Areas in central projection coordinates We have already seen that on the R-sphere, the area ofan cc-lune is 26: - R2. Let's compute this a different way. Problem 6 Let K 2: {} and consider the following region in central projection J'c Use the fact that the area of an a-lune is 2:2: ~ R2 to compute the area of the region. On the other hand, we know the area of the parallelogram spanned by two vectors in K-warped space is det V .K V W . F V det c'c Ve We 'd Ve V . * W W . RW Ve"c We We'd We In particular, if we want to know the area spanned by unit vectors Vc = and = we find the area is exactly vdet P.. This means that to compute area of a region L. using integrals, but in central projection, we use I VaetP dx. dy. Problem 7 Compute \\ det Pc. Hint: As a gesture of friendship, we'll remind you that Pc = (Ky2 + 1) 24 -KXcyc24 -KXcyc24 (Kx2 + 1)14At this point we see we are interested in the following integral ( K(x2 + yz) +1)-3/2dx, dyc. Problem 8 Examining the following diagram yc Xc convert to polar coordinates and compute the integral. Hint: Recall that to convert to polar coordinates, set r = Xe + Vc, 0 = arctan()c/Xc), and replace dxe dy with r dr de