Stereographic Projections on Some Quadric Surfaces

R.B. Leal; T.F. Cruz; W.F.C. Barboza

REVIEW 2 major objections 5 minor 9 references

Stereographic Projections on Some Quadric Surfaces

T0 review · 2 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper adapts stereographic projection to an ellipsoid and an elliptic paraboloid and proves that every horizontal elliptical section projects to an ellipse of the same eccentricity, with curvature, arc length, and area changing by…

desk verdict A correct but modest exposition: the generalized stereographic projection is just a slicewise homothety, and the theorems follow from that; the paper needs a few proof-completeness fixes before it is publishable. read the letter →

arxiv 2506.08289 v1 pith:LMKDGE4F submitted 2025-06-09 math.DG

W.F.C. Barboza , T.F. Cruz , R.B. Leal This is my paper

classification math.DG MSC 53C4253A1053C2053C50

keywords StereographicprojectionEllipsoidEllipticparaboloidEccentricitypreservationSurfaceparametrizationCurvatureArclengthAreascaling

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper adapts the classical stereographic projection of a sphere to two quadric surfaces, an ellipsoid and an elliptic paraboloid, by drawing a line from the north pole through each plane point and taking its second intersection with the surface. It proves that the resulting maps are regular parametrizations of the surface minus the north pole, and that their inverses project horizontal elliptical sections to ellipses in the plane. The main results compare a horizontal section with its planar image: eccentricity is unchanged, while curvature scales by $(c-d)/c$, arc length by $c/(c-d)$, and area by $(c/(c-d))^2$. If correct, this yields explicit planar coordinates for these surfaces in which the shape of every horizontal slice is preserved and only a fixed scale factor separates the section from its projection.

What carries the argument

The object that carries the argument is the line-from-north-pole construction: each planar point $Q(u,v)$ defines a ray $N+t(Q-N)$ from the north pole, and solving for the second intersection with the quadric fixes $t$ algebraically, producing the rational parametrizations (3.4) and (4.4). Their inverses both take $(x,y,z)$ to $(cx/(c-z), cy/(c-z))$, so on a horizontal slice $z=d$ the inverse is a homothety with factor $c/(c-d)$, carrying the entire section ellipse to a homothetic ellipse. Comparing the section's semi-axes $A,B$ with the projected semi-axes $A_0,B_0$ yields all of the eccentricity, curvature, arc length, and area relations.

What would settle it

Take any non-axial point on the section ellipse, say $(A\cos t, B\sin t, d)$, and apply the inverse projection (3.7) to it; the result must satisfy the equation of the projected ellipse (5.5) identically. If the paper's two-point computation missed a distorted image for such a point, then the eccentricity equality and the corresponding curvature, length, and area ratios would fail.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the sphere's stereographic projection has working analogues on an ellipsoid and an elliptic paraboloid. For the ellipsoid, the map (3.4) sends each planar point $(u,v)$ to the second intersection of the line through the north pole and $(u,v,0)$, with inverse (3.7); for the paraboloid, (4.4) and (4.7) do the same. Theorems 1 and 2 prove that the inverse projection takes every horizontal elliptical section to an ellipse with the same eccentricity, so the projected shape does not depend on the height $d$ of the section. Theorems 3 through 8 prove exact ratios for the two surfaces: the projected ellipse's curvature is $(c-d)/c$ times the section's curvature, its arc length is $c/(c-d)$ times the section's arc length, and its area is $(c/(c-d))^2$ times the section's area.

Load-bearing premise

The load-bearing premise is that the inverse projection restricted to the horizontal plane $z=d$ acts as a uniform scaling by $c/(c-d)$, so the entire ellipse, not just the two axial points checked in the proof, maps to a homothetic ellipse.

