Geometry on the Gluing Locus of Two Surfaces

Li Junzhen

REVIEW 2 major objections 3 minor 12 references

Geometry on the Gluing Locus of Two Surfaces

T0 review · 2 major / 3 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper claims that the geometry of a gluing of two surfaces along a curve is fully encoded by three frame invariants and the angle between the two surface normals, with cylinder, cone, cuspidal-edge, and swallowtail cases…

desk verdict The core classification is correct, but the example data are unreliable and must be fixed before the paper is usable as an illustration. read the letter →

arxiv 2506.01397 v1 pith:7GLLQ6X4 submitted 2025-06-02 math.DG

Li Junzhen This is my paper

classification math.DG MSC 53A0553A5558K30

keywords gluingofsurfacesdevelopablemovingframelocusfrontalscuspidaledgeswallowtailcylinderandcone

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 studies two surfaces glued along a common curve, the gluing locus. A moving frame is attached to the curve using the two surface normals, and from this frame two developable surfaces are constructed for each side. The central claim is that the local geometry of the gluing is completely controlled by three frame invariants and the angle between the normals: the auxiliary developable surfaces are cylinders, cones, cuspidal edges, or swallowtails exactly when explicit algebraic conditions on those invariants hold. If this is right, special gluings can be detected purely from frame data, which matters for gluing surfaces in discrete and singular geometry.

What carries the argument

The key machinery is the Frenet–Serret type frame {e, nu_i, b_i} attached to the gluing locus gamma, with gamma' = l e, and the two developable surfaces S_nu_i and S_b_i defined as envelopes of planes orthogonal to nu_i and b_i. Their cylinder and cone conditions are captured by the functions beta_nu_i and rho_nu_i, and their singularities by the same rho_nu_i together with its derivative. Lemma 4.2 shows how the second frame's invariants relate to the first via the angle theta, so the entire classification reduces to four scalar data on one side of the gluing.

What would settle it

At a point where l(t0) = 0 (for example the cusp of gamma(t) = ($t^{2}$, $t^{3}$, 0)), take two different smooth unit vector fields e1 and e2 that both satisfy gamma' = l e, compute the resulting kappa_i1, kappa_i2, kappa_i3 and hence beta_nu and rho_nu for S_nu1, and check whether the predicted singularity type at t0 changes. If it does, the classification of special gluings at singular points is not intrinsic; if it never does, the choice of e is immaterial.

Watch

Extended reading notes

Core claim

The central claim is Theorem 4.5 and Theorem 4.8: for each side i=1,2, the developable surface S_nu_i is a cylinder if and only if beta_nu_i is identically zero, a cone if and only if beta_nu_i is nonzero and rho_nu_i is identically zero, and at a striction point it has a cuspidal edge if and only if beta_nu_i is nonzero and rho_nu_i is nonzero, and a swallowtail if and only if beta_nu_i is nonzero, rho_nu_i is zero, and rho'_nu_i is nonzero. Here beta_nu_i and rho_nu_i are built from the frame invariants kappa_i1, kappa_i2, kappa_i3. Lemma 4.2 shows that the second frame's invariants are obtained from the first by rotating the normal plane by the angle theta, so all four developable surfaces S_nu1, S_b1, S_nu2, S_b2 are governed by three invariants plus theta.

Load-bearing premise

The classification requires a smooth unit vector field e along the gluing locus with gamma' = l e; at points where l = 0 the vector e is not uniquely determined, so the invariants (and the verdicts based on them) can change unless a specific choice is adopted, and the nondegeneracy conditions (kappa_i1, kappa_i3) not both zero and (kappa_i2, kappa_i3) not both zero must also hold.

Editorial extensions

If this is right

If theta = k pi/2, then S_nu2 is a cylinder (or cone) exactly when S_b1 is a cylinder (or cone), so the two sides of the gluing exchange roles under a quarter-turn of the normals.
If theta = k pi, then S_nu2 and S_nu1 have the same cylinder/cone status, so the gluing is symmetric under swapping the two normals.
At a point where the gluing locus is singular (l = 0), the cuspidal-edge and swallowtail conditions simplify to conditions involving l', l'', and the rotated invariants, as in Corollary 4.9.
The four developable surfaces are not independent: S_b1 is obtained from S_nu1 by rotating the rulings by pi/2, and S_nu2 is obtained from S_nu1 by rotating by theta, so any special gluing is classified by the same three invariants plus theta.

