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arxiv: 2606.22545 · v1 · pith:5BYWZNQRnew · submitted 2026-06-21 · 🧮 math.DG

On the contact of surfaces in 4-space with 2-planes and their apparent contours

Jorge Luiz Deolindo-Silva , Mostafa Salarinoghabi This is my paper

Pith reviewed 2026-06-26 09:50 UTC · model grok-4.3

classification 🧮 math.DG
keywords surfaces in 4-spaceparallel projectionapparent contourA-singularitybutterfly singularityLipschitz-Killing curvatureasymptotic directiondiscriminant locus
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The pith

When the projection plane avoids asymptotic directions, generic surfaces in four-space have up to ten tangent directions at hyperbolic or elliptic points and seven at parabolic points that produce butterfly singularities or worse in paralle

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

The paper examines the local geometry of generic smooth surfaces in four-dimensional space through their contact with two-planes and the apparent contours produced by parallel projections along those planes. It shows that, provided the projection plane contains no asymptotic direction, hyperbolic or elliptic points admit up to ten tangent directions for which the projection has A-singularities of butterfly type or more complex, while parabolic points admit up to seven. The work also proves that the discriminant of the equation for these directions vanishes along regular curves in the hyperbolic and elliptic regions and at isolated points in the parabolic set. When the projection is a fold, the Lipschitz-Killing curvature of the surface relates directly to the curvature of the apparent contour, with elliptic and inflection points producing vertices and hyperbolic and parabolic points producing inflections.

Core claim

When the projection plane does not contain an asymptotic direction, at hyperbolic or elliptic points of a generic surface in R^4 there exist up to ten tangent directions determining planes along which the parallel projection exhibits A-singularities of type butterfly or worse; at parabolic points there are up to seven such directions. The locus where the discriminant of the equation defining these directions vanishes forms regular curves in the hyperbolic and elliptic regions and isolated points in the parabolic set. When the projection is a fold the Lipschitz-Killing curvature of the surface is related to the curvature of the apparent contour. Elliptic or inflection points give rise to vert

What carries the argument

The equation whose roots give the tangent directions for planes along which the parallel projection is taken, together with the discriminant of that equation and the A-equivalence classification of the resulting map singularities.

If this is right

  • The discriminant locus forms regular curves in the hyperbolic and elliptic regions and isolated points in the parabolic set.
  • Elliptic and inflection points of the surface produce vertices in the apparent contour.
  • Hyperbolic and parabolic points produce inflections in the apparent contour.
  • Apparent contours from non-fold projections admit (t^k, t^ℓ)-cusp singularities associated with the projection singularities.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The finite counts of directions could support enumerative calculations for explicit surfaces such as quadrics embedded in four-space.
  • The curvature relation between surface and contour may extend to other invariants when the projection is observed in applications.
  • Links between parallel projections, orthogonal projections, and height functions when asymptotic directions are present could inform global questions about contour topology.

Load-bearing premise

The surface must be generic and the projection plane must not contain an asymptotic direction.

What would settle it

A generic surface in R^4 at a hyperbolic point with a projection plane free of asymptotic directions having more than ten tangent directions that yield A-singularities of butterfly type or worse, or a discriminant locus that fails to be a regular curve in the hyperbolic region.

Figures

Figures reproduced from arXiv: 2606.22545 by Jorge Luiz Deolindo-Silva, Mostafa Salarinoghabi.

Figure 1
Figure 1. Figure 1: Curves and special points on M ([15, 19]). Let E = {e1, e2, e3, e4} be the standard basis of R 4 , and let M be parametrised in Monge form by F(x, y) = (x, y, f 1 (x, y), f 2 (x, y)), where (3) f 1 (x, y) = X i+j≥2 aijx i y j , f 2 (x, y) = X i+j≥2 bijx i y j . Then, at the origin p = (0, 0), the tangent space TpM is generated by e1 and e2. The coefficients of (Q1, Q2) at p are given by a = 1 2 f 1 xx, b =… view at source ↗
Figure 2
Figure 2. Figure 2: Stratification of the space of planes Π according to the singularities of Pπ at elliptic points. In the gray region, Pπ is a fold. In the cream-colored region, Pπ is a cusp. The red curve corresponds to swallowtail singularities, while at the isolated black points on the red curve, Pπ has a butterfly singularity or worse. 3.1.2. u is an asymptotic direction in π. In this case, the assumptions of Remark 3.1… view at source ↗
Figure 3
Figure 3. Figure 3: Curves and special points on M via contact with 2- planes. In [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
read the original abstract

