As proje c oes de superf Jorge Luiz Deolindo Silva Universidade - - PowerPoint PPT Presentation

Cross-ratio Motivation in R3 Surfaces in R4 Cross-ratio for surfaces in R4

As proje¸ c˜

• es de superf´

ıcies em R4

Jorge Luiz Deolindo Silva

Universidade Federal de Santa Catarina - Blumenau

28/04/2017 Col´

• quio de Matem´

atica - UFSC

Cross-ratio Motivation in R3 Surfaces in R4 Cross-ratio for surfaces in R4

Cross-ratio Motivation in R3 Surfaces in R4 Cross-ratio for surfaces in R4

Contents

1986 2002 1995 80's 2015

• R. Thom

and I. Porteous James Montaldi Bruce and T ari Mochida, Romero-Fuster and Ruas We are here

1955

• H. Whitney

Cross-ratio Motivation in R3 Surfaces in R4 Cross-ratio for surfaces in R4

Contents

1986 2002 1995 80's 2015

• R. Thom

and I. Porteous James Montaldi Bruce and T ari Mochida, Romero-Fuster and Ruas We are here

1955

• H. Whitney

Singularity Theory

Cross-ratio Motivation in R3 Surfaces in R4 Cross-ratio for surfaces in R4

Contents

1986 2002 1995 80's 2015

• R. Thom

and I. Porteous James Montaldi Bruce and T ari Mochida, Romero-Fuster and Ruas We are here

1955

• H. Whitney

Cross-ratio Motivation in R3 Surfaces in R4 Cross-ratio for surfaces in R4

Contents

1986 2002 1995 80's 2015

• R. Thom

and I. Porteous James Montaldi Bruce and T ari Mochida, Romero-Fuster and Ruas We are here

1955

• H. Whitney

Contact geometry with special submanifolds

Height function, orthogonal projections, distance square function, etc.

Cross-ratio Motivation in R3 Surfaces in R4 Cross-ratio for surfaces in R4

Contents

• Cross-ratio

Cross-ratio Motivation in R3 Surfaces in R4 Cross-ratio for surfaces in R4

Contents

• Cross-ratio
• Motivation in R3

Cross-ratio Motivation in R3 Surfaces in R4 Cross-ratio for surfaces in R4

Contents

• Cross-ratio
• Motivation in R3
• Surfaces in R4

Cross-ratio Motivation in R3 Surfaces in R4 Cross-ratio for surfaces in R4

Contents

• Cross-ratio
• Motivation in R3
• Surfaces in R4
• Cross-ratio for surfaces in R4

Cross-ratio Motivation in R3 Surfaces in R4 Cross-ratio for surfaces in R4

Cross-ratio

Cross-ratio Motivation in R3 Surfaces in R4 Cross-ratio for surfaces in R4

Cross-ratio

A B C D A' B' C' D'

l1 l2 l3 l4

Cross-ratio Motivation in R3 Surfaces in R4 Cross-ratio for surfaces in R4

Cross-ratio

A B C D A' B' C' D'

l1 l2 l3 l4

ρ = C − A C − B · D − B D − A

Cross-ratio Motivation in R3 Surfaces in R4 Cross-ratio for surfaces in R4

Cross-ratio

A B C D A' B' C' D'

l1 l2 l3 l4

ρ = C − A C − B · D − B D − A = C ′ − A′ C ′ − B′ · D′ − B′ D′ − A′

Cross-ratio Motivation in R3 Surfaces in R4 Cross-ratio for surfaces in R4

Cross-ratio

A B C D A' B' C' D'

l1 l2 l3 l4

ρ = C − A C − B · D − B D − A = C ′ − A′ C ′ − B′ · D′ − B′ D′ − A′ if l2 has infinite ρ = C − A D − A

Cross-ratio Motivation in R3 Surfaces in R4 Cross-ratio for surfaces in R4

Surfaces in R3

Definition Given a surface S ⊂ R3. The Gauss map is given by N : S → S2 p → N(p) =

φx×φy ||φx×φy||(p),

where φ is a parametrization of S.