Editorial extensions

If this is right

For every horizontal section of the ellipsoid or paraboloid below the north pole, the projected ellipse has the same eccentricity as the section, so eccentricity is independent of section height $d$.
As the section approaches the north pole, $d\to c$, the projected curvature tends to zero while the projected arc length and area diverge at reciprocal rates.
The explicit formulas (3.4) and (4.4), together with their inverses, give rational parametrizations of both surfaces minus the north pole, so local geometric quantities can be computed in planar coordinates using fixed scaling factors.
The sphere is recovered as the special case $a=b=c$, giving the classical stereographic projection and its known geometry as a limiting case.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The eccentricity result points to a slice-wise analogue of conformality: although the whole projection is not conformal, each horizontal level set is mapped by a homothety; a natural test is whether other families of parallel sections on these quadrics also project to similar families of conics.
The same north-pole-line construction should extend to other quadrics, such as two-sheeted hyperboloids, whenever the defining equation is quadratic in $z$; the eccentricity and scaling relations would hold whenever the inverse map restricts to a homothety on a family of parallel sections.
The reciprocal relation between curvature and length or area noted in the paper suggests a general pattern for central projections of surfaces of revolution, which could be checked on higher-dimensional analogues of the ellipsoid.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper does what it claims: it writes down explicit stereographic parametrizations for the ellipsoid and the elliptic paraboloid, gives the inverses, and proves that each horizontal elliptical section projects to an ellipse with the same eccentricity while curvature, arc length, and area scale by the expected factors. I checked the algebra in Sections 3 and 4 and in Theorems 1, 3, 5, and 7; it is correct. The regularity proofs are careful, and the presentation is readable apart from minor typos. This is genuinely a useful teaching resource: the formulas (3.4), (4.4), and their inverses are not in the cited textbooks, and the scaling relations are clean enough to show students.

The deepest issue is also the simplest one: the whole story reduces to the observation that on the slice z=d, the inverse map φ^{-1}(x,y,d) = (c/(c-d))(x,y) is a homothety. That is why eccentricity is preserved and why the other quantities scale the way they do. The proof of Theorem 1 computes only the two axial points and then asserts the projected ellipse, without naming the homothety. The same gap is imported into Theorem 2. This is a one-line fix, but it is the actual content, so it should be stated.

Two theorems, 6 and 8, are effectively unproved: the paraboloid arc-length and area results are dismissed with “accounts analogous.” They follow immediately by substitution from (5.10) and (4.7), so the omission is not fatal, but it is a real completeness gap for a paper that otherwise takes care to prove things.

A smaller issue: the introduction says the construction “opens up possibilities for conformal visualization of ellipsoids.” The map is not conformal except in the sphere case, and the paper never proves or uses conformality. That sentence should be removed or rewritten. The reference list is also thin; no prior work on stereographic projections for quadrics is cited, so the authors do not demonstrate that the formulas are new, even if they may be.

Overall, the mathematical content is sound and the scope is modest. This is a note for differential-geometry teachers and for people who want explicit parametrizations for GeoGebra-style visualization. It deserves a serious referee, not a desk reject, but the referee should require the authors to fill in the missing proofs and correct the conformality overreach.

Referee Report

2 major / 5 minor

Summary. The paper adapts the classical stereographic projection from the sphere to two quadric surfaces: an ellipsoid and an elliptic paraboloid. Sections 3 and 4 introduce explicit parametrizations (3.4) and (4.4), verify that they are regular, and derive the inverse maps (3.7) and (4.7). Section 5 studies horizontal sections z = d of the two surfaces, computes the images of those elliptical sections under the inverse maps, and states eight theorems: eccentricity is preserved (Theorems 1 and 2), curvature scales by (c-d)/c (Theorems 3 and 4), arc length scales by c/(c-d) (Theorems 5 and 6), and area scales by (c/(c-d))^2 (Theorems 7 and 8). The paper ends with a remark relating these scaling behaviors. The main algebraic derivations and the stated formulas are correct; the principal weaknesses are proof-completeness gaps in the identification of projected ellipses and in the paraboloid arc-length and area theorems.

