Projections of small areas of the unit sphere preserve distances. Left A projection of the 2D semicircle to 1D. For the points x1, x2,x3, X4,5 at x coordinates -09,-02, 0,02, 0.9 the distance lx2xal 0.201 only differs by 0.5% from 1x341 1.06/0.9 ~ 1.18 is an example of a large distortion (18%) when projecting a large area. Right: The corresponding projection of the 3D hemisphere to 2D x5 projectedH Figure 14.2 Projections of small areas of the unit sphere preserve distances. Left A projection of the 2D semicircle to 1D. For the points x1, x2,x3, X4,5 at x coordinates -09,-02, 0,02, 0.9 the distance lx2xal 0.201 only differs by 0.5% from 1x341 1.06/0.9 ~ 1.18 is an example of a large distortion (18%) when projecting a large area. Right: The corresponding projection of the 3D hemisphere to 2D