Cross-ratio Motivation in R3 Surfaces in R4 Cross-ratio for surfaces in R4

Surfaces in R3

Let p ∈ S and dNp : TpS → TpS. The Gaussian curvature is K = det(dNp). A point p ∈ S is

• 1. Elliptic, if K > 0;
• 2. Hyperbolic, if K < 0;
• 3. Parabolic, is K = 0.

Cross-ratio Motivation in R3 Surfaces in R4 Cross-ratio for surfaces in R4

Surfaces in R3

Elliptic Hyperbolic Parabolic

Gaussian Curvature: K = det(dNp).

Cross-ratio Motivation in R3 Surfaces in R4 Cross-ratio for surfaces in R4

Surfaces in R3

Elliptic Hyperbolic Parabolic

Asymptotic directions: ∃ 2 directions in hyperbolic region

Cross-ratio Motivation in R3 Surfaces in R4 Cross-ratio for surfaces in R4

Surfaces in R3

Field of pairs

• f directions

Elliptic Hyperbolic Parabolic

Asymptotic directions: ∃! direction in parabolic region

Cross-ratio Motivation in R3 Surfaces in R4 Cross-ratio for surfaces in R4

Surfaces in R3

Field of pairs

• f directions

Elliptic Hyperbolic Parabolic

Asymptotic curves

Cross-ratio Motivation in R3 Surfaces in R4 Cross-ratio for surfaces in R4

Surfaces in R3

Field of pairs

• f directions

Elliptic Hyperbolic Parabolic

Asymptotic curves

Cross-ratio Motivation in R3 Surfaces in R4 Cross-ratio for surfaces in R4

Surfaces in R3

Field of pairs

• f directions

Elliptic Hyperbolic Parabolic asymptotic curve

Cross-ratio Motivation in R3 Surfaces in R4 Cross-ratio for surfaces in R4

Surfaces in R3

Field of pairs

• f directions

Elliptic Hyperbolic Parabolic asymptotic curve

Definition A Cusp of Gauss is a parabolic point at which the single asymptotic direction is tangent to the parabolic curve.

Cross-ratio Motivation in R3 Surfaces in R4 Cross-ratio for surfaces in R4

Surfaces in R3

Why the name Cusp of Gauss?

Cross-ratio Motivation in R3 Surfaces in R4 Cross-ratio for surfaces in R4

Monge Form and Height Functions

We can take the parametrization locally at origin in Monge form φ : R2, 0 → M ⊂ R3 (x, y) → (x, y, f (x, y)) where df (0, 0) = 0.

Cross-ratio Motivation in R3 Surfaces in R4 Cross-ratio for surfaces in R4

Monge Form and Height Functions

We can take the parametrization locally at origin in Monge form φ : R2, 0 → M ⊂ R3 (x, y) → (x, y, f (x, y)) where df (0, 0) = 0. The contact of M with planes is determined by singularities of H : M × S2 → R (x, y, v) → hv = φ(x, y), v

Cross-ratio Motivation in R3 Surfaces in R4 Cross-ratio for surfaces in R4

Monge Form and Height Functions

We can take the parametrization locally at origin in Monge form φ : R2, 0 → M ⊂ R3 (x, y) → (x, y, f (x, y)) where df (0, 0) = 0. The contact of M with planes is determined by singularities of H : M × S2 → R (x, y, v) → hv = φ(x, y), v If v = (0, 0, 1) then Hv = f (x, y).

