Radial Projection: Definition and Relation to Central Projection

Technical Note. We define the radial projection sigma_R(x) = (R/||x||) x from R^(n+1) minus origin to S^n(R) and establish its basic properties (smoothness, idempotence, quotient structure, kernel of differential, angle preservation, scale invariance). We show that the central projection Phi_R of the authors prior works coincides with sigma_R restricted to the tangent hyperplane Pi_R = {x : x_(n+1) = R}, with image equal to the open upper hemisphere S^n_+(R) where Phi_R is a diffeomorphism. The contrast between the non-injective sigma_R (radial fibers collapsed) and the injective Phi_R (full geometric structure) clarifies what is gained and lost in extending the domain. This note serves as a foundational reference for the central projection series. Version v3.2 (2026-06-02): copy-editing corrections with no change to the mathematical content — resolved a duplicate equation number (the angle-preservation identity in Proposition 2.5 was renumbered to (2.5)); corrected the cross-reference in Section 1.2 to Lemma 3.2; added the image argument (Im = x-perp, via x.Dsigma=0 and the rank-nullity theorem) to the proof of Proposition 2.4; renamed the heading of Proposition 2.4 to 'Kernel and image of the differential'; tightened the transversality wording in the Section 3.4 table; and clarified the directional-derivative notation in the proof.