By Peter Freyd

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3 Deﬁnition. ” 13) The notions of elliptic versus hyperbolic pencils are not used consistently in the literature. , [162]). , [66]). 5ﬀ) we agreed on describing m-dimensional spheres in S n as the intersection of n − m hyperspheres. Now, we see that an m-sphere can be identiﬁed with ∗ an (m + 1)-plane P m+1 ⊂ P n+1 that intersects S n transversally, that is, with a Minkowski (m + 2)-subspace in the coordinate Minkowski ; space n+2 1 or, via orthogonality14) (polarity P m+1 → P n−m−1 = polP m+1 ) with ∗ an (n − m − 1)-plane P n−m−1 ⊂ P n+1 that does not meet S n , that .

6) Given a hypersphere S ⊂ S n in terms of a center7) m ∈ S n and the corresponding radius ∈ (0, π), we have S = [(cos , m)] ∈ P n+1 . Note that, with the above choice of scaling, S, S = sin2 > 0. S . n . ρ . m 1 cos ρ S Fig. 2. , . be a Minkowski product on n+2 = n+2 , that 1 2 2 is, we rule out negative scalings of −dx0 + dxi in the above setup. 1). 3 Later on, we will need to provide the space of hyperspheres with a metric: Similar to the way the Fubini-Study metric is introduced on a real projective space by considering the (metric) sphere as a Riemannian double cover, we obtain a metric on n+1 PO := { v ∈ P n+1 | v ∈ n+2 , 1 v, v > 0} by considering the Lorentz sphere S1n+1 := {v ∈ n+2 1 | v, v = 1} n+1 as a Lorentzian double cover of PO .

4). To ﬁnd the angle formula, suppose p ∈ S1 ∩ S2 and the Si ’s are given in terms of their centers mi ∈ S n and their radii i ∈ (0, π), that is, we have Si = (cos i , mi ). Then, the intersection angle of S1 and S2 at p (1, p) is the angle (in n+1 ) between the vectors mi − cos i p. Since p · mi = cos i , we ﬁnd S 1 , S2 S i , Si = (m1 − cos = (mi − cos p) · (m2 − cos 2 p) , i 1 2 p) and which proves the claim. , . on n+2 . 1 12) n+1 by p · m, to distinguish it from the Minkowski We consider the unoriented angle of intersection.