Cross-ratio Motivation in R3 Surfaces in R4 Cross-ratio for surfaces in R4

Height Functions

Fixed v ∈ S2, Hv is equivalent to:

• A±

1 : x2 ± y2 ⇔ p is not parabolic

• A2 : x2 + y3 ⇔ p is parabolic point
• A±

3 : x2 ± y4 ⇔ p is parabolic and a Cusp of Gauss

Field of pairs

• f directions

Elliptic Hyperbolic Parabolic asymptotic curve

A+

1

A

• 1

A3 A>2

Cross-ratio Motivation in R3 Surfaces in R4 Cross-ratio for surfaces in R4

Height Functions

Figure: A+

1

Figure: A2 Figure: A−

1

Figure: A+

3

Figure: A−

3

Cross-ratio Motivation in R3 Surfaces in R4 Cross-ratio for surfaces in R4

Asymptotic Curves

(Platonova, 1981) Monge-form (x, y, f (x, y)), the 4-jet of a surface at A3 can be sent by projective transformation to the normal form j4f (x, y) = y 2 2 − x2y + λx4, λ = 0, 1 2

Cross-ratio Motivation in R3 Surfaces in R4 Cross-ratio for surfaces in R4

Asymptotic Curves

(Platonova, 1981) Monge-form (x, y, f (x, y)), the 4-jet of a surface at A3 can be sent by projective transformation to the normal form j4f (x, y) = y 2 2 − x2y + λx4, λ = 0, 1 2

parabolic curve A3

Discriminant: f 2

xy − f xxf yy = 0 (parabolic curve)

Cross-ratio Motivation in R3 Surfaces in R4 Cross-ratio for surfaces in R4

Parabolic Curves

(Platonova, 1981) Monge-form (x, y, f (x, y)), the 4-jet of a surface at A3 can be sent by projective transformation to the normal form j4f (x, y) = y 2 2 − x2y + λx4, λ = 0, 1 2

parabolic curve A3

y = 2(3λ − 1)x2 + h.o.t

Cross-ratio Motivation in R3 Surfaces in R4 Cross-ratio for surfaces in R4

Inflection of asymptotic curve

(Platonova, 1981) Monge-form (x, y, f (x, y)), the 4-jet of a surface at A3 can be sent by projective transformation to the normal form j4f (x, y) = y 2 2 − x2y + λx4, λ = 0, 1 2

parabolic curve flecnodal A3

y = 2λ(4λ − 1)x2 + h.o.t

Cross-ratio Motivation in R3 Surfaces in R4 Cross-ratio for surfaces in R4

Inflection of asymptotic curve

(Platonova, 1981) Monge-form (x, y, f (x, y)), the 4-jet of a surface at A3 can be sent by projective transformation to the normal form j4f (x, y) = y 2 2 − x2y + λx4, λ = 0, 1 2

parabolic curve flecnodal A3

y = 2λ(4λ − 1)x2 + h.o.t

Cross-ratio Motivation in R3 Surfaces in R4 Cross-ratio for surfaces in R4

Multi-local curve

(Platonova, 1981) Monge-form (x, y, f (x, y)), the 4-jet of a surface at A3 can be sent by projective transformation to the normal form j4f (x, y) = y 2 2 − x2y + λx4, λ = 0, 1 2 v

multi-local points

Cross-ratio Motivation in R3 Surfaces in R4 Cross-ratio for surfaces in R4

Multi-local curve

(Platonova, 1981) Monge-form (x, y, f (x, y)), the 4-jet of a surface at A3 can be sent by projective transformation to the normal form j4f (x, y) = y 2 2 − x2y + λx4, λ = 0, 1 2

parabolic curve flecnodal conodal ( ) A A

1 1

A3

y = 2λx2 + h.o.t

Cross-ratio Motivation in R3 Surfaces in R4 Cross-ratio for surfaces in R4

Lifted field

p a r a b

• l

i c f l e c n

• d

a l conodal ( ) A A

1 1

A3 e l l i p t i c h y p e r b

• l

i c

p

3-manifold of contact elements of M, PT ∗M ≃ M × RP1 Lift: π : PT ∗M → M is a difeomorphism in hyperbolic points

