Stereographic projection and inverse geometryThe conformal property of stereographic projections can be established fairly effi-ciently using the concepts and methods of inverse geometry. This topic is relativelyelementary, and it has important connections to complex variables and hyperbolic (=Bolyai-Lobachevsky noneuclidean) geometry.Definition. Let r > 0 be a real number, let y ∈ Rn, and let S(r; y) be the set of allpoints x ∈ Rnsuch that |x − y| = r. The inversion map with respect to the sphere S(r; y)is the map T on Rn− {0} defined by the formulaT (x) = y +r2|x − y|·x − y.Altternatively, T (x) is defined so that T (x) − y is the unique positive scalar multiple ofx − y such thatT (x) − y·x − y= r2.Another way of saying this is that inversion interchanges the exterior points to S(r; y) andthe interior points with the center deleted. If r = 1 and y = 0 then inversion simply takesa nonzero vector x and sends it to the nonzero vector pointing in the same direction withlength equal to the reciprocal of |x| (this should explain the term “inversion”).If n = 2 then inversion corresponds to the conjugate of a complex analytic function.Specifically, if a is the center of the circle and r is the radius, then inversion is given incomplex numbers by the formulaT (z) = r2·z − a−1= r2·z − a−1where the last equation holds by the basic properties of complex conjugation. If r = 1 anda = 0 then inversion is just the conjugate of the analytic map sending z to z−1.The geometric properties of the analytic inverse map on the complex plane are fre-quently discussed in complex variables textbooks. In particular, this map has nonzeroderivative wherever the function is defined, and accordingly the map is conformal. Fur-thermore, the map z → z−1is an involution (its composite with itself is the identity), andit sends circles not containing 0 to circles of the same type. In addition, it sends lines notcontaining the origin into circles containing the origin and vice versa (Note: This meansthat 0 lies on the circle itself and NOT that 0 is the center of the circle!). Since complexconjugation sends lines and circles to lines and circles, preserves the angles at which curvesintersect and also sends 0 to itself, it follows that the inversion map with respect to theunit circle centered at 0 also has all these properties).It turns out that all inversion maps have similar properties. In particular, they sendthe every point of sphere S(r; y) to itself and interchange the exterior points of that sphere1with all of the interior points except y (where the inversion map is not defined), theypreserve the angles at which curves intersect, they are involutions, they send hyperspheresnot containing the central point y to circles of the same type, and they send hyperplanesnot containing {y} into hyperspheres containing {y} and vice versa. We shall limit ourproofs to the properties that we need to study stereographic projections; the reader isencouraged to work out the proofs of the other assertions.Relating stereographic projections and inversionsWe begin by recalling(?) some simple observations involving isometries and similaritytransformations from Rnto itself.FACT 1. If b ∈ Rnand F is translation by b (formally, F (x) = x + b), then F is anisometry from Rnto itself and F sends the straight line curve from x to y defined byα(t) = ty + (1 − t)xto the straight line curve from F (x) to F (y) defined byα(t) = tF (y) + (1 − t)F (x) .Definition. If r > 0 then a similarity transformation with ratio of similitude r ona metric space X is a 1–1 correspondence f from X to itself such that df(x), f(y)=r · d(x, y) for all x, y ∈ X.Note that every isometry (including every identity map) is a similarity transformationwith ration of similitude 1 and conversely, the inverse of a similarity transformation withratio of similitude r is a similarity transformation with ratio of similitude r−1, and thecomposite of two similarity transformations with ratios of similitude r and s is a similaritytransformation with ratio of similitude rs. Of course, if r > 0 then the invertible lineartransformation rI on Rnis a similarity transformation with ratio of similitude r.FACT 2. In Rnevery similarity transformation with ratio of similitude r satisfyingF (0) = 0 has the form F (x) = rA(x) where A is given by an n × n orthogonal matrix.FACT 3. In Rnevery similarity transformation F is conformal; specifically, if α andβ are differentiable curves in Rnthat are defined on a neighborhood of 0 ∈ R such thatα(0) = β(0) such that both α0(0) and β0(0) are nonzero, then the angle between α0(0) andβ0(0) is equal to the angle between [F ◦ α]0(0) and [F ◦ β]0(0).The verification of the third property uses Facts 1 and 2 together with the additionalobservation that if A is given by an orthogonal transformation then A preserves innerproducts and hence the cosines of angles between vectors.The key to relating inversions and stereographic projections is the following result:2PROPOSITION. Let e ∈ Rnbe a unit vector, and let T be inversion with respect to thesphere S(1; 0). Then T interchanges the hyperplane defined by the equation hx, ei = −1with the nonzero points of the sphere S(12; −12e).Proof. By definition we haveT (x) =1hx, xi· xand therefore the proof amounts to finding all x such thatT (x) +12e=12.This equation is equivalent to14=T (x) +12e2= hT (x) +12e, T (x) +12eiand the last expression may be rewritten in the formhT (x), T (x)i + hT (x), ei +14=hx, xihx, xi2+hx, eihx, xi+14which simplifies to1hx, xi+hx, eihx, xi+14.Our objective was to determine when this expression is equal to14, and it follows imme-diately that the latter is true if and only if 1 + hx, ei = 0; i.e., it holds if and only ifhx, ei = −1.COROLLARY. Let e be as above, and let W be the (n − 1)-dimensional subspace ofvectors that are perpendicular to e. Then the stereographic projection map from S(1; 0)−{e}to W is given by the restriction of the compositeG ◦ T ◦ Hto S(1; 0) − {e}, where H is the similarity transformation H(u) =12(u − e) and G is thetranslation isometry G(v) = v + e.Proof. It will be convenient to talk about the closed ray starting at a vector a and pasingthrough a vector b; this is the image of the parametrized curveγ(t) = (1 − t)a + tb = a + t(b − a)where t ≥ 0.First note that the composite T ◦ H sends the ray