Technical Note. We define the radial projection sigmaR (x) = (R/||x||) x from R^ (n+1) minus origin to Sⁿ (R) and establish its basic properties (smoothness, idempotence, quotient structure, kernel of differential, angle preservation, scale invariance). We show that the central projection PhiR of the authors prior works coincides with sigmaR restricted to the tangent hyperplane PiR = x: x_ (n+1) = R, with image equal to the open upper hemisphere Sⁿ_+ (R) where PhiR is a diffeomorphism. The contrast between the non-injective sigmaR (radial fibers collapsed) and the injective PhiR (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.
Noriaki Kihara (Sat,) studied this question.
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