Cross-ratio Motivation in R3 Surfaces in R4 Cross-ratio for surfaces in R4

Lifted field

p a r a b

• l

i c f l e c n

• d

a l conodal ( ) A A

1 1

A3 e l l i p t i c h y p e r b

• l

i c

p

Cross-ratio Motivation in R3 Surfaces in R4 Cross-ratio for surfaces in R4

Lifted field

p a r a b

• l

i c f l e c n

• d

a l conodal ( ) A A

1 1

A3 e l l i p t i c h y p e r b

• l

i c

p

Cross-ratio Motivation in R3 Surfaces in R4 Cross-ratio for surfaces in R4

Lifted field

p a r a b

• l

i c f l e c n

• d

a l conodal ( ) A A

1 1

A3 e l l i p t i c h y p e r b

• l

i c

p

Cross-ratio Motivation in R3 Surfaces in R4 Cross-ratio for surfaces in R4

Lifted field

p a r a b

• l

i c f l e c n

• d

a l conodal ( ) A A

1 1

A3 e l l i p t i c h y p e r b

• l

i c

p

Cross-ratio Motivation in R3 Surfaces in R4 Cross-ratio for surfaces in R4

Lifted field

p a r a b

• l

i c f l e c n

• d

a l conodal ( ) A A

1 1

A3 e l l i p t i c h y p e r b

• l

i c

p

Cross-ratio Motivation in R3 Surfaces in R4 Cross-ratio for surfaces in R4

Lifted field

p a r a b

• l

i c f l e c n

• d

a l conodal ( ) A A

1 1

A3 e l l i p t i c h y p e r b

• l

i c

p

Cross-ratio Motivation in R3 Surfaces in R4 Cross-ratio for surfaces in R4

Lifted field

p a r a b

• l

i c f l e c n

• d

a l conodal ( ) A A

1 1

A3 e l l i p t i c h y p e r b

• l

i c

p

Cross-ratio Motivation in R3 Surfaces in R4 Cross-ratio for surfaces in R4

Uribe-Vargas

(Platonova, 1981) Monge-form (x, y, f (x, y)), 4-jet in A3 j4f (x, y) = y 2 2 − x2y + λx4, λ = 0, 1 2

Theorem (Uribe-Vargas, 2006) Cross-Ratio: ρ = 2λ(4λ − 1) − 2λ 3(4λ − 1) − 2λ = 2 · λ

Cross-ratio Motivation in R3 Surfaces in R4 Cross-ratio for surfaces in R4

Surfaces in R4

Given a surface M ֒ → R4 and p ∈ M in Monge form φ : R2, 0 → M ⊂ R4 (x, y) → (x, y, f1(x, y), f2(x, y)) where df1(0, 0) = df2(0, 0) = 0.

Cross-ratio Motivation in R3 Surfaces in R4 Cross-ratio for surfaces in R4

Surfaces in R4

Given a surface M ֒ → R4 and p ∈ M in Monge form φ : R2, 0 → M ⊂ R4 (x, y) → (x, y, f1(x, y), f2(x, y)) where df1(0, 0) = df2(0, 0) = 0. The contact of M with lines is determines by singularities of

The family of orthogonal projections in 3-space is given by P : M × S3 → TS3 with P(x, y, v) = (v, φ(x, y) − φ(x, y), vv).

Cross-ratio Motivation in R3 Surfaces in R4 Cross-ratio for surfaces in R4

Orthogonal Projection and the Surface

Fixed v, the projection is locally at Pv : R2, 0 → R3, 0.