Reading between the lines

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

Because the frame depends on the choice of the unit vector field e wherever l = 0, the classification at singular points of the gluing locus is well-posed only after a definite choice is fixed; an intrinsic reformulation would need to show the verdicts are independent of that choice or to quotient out this freedom.
The conditions are numerically testable: for any pair of fronts glued along a curve, one can compute kappa_i1, kappa_i2, kappa_i3 and theta, predict whether each S_nu_i or S_b_i is a cylinder, cone, cuspidal edge, or swallowtail, and compare directly with the computed surface.
The same frame-and-envelope construction may extend to gluings of higher-codimension submanifolds or to other ambient spaces, where the role of developable surfaces is played by envelopes of families of hyperplanes.

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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

Core take: the classification of special gluings via the frame invariants is right. The reported problems in the examples are real but do not affect the theorems.

What's actually new is Section 4: setting up the gluing locus with two frames, the rotation formula (4.2) relating the two normal frames, and the definitions of Sνi- and Sbi-cylindrical/conical/cuspidal-edgy/swallowtailed gluings. Theorems 4.5 and 4.8/4.9 are genuine extensions of Izumiya-Otani's flat approximation to the gluing context, and the criteria in terms of βνi and ρνi are computable from the local frame data. The attribution to [7] is honest, and the Section 3 machinery is reproduced carefully.

The soft spots are in the examples. Example 5.6 is internally inconsistent: ν1=(0,0,1) is not a unit normal to f1=(0,4u^3+v,3u^4) because ∂_u f1·ν1 = 12u^3, which is nonzero away from u=0. The proposed θ also fails to satisfy the rotation law (4.2). Similar normalization slips appear in other examples. These do not invalidate the central derivation; the theorems are direct substitutions from Section 3. But the illustrations should be treated as unreliable until recomputed. There is also a typo in the proof of Theorem 3.4 where s'(t) uses βν instead of βb, and Theorem 4.8/Corollary 4.9 are stated without showing the substitution; that missing detail should be added.

The l=0 frame ambiguity that might worry a reader turns out to be a non-issue: if a smooth unit e exists with γ'=le, then e is pointwise determined up to sign, and the quantities βνi and ρνi are invariant under the induced sign changes. So the classification is well-posed.

Who this is for: people working on discrete surface gluing, curved foldings, or developable surfaces along curves. It does not solve a major open problem, but it gives a clean local taxonomy that could be useful.

Recommendation: send to peer review. The central math holds up and the extension is legitimate. The referee report should demand corrected examples and the missing proof details. A serious referee will find this fixable.

Referee Report

2 major / 3 minor

Summary. The paper studies the local geometry of two surfaces glued along a curve. For each surface, a moving frame {e, ν_i, b_i} is introduced along the gluing locus, and two developable surfaces S_{ν_i} and S_{b_i} are associated with the frame. The main results give explicit criteria, in terms of the frame invariants κ_{i1}, κ_{i2}, κ_{i3} and the angle θ between the two normal vector fields, for these developable surfaces to be cylinders, cones, cuspidal edges, or swallowtails (Theorems 3.1, 3.2, 3.4, 3.5 and the gluing versions Theorems 4.5 and 4.8, with Corollaries 3.3, 3.6, and 4.9). The paper then presents five examples intended to illustrate the classification.