Significance. The contribution is elementary but cleanly executed. The explicit formulas for the stereographic parametrizations and their inverses are parameter-free, reduce to the classical sphere case when a=b=c, and the scaling relations in Section 5 are simple and potentially useful. The central claims are mathematically sound: I verified the inverse compositions and the scale factors independently, and there is no circular reasoning or hidden fitted parameter. The main value of the paper is pedagogical and as a reference for explicit computations. The missing justifications in the proofs of Theorems 1, 2, 6, and 8 are fixable and do not affect the truth of the statements. If these completeness issues are addressed, the paper would be suitable for publication.

major comments (2)

[§5.1 (Theorem 1)] The identification of the projected ellipse E0 as (5.5) with semiaxes (5.6) is not established by the displayed calculation. The authors compute φ^{-1}(P1) and φ^{-1}(P2) for the two axial points and then assert that 'from these points' one obtains the ellipse (5.5). Two points do not determine an ellipse. The missing ingredient is the elementary fact that on the plane z=d, the map (3.7) restricts to the homothety (x,y) ↦ (c/(c-d))(x,y). Once stated, this immediately implies that the image of Ed is the homothetic ellipse (5.5)-(5.6). This observation should be stated and proved, and the same repair is needed in the proof of Theorem 2, which invokes 'proceeding analogously'.
[§5.3 and §5.4 (Theorems 6 and 8)] The proofs of Theorems 6 and 8 are omitted. After the statement of Theorem 6 the proof reads only 'With accounts analogous to the theorem 5...', and Theorem 8 is treated similarly. Since these theorems are part of the paper's central set of results, the derivations should be given at least in condensed form. For example, from (5.10) and (4.7) one obtains A0 = ca/√(c-d) and B0 = cb/√(c-d), so the arc-length integrand scales by c/(c-d), giving (5.27); similarly A(Pd) = πab(c-d) and A(P0) = πabc^2/(c-d), giving (5.31). The results are correct, but the proofs as written are not complete.

minor comments (5)

[Theorem 4] The statement of Theorem 4 refers to 'ellipses (5.1) and (5.5)' but should refer to (5.9) and (5.12).
[§5.2 (Theorem 3 proof)] In the proof of Theorem 3, the intermediate expression for k_E0(t) as k_Ed(t) minus the term (abd/√(c^2-d^2))(a^2 sin^2 t + b^2 cos^2 t)^{3/2} is dimensionally inconsistent; the correct simplification is k_E0(t) = k_Ed(t) - (d/c) k_Ed(t).
[§5.3-5.4 (Theorems 5-8)] The notation L_E0(t) and L_P0(t) suggests an arc-length function of the parameter t, but the integrals are taken over the full period [0,2π]. Use L(E0) or a similar notation for total arc length.
[Equation (4.7)] The codomain in (4.7) is written as R^2 \ {(0,0} with a missing closing brace; it should be R^2 \setminus {(0,0)}.
[Throughout] The phrase 'plane-xy' appears in the abstract and elsewhere; the standard term 'xy-plane' would be clearer.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the stereographic maps and all eight theorems are derived from first principles; the proof-completeness gaps identified do not make any claim depend on its own conclusion.

full rationale

The paper's central claims are the parametrization theorems for maps (3.4) and (4.4), and the eccentricity, curvature, arc-length and area relations of Theorems 1 through 8. Every one of these is derived from the explicit definitions of the maps, which are constructed by intersecting the line through the north pole and a plane point with the quadric: (3.4) and (4.4) are obtained by substituting the line equation into the quadric equation and solving for t; the inverses (3.7) and (4.7) are obtained by intersecting the line with z = 0 and are verified directly by composition. The eccentricity theorems reduce to algebra with the semiaxes (5.2)-(5.6) and (5.10)-(5.13); the curvature theorems compute k from Definition 4; the arc-length and area theorems follow from the same semiaxes. No parameter is fitted, no benchmark is predicted from a subset of itself, and no load-bearing result is imported solely from a self-citation. The reader's noted gaps are real proof-completeness issues, not circularity: in Theorem 1 the projected ellipse (5.5) is identified from the images of two axial points, which silently uses that phi^{-1}(x,y,d) = (cx/(c-d), cy/(c-d)) is a homothety on the slice z=d; this fact is true and is an immediate consequence of (3.7), so the proof is incomplete but not circular. Similarly, Theorems 6 and 8 are stated with only 'With accounts analogous...' as proof; this is an omitted-detail issue, not a circular reduction. The paper is therefore self-contained against its own definitions and the central derivation is independent of its conclusions.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