Cross-ratio Motivation in R3 Surfaces in R4 Cross-ratio for surfaces in R4

Orthogonal Projection and the Surface

Fixed v, the projection is locally at Pv : R2, 0 → R3, 0. The singularity of Pv at p is worse than a cross-cap iff v ∈ TpM is a special direction, called the asymptotic direction at p. (Bruce, Nogueira, 1998) The contact of surface M with lines is also determined by j2(f1(x, y), f2(x, y)) such that the asymptotic directions satisfied:

hyperbolic region parabolic region elliptic region

Cross-ratio Motivation in R3 Surfaces in R4 Cross-ratio for surfaces in R4

Classification of D. Mond

The generic singularities of Pv are Ae-equivalent to the singularities below classified by [Mond, 1985] Name Normal Form Ae-codim immersion (x, y, 0) Cross-cap (x, y2, xy) S±

k

(x, y2, y3 ± xk+1y) k = 1, 2, 3 B±

k

(x, y2, x2y ± y2k+1) k = 2, 3 C ±

3

(x, y2, xy3 ± x3y) 3 H±

k

(x, xy ± y3k−1, y3) k = 2, 3 P3(c) (x, xy + y3, xy2 + cy4), c = 0, 1

2, 1, 3 2

3 The codimension of P3(c) is that of its stratum.

Cross-ratio Motivation in R3 Surfaces in R4 Cross-ratio for surfaces in R4

Special curves and points on a surface

Proposition (Bruce and Nogueira, 1998. Bruce and Tari,

• 2002. Mochida, Romero-Fuster and Ruas, 1995)

P

(c)

3

H

3

B

3

S

3

C

3

B

1

S

1

= B

1

S

1

= B -curve

2

S

2

H

2

 parabolic curve hyperbolic elliptic

Cross-ratio Motivation in R3 Surfaces in R4 Cross-ratio for surfaces in R4

The point P3(c) have a behavior similar to Cusp of Gauss

• Looking for “flecnodal curve”
• Looking for multi-local curves.

Cross-ratio Motivation in R3 Surfaces in R4 Cross-ratio for surfaces in R4

Theorem

Theorem At a P3(c)-point passe the following curves: the flecnodal curve Fl and the curves obtained from the local and multi-local singularities

• f Pv. All these curves generically have tangency of order 2.

P

(c)

3

B

1

S

1

= B -curve

2

(A

0S 0) 2

A

0S 1

A

0S 1

| A

+

• 

parabolic curve hyperbolic elliptic

S

2

F

l

Cross-ratio Motivation in R3 Surfaces in R4 Cross-ratio for surfaces in R4

Questions???

Surfaces in R3

A3



parabolic curve hyperbolic elliptic Conodal curve

Flecnodal curve

(A )

1A1

P

(c) 3

B

S

= B -curve

(A

A

A

| A

• 

parabolic curve hyperbolic elliptic

S

F

l

Cross-ratio Motivation in R3 Surfaces in R4 Cross-ratio for surfaces in R4

Questions???

Surfaces in R3

A3



parabolic curve hyperbolic elliptic Conodal curve

Flecnodal curve

(A )

1A1

[Platonova, 1981] j4f = y 2 2 −x2y+λx4, λ = 0, 1 2

P

(c) 3

B

S

= B -curve

(A

A

A

| A

• 

parabolic curve hyperbolic elliptic

S

F

l

Cross-ratio Motivation in R3 Surfaces in R4 Cross-ratio for surfaces in R4

Questions???

Surfaces in R3

A3



parabolic curve hyperbolic elliptic Conodal curve

Flecnodal curve

(A )

1A1

[Platonova, 1981] j4f = y 2 2 −x2y+λx4, λ = 0, 1 2

[Uribe-Vargas, 2006] Cross-Ratio: ρ = 2 · λ

P

(c) 3

B

S

= B -curve

(A

A

A

| A

• 

parabolic curve hyperbolic elliptic

S

F

l

Cross-ratio Motivation in R3 Surfaces in R4 Cross-ratio for surfaces in R4

Questions???

Surfaces in R3

A3



parabolic curve hyperbolic elliptic Conodal curve

Flecnodal curve

(A )

1A1

[Platonova, 1981] j4f = y 2 2 −x2y+λx4, λ = 0, 1 2

[Uribe-Vargas, 2006] Cross-Ratio: ρ = 2 · λ Surfaces in R4

P

(c) 3

B

S

= B -curve

(A

A

A

| A

• 

parabolic curve hyperbolic elliptic

S

F

l

Cross-ratio Motivation in R3 Surfaces in R4 Cross-ratio for surfaces in R4

Questions???