Significance. If the main theorems are correct, the paper provides a complete and explicit local classification of 'special gluings' in terms of standard differential invariants; the formulas for β and ρ are explicit, and no parameters are fitted. The proof strategy is direct and largely computational, and the rotation formula (4.2) correctly links the second surface's invariants to the first. However, the conceptual novelty is modest: the results are a direct translation of the known developable-surface theory of Izumiya-Otani into the gluing setting, and several of the illustrative examples contain mathematical errors that must be repaired before the paper can be accepted.

major comments (2)

[Section 5, Example 5.6] The example is internally inconsistent. The vector ν_1=(0,0,1) is not a unit normal of f_1(u,v)=(0,4u^3+v,3u^4), because ∂_u f_1·ν_1 = 12u^3, which is not identically zero. Moreover, the stated angle θ satisfies sinθ(0)=1 and cosθ(0)=0, so θ(0)=π/2, while ν_1(0)=(0,0,1) and ν_2(0)=(0,0,-1) are antiparallel. Consequently equation (4.2) is violated at u=0: the listed κ_{21}(0) is -1, whereas κ_{11}(0)cosθ(0)+κ_{12}(0)sinθ(0) = 1·0+0·1 = 0. The claimed verification of the swallowtail condition (κ_{11}cosθ+κ_{12}sinθ)l''|_0 = -24 is also inconsistent with the displayed formulas, since the expression is 0. Thus Example 5.6 does not illustrate Corollary 4.9 and must be corrected or replaced.
[Section 5, Example 5.2] The vector ν_1=(cos u, sin u, 1) is not a unit normal of f_1(u,v)=(cos u, sin u, v), because ∂_v f_1·ν_1 = 1 ≠ 0. The invariants (κ_{11},κ_{12},κ_{13})=(-1,0,0) given in the example are exactly those computed from the unit normal (cos u, sin u, 0), so the example as written is inconsistent. The normal vector should be corrected, and the subsequent verification of β_{ν1}=0 should be redone with the corrected normal.

minor comments (3)

[Section 2.2 and Corollaries 3.3, 3.6, 4.9] At a point where l(t)=0, the unit vector e is not uniquely determined by the condition γ'=le. The paper should state explicitly that a smooth unit vector field e is chosen on the whole interval I and that the classification is invariant under the global sign change (e,l,κ_1,κ_3,θ) → (-e,-l,-κ_1,-κ_3,-θ), under which β_ν, ρ_ν, β_b, ρ_b and the conditions in the corollaries are unchanged. Without this remark, the well-posedness of the invariants at isolated zeros of l is not manifest.
[Section 5, Examples 5.2 and 5.4] There are small typos in the examples: in Example 5.2, 'cosu.sinu' should be 'cosu, sinu', and in Example 5.4 the symbol 'a+' appears to be a typo for 'v+'.
[Section 2.2] The notation overloads γ: it is first introduced as a curve in the parameter domain U and then used for its image f∘γ on the surface. This makes Definition 4.1 and the subsequent formulas harder to parse; using a separate symbol for the image curve would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the gluing classification is derived by direct computation from the frame equations, with the standard singularity theory of [11] as the only external input.

full rationale

This paper contains no derived claim that reduces to its own inputs. The derivation chain is: Section 2.3 constructs the developable surfaces S_nu and S_b as envelopes of the normal-plane families along the gluing curve, and Lemma 2.5 proves the ruled-surface parametrizations in-paper without importing the result. Section 3 then computes, from the frame equations (2.1) alone, the cylinder condition delta'_nu = beta_nu w (3.1), the cone condition via the striction-curve derivative sigma'_nu = (rho_nu/beta_nu^2)(kappa_3 e + kappa_1 b), and the cuspidal-edge/swallowtail criteria through the identities eta_nu lambda_nu = -rho_nu sqrt(kappa_3^2+kappa_1^2)/beta_nu; these are closed-form computations, not assumed classifications. Section 4 applies these to the gluing of two surfaces: (4.1) and (4.2) are consistency relations among the defined frame invariants and the rotation angle theta, and the displayed expressions for beta_nu2, rho_nu2 in Theorem 4.5, together with Corollary 4.9, are obtained by substituting (4.2) into the Section 3 formulas; the substitution is confirmed term by term (for example, the swallowtail branch kappa_22 kappa_23 + 3 kappa'_21 of Corollary 3.3 expands to exactly the expression (kappa_12 cos theta - kappa_11 sin theta)(kappa_13 + 4 theta') + 3(kappa'_11 cos theta + kappa'_12 sin theta) in Corollary 4.9). No parameter is fitted and no empirical quantity is predicted. The only external inputs are Theorem 2.10, the standard singularity criteria from the independent reference [11], and the construction method from [7], which is re-proved in Lemma 2.5; there are no self-citations and no imported uniqueness theorem. The special-gluing Definitions 4.3-4.4 are phrased geometrically ('S_nu i is a cylinder', 'S_nu i has a cuspidal edge'), not analytically, so the theorems characterize the definitions rather than being true by definition. The remaining concern, that e is not unique at points with l(t) = 0, is a well-posedness matter, not a circularity: existence of a smooth e is assumed, the choice is unique up to the global gauge (l, e, kappa_1, kappa_3, theta) -> (-l, -e, -kappa_1, -kappa_3, -theta), and every condition in Corollaries 3.3, 3.6 and 4.9 is invariant under this gauge, so the classification is well-defined.

