Notes on Differentiation - Michigan State University

Math 829 Spring 19994.3 Analytic statement of the problem. It’s pretty clear that the stereographicprojection preserves the sense of angles, so we will concentrate on preservation ofmagnitudes of angles. Here is the analytic expression of what needs to be done.Suppose p ∈ S 2 \{N} and v, w ∈ R 3 with〈p, v〉 =0 and 〈p, w〉 =0, (9)(i.e. v and w are orthogonal to the line from the origin to p, and hence tangent to S 2at p). Let A =[Dσ(p)], the matrix of Df(p) with respect to the standard basis. Wedesire to show that〈Av, Aw〉|Av||Aw|=〈v, w〉|v||w| . (10)The quantities that show up on the left and right hand sides of this last equationare, you will recall, the cosines of the angles between the respective pairs of vectors 1 .4.4 Exercise—reduction of problem. Show that in order to prove (10) you needonly show that there is a positive constant c such thatfor all v, w ∈ R 3 satisfying (9).In order to prove (11) we observe that〈Av, Aw〉 = c〈v, w〉 (11)〈Av, Aw〉 = 〈A ∗ Av, w〉where A ∗ is the linear transformation R 2 → R 3 whose matrix is the transpose of thematrix of A. This is just a matrix calculation based on the fact that 〈x, y〉 =[x] T [y](matrix product), where x and y are any vectors in the same Euclidean space, [x]and [y] are their respective column vectors, and the superscript “T ” denotes matrixtranspose. Thus:〈Aw, Av〉 = [Aw] T [Av] =([A][w]) T [A][v]= [w] T [A] T [A][v] =[w] T [A ∗ A][v]= 〈w, A ∗ Av〉from which the desired result follows by the symmetry of the inner product (it isunchanged if the order of its entries is reversed). Note that this calculation works aswell for any m × n matrix and any pair of vectors in R n .1 Remember that the inner product on the left-hand side of the equation is the one for R 3 , andthe one on the right is that of R 2 .-6-