All assumptions are standard geometric inputs; no ad hoc parameters or invented entities appear. The only constants a, b, c and d are the defining parameters of the surfaces and sections.

assumptions (4)

domain assumption a, b, c are positive real constants defining the quadrics (3.1) and (4.1).
The parametrizations and all theorems assume these constants are positive; no dependence on their fitted values is used.
domain assumption d < c for the horizontal sections.
Theorems use division by c-d and sqrt(c^2-d^2), requiring d<c and c>0.
standard math Without loss of generality a > b for eccentricity computations.
Theorems 1 and 2 assume a > b so that the x-axis is the major axis; the opposite case is symmetric.
standard math Standard definitions of regular surfaces, curvature, arc length, and area from do Carmo.
Section 2 imports standard definitions; the paper relies on these without proof.

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Cite this review

Pith. "Pith review of Stereographic Projections on Some Quadric Surfaces." pith.science (2026). https://pith.science/paper/LMKDGE4F

@misc{pith2026250608289,
  author       = {Pith},
  title        = {Pith review of: Stereographic Projections on Some Quadric Surfaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LMKDGE4F}},
  note         = {Machine review of arXiv:2506.08289}
}

read the original abstract

In this work, we present an adaptation of the classical stereographic projection, originally formulated for the sphere, now considering the context of the ellipsoid and the elliptic paraboloid. We begin by constructing the stereographic projections for both quadric surfaces separately, analyzing the geometric particularities of each surface and the challenges arising from their variable curvatures and, in the case of the paraboloid, its non-compactness. In the final part of the work, we establish results concerning the eccentricities, curvatures, arc length, and areas of the ellipses formed by the intersection of the quadrics with horizontal sections and their corresponding projections onto the plane-xy.

Figures

Figures reproduced from arXiv: 2506.08289 by the authors.

**Figure 1.** Ellipsoid stereographic projection equation. Source: Prepared by the authors. Note that the vector equation of the line passing through the point N(0, 0, c) and the projected point on the plane Q(u, v, 0) is given by r : (x, y, z) = (0, 0, c) + t(u, v, −c), t ∈ R, in the parametric equation, we obtain the equation of the line as (3.2) r :    x = tu y = tv z = (1 − t)c , t ∈ R. Thus, for each point (u, v) ∈ R 2 … view at source ↗

**Figure 2.** Elliptic Paraboloid stereographic projection equation. Source: Prepared by the authors. Thus, for each point (u, v) ∈ R 2 , we can find the intersection point of the paraboloid with the stereographic projection. We then define the following map: (4.3) ϕ : R 2 − {(0, 0)} −→ P \ {N} (u, v) 7−→ ϕ(u, v) = N + t(Q − N) = (tu, tv,(1 − t)c). To find the parameter t for the point P(x, y, z) on the paraboloid projected from … view at source ↗

**Figure 3.** Intersection of the ellipsoid with a plane. Source: Prepared by the authors. This intersection yields an ellipse with the equation: (5.1) Ed : x 2 A2 + y 2 B2 = 1, where the values of A and B are given by: (5.2) A = [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: Projection of the ellipse Ed onto the xy-plane. Source: Prepared by the authors. Theorem 1. The projection map (3.7) preserves eccentricity, that is, e(E0) = e(Ed), where e(Ed) and e(E0) denote the eccentricities of the ellipses Ed and E0, respectively. Proof. We need …

**Figure 5.** Figure 5: Projection of the ellipse Pd onto the xy-plane. Source: Prepared by the authors. This intersection yields an ellipse with the equation: (5.9) Pd : x 2 A2 + y 2 B2 = 1, where the values of A and B are given by: (5.10) A = a √ c − d and B = b √ c − d. Let P0 = ϕ −1 (Pd) …

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Reference graph

Works this paper leans on

9 extracted references · 9 canonical work pages

[1]

Carroll, T.,The Riemann Sphere, Geometric Function Theory, Springer Undergraduate Mathematics Series, Spring (2024)

work page 2024
[2]