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

No free parameters are fitted. The paper rests on standard differential geometry of developables and fronts, plus three explicitly stated domain assumptions: the gluing locus is a curve, the frames have unit normals with a chosen e, and certain nondegeneracy conditions hold. The novel definitions of S_nu_i-cylindrical and related classes are analytic classification labels, not new physical entities.

assumptions (4)

standard math Standard classification of developable surfaces in R^3 into cylinders, cones, tangent developables, or combinations, used throughout Sections 3 and 4.
Invoked in Section 2.1 and in Definitions 2.1 through 2.3; the paper uses this classification to define what special gluings mean.
domain assumption The envelope of the family of planes H_v(t,X)=0 is the ruled surface S_v, and S_v is a developable frontal, as constructed in Section 2.3.
This identification relies on [7] and on the nondegeneracy assumptions (kappa1,kappa3) not equal (0,0) or (kappa2,kappa3) not equal (0,0). If these fail, the surfaces S_nu or S_b are not defined.
domain assumption The gluing locus admits a smooth unit vector field e with gamma prime equals l e and a smooth angle function theta between the two surface normals.
Used in Definition 4.1, Lemma 4.2 and all subsequent formulas. At points where l equals 0, the choice of e is not canonical.
standard math Singularity criteria for cuspidal edges and swallowtails of fronts (Theorem 2.10 from [11]) are applicable to S_nu and S_b.
Used in Theorem 3.2 and Theorem 3.5; relies on the surfaces being fronts, which is proven in Lemma 2.6.

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Pith. "Pith review of Geometry on the Gluing Locus of Two Surfaces." pith.science (2026). https://pith.science/paper/7GLLQ6X4

@misc{pith2026250601397,
  author       = {Pith},
  title        = {Pith review of: Geometry on the Gluing Locus of Two Surfaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7GLLQ6X4}},
  note         = {Machine review of arXiv:2506.01397}
}

read the original abstract

In this paper, we deal with the gluing of two surfaces, where the gluing locus is assumed to be a curve. We consider a moving frame along the gluing locus, and define developable surfaces with respect to the frame. Considering geometric properties of these developable surfaces, we study the geometry of gluing two surfaces.

Figures

Figures reproduced from arXiv: 2506.01397 by the authors.

**Figure 3.** From left to right, image of (f1, Sν1 ), image of (f2, Sν2 ), image of Glued (Sν1 , Sν2 ). Example 5.2. We give an example of Sν1 -cylindrical glue and also Sν2 -conical glue. Let us set γˆ(u, 0) = (cos u,sin u, 1), and let us set f1 and f2 by f1(u, v) = cos u,sin u, v , f2(u, v) = v cos u, v sin u, v where the gluing locus is γˆ. They are shown in the [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗

**Figure 4.** From left to right, image of (f1, Sν1 ), image of (f2, Sν2 ), image of Glued (Sν1 , Sν2 ). Example 5.3. We give an example of Sν1 -cylindrical glue. Let us set γˆ(u, 0) = (cos u,sin u, 0) and let us set f1 and f2 by f1(u, v) = sin(v + π 2 ) cos u,sin(v + π 2 ) sin u, cos(v + π 2 ) , f2(u, v) = cos u, v + sin u, 0 where the gluing locus is γˆ. They are shown in the [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗

Figures from the paper (4 more)

**Figure 5.** Figure 5: From left to right, image of f1, image of Sν1 , image of f2, image of Glued (f1, Sν1 , f2) along γˆ. Example 5.4. We give an example of Sν2 -conical glue. Let us set γˆ(u, 0) = √ 2 2 (cos u,sin u, 1) and let us set f1 and f2 by f1(u, v) = √ 2 2 cos u, a + √ 2 2 sin u,…

**Figure 6.** Figure 6: From left to right, image of f1, image of Sν1 , image of f2, image of Glued (f1, Sν1 , f2) along γˆ. Example 5.5. We give an example of Sν2 -cuspidal edgy glue and it is obtained by rotating the unit normal vector of the plane. Let us set γˆ(u, 0) = (u 2 , u3 , 0) and …

**Figure 7.** Figure 7: From left to right, image of f1, image of Sν1 , image of f2, image of Glued (f1, Sν1 , f2) along γˆ. Example 5.6. We give an example of Sν2 -swallowtailed glue and it is obtained by rotating the unit normal vector of the plane. Let us set γˆ(u, 0) = (0, 4u 3 , 3u 4 ) a…

**Figure 8.** Figure 8: From left to right, image of f1, image of Sν1 , image of f2, image of Glued (f1, Sν1 , f2) along γˆ. References [1] S. Fujimori, K. Saji, M. Umehara and K. Yamada, Singularities of maximal surfaces. Mathematische Zeitschrift, 259(4), 827–848. https://doi.org/10.1007/s0…

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Works this paper leans on

12 extracted references · 9 canonical work pages

[7]

Izumiya and S

S. Izumiya and S. Otani, Flat approximations of surfaces along curves.Demonstra- tio Mathematica,48(2015), 217–241. https://doi.org/10.1515/dema-2015-0018

work page doi:10.1515/dema-2015-0018 2015
[1]

Fujimori, K

S. Fujimori, K. Saji, M. Umehara and K. Yamada, Singularities of maximal surfaces.Mathematische Zeitschrift,259(4), 827–848. https://doi.org/10.1007/s00209-007-0250-0 20

doi:10.1007/s00209-007-0250-0
[2]

Gray,Modern Differential Geometry of Curves and Surfaces with Mathematica, Second edition

A. Gray,Modern Differential Geometry of Curves and Surfaces with Mathematica, Second edition. CRC Press, Boca Raton, FL, 1998. ISBN: 9780849371646

work page 1998
[3]

Honda, K

A. Honda, K. Naokawa, K. Saji, M. Umehara and K. Yamada, Curved foldings with common creases and crease patterns.Advances in Applied Mathematics,121 (2020), 102083, 10 pp. https://doi.org/10.1016/j.aam.2020.102083

arXiv 2020
[4]

Honda, K

A. Honda, K. Naokawa, K. Saji, M. Umehara and K. Yamada, On the existence of four or more curved foldings with common creases and crease patterns.Beiträge zur Algebra und Geometrie,63(2022), no.4, 723–761.https://doi.org/10.1007/s13366- 021-00602-2

doi:10.1007/s13366- 2022
[5]

Ishikawa, Singularities of flat extensions from generic surfaces with bound- aries.Differential Geometry and its Applications,28(2010), no

G. Ishikawa, Singularities of flat extensions from generic surfaces with bound- aries.Differential Geometry and its Applications,28(2010), no. 3, 341–354. https://doi.org/10.1016/j.difgeo.2010.02.001

work page doi:10.1016/j.difgeo.2010.02.001 2010
[6]

Izumiya, Ruled fronts and developable surfaces.Publicationes Mathematicae Debrecen,61(2002), 139–144

S. Izumiya, Ruled fronts and developable surfaces.Publicationes Mathematicae Debrecen,61(2002), 139–144. https://doi.org/10.5486/PMD.2002.2621

work page arXiv 2002
[8]