We investigate the local geometry of generic smooth surfaces in $\mathbb R^4$ via the contact with $2$-planes and the associated apparent contour. We study the $\mathcal A$-singularities of parallel projections of such surfaces along planes to transverse planes. When the projection plane does not contain an asymptotic direction, we show that, at hyperbolic or elliptic (respectively, parabolic) points, there exist up to ten (respectively, seven) tangent directions determining planes along which the projection exhibits $\mathcal A$-singularities of type butterfly or worse. Moreover, we prove that the locus of points where the discriminant of the equation defining these directions vanishes generically forms regular curves in the hyperbolic and elliptic regions, and isolated points in the parabolic set of $M$. When the projection plane contains an asymptotic direction, we establish connections between the singularities of parallel projections, orthogonal projections to hyperplanes, and height functions. We further study the apparent contour associated with the parallel projection. When the projection is a fold, we prove a Koenderink-type theorem relating the Lipschitz--Killing curvature of the surface to the curvature of the apparent contour. Moreover, we show that elliptic or inflections points of the surface give rise to vertices of the apparent contour, whereas hyperbolic and parabolic points give rise to inflections. These phenomena are then characterized in terms of the singularities of the projection. In the non-fold case, we show that the apparent contour admits singularities of type $(t^k,t^\ell)$-cusp associated with the singularities of the projection.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript studies the local geometry of generic smooth surfaces in Euclidean 4-space through their contact with 2-planes and the A-singularities of parallel projections onto transverse planes, together with the geometry of the resulting apparent contours. When the projection plane avoids asymptotic directions, it establishes that at hyperbolic or elliptic points there are at most ten tangent directions yielding A-singularities of butterfly type or worse, while at parabolic points the bound is seven; the discriminant locus of the associated equation is shown to be generically a regular curve in the hyperbolic/elliptic regions and isolated points on the parabolic set. In the asymptotic case the paper relates the singularities to those of orthogonal projections and height functions. For fold projections it proves a Koenderink-type theorem linking the Lipschitz-Killing curvature of the surface to the curvature of the apparent contour, shows that elliptic and inflection points produce vertices while hyperbolic and parabolic points produce inflections, and classifies the non-fold case by (t^k, t^ℓ)-cusp singularities of the contour.

Significance. If the derivations hold, the work supplies explicit, falsifiable counts (10 and 7) and a curvature relation that extend classical singularity-theoretic results on projections from 3-space to 4-space. The absence of free parameters or circular reductions, together with the use of standard A-equivalence and transversality tools, makes the numerical claims directly testable on low-degree model surfaces. The Koenderink-type statement and the vertex/inflection classification provide concrete geometric consequences that could be checked against explicit examples.

minor comments (3)
  1. [Abstract] The abstract states the counts 'up to ten (respectively, seven)' without indicating whether these maxima are attained for a dense open set of surfaces; a brief remark in the introduction or §2 on the degree of the defining equation would clarify this.
  2. The transition between the non-asymptotic and asymptotic cases (around the statement of the connections to orthogonal projections) would benefit from an explicit sentence recalling the standing genericity hypothesis on the surface.
  3. Notation for the Lipschitz-Killing curvature and the precise definition of 'fold' projection should be recalled or cross-referenced at the point where the Koenderink-type theorem is stated.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the detailed summary and positive evaluation of the manuscript. The recommendation for minor revision is appreciated, and we will incorporate any necessary clarifications in the revised version.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper derives its counts of tangent directions (up to 10 or 7) and the Koenderink-type curvature relation from standard A-equivalence classifications and transversality arguments in singularity theory applied to generic surfaces in R^4. These rest on external background results about map-germs and height functions rather than any self-definitional loop, fitted parameter renamed as prediction, or load-bearing self-citation chain. The genericity assumptions are invoked explicitly to guarantee finiteness and regularity of loci, but do not reduce the stated theorems to the paper's own equations by construction. The derivation chain remains self-contained against external benchmarks in differential geometry.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard background results in singularity theory and the differential geometry of surfaces in Euclidean 4-space; no free parameters, invented entities, or ad-hoc axioms are introduced in the abstract.

axioms (2)
  • standard math A-equivalence classification of map-germs from R^2 to R^2 and the associated normal forms (fold, cusp, butterfly, etc.)
    Invoked throughout the study of parallel projections and their singularities.
  • domain assumption The standard classification of points on a surface in R^4 into elliptic, hyperbolic, and parabolic types via the second fundamental form or asymptotic directions
    Used to stratify the surface and state the differing counts of special directions.

pith-pipeline@v0.9.1-grok · 5817 in / 1621 out tokens · 27553 ms · 2026-06-26T09:50:46.188823+00:00 · methodology

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Reference graph

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24 extracted references · 5 canonical work pages

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