Carmo, M.P.,Differential Geometry of curves and surfaces, Precentice-Hall, 1976

work page 1976
[3]

A., Candido, L., Garcia, R., Mathias, C.V.,O Geogebra como ferramenta de apoio ao entendimento de demonstra¸ c˜ oes em Geometria, PMO, v

Da Silva, H. A., Candido, L., Garcia, R., Mathias, C.V.,O Geogebra como ferramenta de apoio ao entendimento de demonstra¸ c˜ oes em Geometria, PMO, v. 10, pages 482-501, 2022

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Gentili, G., Simonutti, L., Struppa, D.C.,The Mathematics of the Astrolabe and its History, Journal of Humanistic Mathematics, Volume 10 Issue 1 (January 2020), pages 101-144

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[5]

Lima, E.L.,Curso de An´ alise, Volume 2, 11ªed., Projeto Euclides, IMPA, Rio de Janeiro, 2012

work page 2012
[6]

Mathias, C.V.,Geogebra’s potential as a tool to aid visualization skills, Revista do Instituto Geogebra Internacional de S˜ ao Paulo, v.12, pages 044-066, 2023

work page 2023
[7]

Santos, J.M.S., Breda, A.M.D.,The stereografic Projection in Geogebra, Revista do Instituto GeoGebra de S˜ ao Paulo, INSS 2233-9657, v.7, n.1, pages 31-40, 2018

work page 2018
[8]

Sidoli, N., Berggren, J.L.,The Arabic Version of Ptolemy’s Planisphere of Flattening the Surface of the Sphere, Text, Translation, Comentary, SCIAMVS, Volume 8 (2007), pages 37-139

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Show all 9 references

[9]

1 Unidade Acad ˆemica de Matem ´atica, Universidade Federal de Campina Grande, 58.429-970 Campina Grande, Para´ıba, Brazil

Yavetz I.,On the Homocentric Sphers of Eudoxus, Archive for History of Exact Sciences, Volume 52, Number 3 (1998), pages 221-278. 1 Unidade Acad ˆemica de Matem ´atica, Universidade Federal de Campina Grande, 58.429-970 Campina Grande, Para´ıba, Brazil. Email address:weiller@m...

1998

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Carroll, T.,The Riemann Sphere, Geometric Function Theory, Springer Undergraduate Mathematics Series, Spring (2024)

work page 2024

[2] [2]

Carmo, M.P.,Differential Geometry of curves and surfaces, Precentice-Hall, 1976

work page 1976

[3] [3]

A., Candido, L., Garcia, R., Mathias, C.V.,O Geogebra como ferramenta de apoio ao entendimento de demonstra¸ c˜ oes em Geometria, PMO, v

Da Silva, H. A., Candido, L., Garcia, R., Mathias, C.V.,O Geogebra como ferramenta de apoio ao entendimento de demonstra¸ c˜ oes em Geometria, PMO, v. 10, pages 482-501, 2022

work page 2022

[4] [4]

Gentili, G., Simonutti, L., Struppa, D.C.,The Mathematics of the Astrolabe and its History, Journal of Humanistic Mathematics, Volume 10 Issue 1 (January 2020), pages 101-144

work page 2020

[5] [5]

Lima, E.L.,Curso de An´ alise, Volume 2, 11ªed., Projeto Euclides, IMPA, Rio de Janeiro, 2012

work page 2012

[6] [6]

Mathias, C.V.,Geogebra’s potential as a tool to aid visualization skills, Revista do Instituto Geogebra Internacional de S˜ ao Paulo, v.12, pages 044-066, 2023

work page 2023

[7] [7]

Santos, J.M.S., Breda, A.M.D.,The stereografic Projection in Geogebra, Revista do Instituto GeoGebra de S˜ ao Paulo, INSS 2233-9657, v.7, n.1, pages 31-40, 2018

work page 2018

[8] [8]

Sidoli, N., Berggren, J.L.,The Arabic Version of Ptolemy’s Planisphere of Flattening the Surface of the Sphere, Text, Translation, Comentary, SCIAMVS, Volume 8 (2007), pages 37-139

work page 2007