Izumiya, K

S. Izumiya, K. Saji and N. Takeuchi, Flat surfaces along cuspidal edges.Journal of Singularities,16(2017), 73–100. https://doi.org/10.5427/jsing.2017.16c

work page doi:10.5427/jsing.2017.16c 2017

Show all 12 references

[9]

1–2, 63–80

S.Izumiya, K.SajiandK.Teramoto, Flatsurfacesalongswallowtails.Kobe Journal of Mathematics,39(2022), no. 1–2, 63–80. https://doi.org/10.24546/0100492104

2022 doi
[10]

Izumiya and N

S. Izumiya and N. Takeuchi, Singularities of ruled surfaces inR 3.Mathe- matical Proceedings of the Cambridge Philosophical Society,130(2001), 1–11. https://doi.org/10.1017/S0305004100004643

2001 doi
[11]

Kokubu, W

M. Kokubu, W. Rossman, K. Saji, M. Umehara and K. Yamada, Singularities of flat fronts in hyperbolic space.Pacific Journal of Mathematics,221(2005), 303–351. https://doi.org/10.2140/pjm.2005.221.303

2005 doi
[12]

Umehara, K

M. Umehara, K. Saji and K. Yamada,Differential Geometry of Curves and Sur- faces with Singularities, Series in Algebraic and Differential Geometry,1, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2022. ISBN: 9789811237133 21

2022

Pith tools

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

[1] [7]

Izumiya and S

S. Izumiya and S. Otani, Flat approximations of surfaces along curves.Demonstra- tio Mathematica,48(2015), 217–241. https://doi.org/10.1515/dema-2015-0018

work page doi:10.1515/dema-2015-0018 2015

[2] [1]

Fujimori, K

S. Fujimori, K. Saji, M. Umehara and K. Yamada, Singularities of maximal surfaces.Mathematische Zeitschrift,259(4), 827–848. https://doi.org/10.1007/s00209-007-0250-0 20

doi:10.1007/s00209-007-0250-0

[3] [2]

Gray,Modern Differential Geometry of Curves and Surfaces with Mathematica, Second edition

A. Gray,Modern Differential Geometry of Curves and Surfaces with Mathematica, Second edition. CRC Press, Boca Raton, FL, 1998. ISBN: 9780849371646

work page 1998

[4] [3]

Honda, K

A. Honda, K. Naokawa, K. Saji, M. Umehara and K. Yamada, Curved foldings with common creases and crease patterns.Advances in Applied Mathematics,121 (2020), 102083, 10 pp. https://doi.org/10.1016/j.aam.2020.102083

arXiv 2020

[5] [4]

Honda, K

A. Honda, K. Naokawa, K. Saji, M. Umehara and K. Yamada, On the existence of four or more curved foldings with common creases and crease patterns.Beiträge zur Algebra und Geometrie,63(2022), no.4, 723–761.https://doi.org/10.1007/s13366- 021-00602-2

doi:10.1007/s13366- 2022

[6] [5]

Ishikawa, Singularities of flat extensions from generic surfaces with bound- aries.Differential Geometry and its Applications,28(2010), no

G. Ishikawa, Singularities of flat extensions from generic surfaces with bound- aries.Differential Geometry and its Applications,28(2010), no. 3, 341–354. https://doi.org/10.1016/j.difgeo.2010.02.001

work page doi:10.1016/j.difgeo.2010.02.001 2010

[7] [6]

Izumiya, Ruled fronts and developable surfaces.Publicationes Mathematicae Debrecen,61(2002), 139–144

S. Izumiya, Ruled fronts and developable surfaces.Publicationes Mathematicae Debrecen,61(2002), 139–144. https://doi.org/10.5486/PMD.2002.2621

work page arXiv 2002

[8] [8]

Izumiya, K

S. Izumiya, K. Saji and N. Takeuchi, Flat surfaces along cuspidal edges.Journal of Singularities,16(2017), 73–100. https://doi.org/10.5427/jsing.2017.16c

work page doi:10.5427/jsing.2017.16c 2017