Surfaces in R3

A3



parabolic curve hyperbolic elliptic Conodal curve

Flecnodal curve

(A )

1A1

[Platonova, 1981] j4f = y 2 2 −x2y+λx4, λ = 0, 1 2

[Uribe-Vargas, 2006] Cross-Ratio: ρ = 2 · λ Surfaces in R4

P

(c) 3

B

S

= B -curve

(A

A

A

| A

• 

parabolic curve hyperbolic elliptic

S

F

l

Question 1: ∃ a Normal form (f1, f2)??

Cross-ratio Motivation in R3 Surfaces in R4 Cross-ratio for surfaces in R4

Questions???

Surfaces in R3

A3



parabolic curve hyperbolic elliptic Conodal curve

Flecnodal curve

(A )

1A1

[Platonova, 1981] j4f = y 2 2 −x2y+λx4, λ = 0, 1 2

[Uribe-Vargas, 2006] Cross-Ratio: ρ = 2 · λ Surfaces in R4

P

(c) 3

B

S

= B -curve

(A

A

A

| A

• 

parabolic curve hyperbolic elliptic

S

F

l

Question 1: ∃ a Normal form (f1, f2)?? Question 2: Cross-Ratio: ???

Cross-ratio Motivation in R3 Surfaces in R4 Cross-ratio for surfaces in R4

Some Results

Theorem (D-S, Kabata, 2015) The 4-jet of surface at P3(c) can be sent by projective transformations to the normal form j4(f1, f2) = (x2 + xy2 + αy4, xy + βy3 + φ4), where 6β2 + 4α − 15β + 5 = 0, α = 0, 1/2, 1, 3/2 and φ4 is a polynomial of degree 4.

Cross-ratio Motivation in R3 Surfaces in R4 Cross-ratio for surfaces in R4

Some Results

Lemma At P3(c):

• ∆-set: x = 3(−3β2 − 2α + 2β)y2 + h.o.t.
• B2-curve: x = 2(12αβ2+8α2−8αβ−4β2−3α+3β)

(2β−1)2

y2 + h.o.t.

• S2-curve: x = 6(36αβ2−72β3+24α2−72αβ+66β2−α+β)

(6β+1)2

y 2 + h.o.t.

• Fl-curve: x = 6(−36β2+24β+1−24α)(−α+β)

(−6β+1)2

y2 + h.o.t.

• (A0S0)2-curve: x = 3(−3β2 − 2α + 4β)y2 + h.o.t.
• A0S1-curve: x = − 3αβ2+2α2−4αβ+3β2

−4β2−3α+4β

y2 + h.o.t.

• A0S0|A1-curve: x = 1

4 (3β2−16αβ+8α212αβ2) −β2−α+β

y2 + h.o.t.

Cross-ratio Motivation in R3 Surfaces in R4 Cross-ratio for surfaces in R4

Theorem

Theorem At a generic P3(c)-point, three cross-ratios of the lines above allow to recover the projective invariants α and β of surface.

β = ρ1 − 1 3(2ρ1 − 1), α = (80ρ2ρ1 − 32ρ2 + 20ρ3ρ1 − 8ρ3 + 42ρ1 − 21)(ρ1 − 1) 9(3ρ3 + 1 + 12ρ2)(2ρ1 − 1)2 . where, ρ1 = (lP, lB : lS, lF) ρ2 = (lP, lg : ls01, ls02) ρ3 = (lP, lg : ls1, ls02) The cross-ratio ρ1 (resp. ρ2, ρ3) is obtained from the local (resp. multi-local) singularities and and depend of the multi-local curves.

Cross-ratio Motivation in R3 Surfaces in R4 Cross-ratio for surfaces in R4

Theorem The configuration of the curves ∆ and B2 at P3(c)-point.

4 2 1 6 5 3 a b cP = 0 cB = 0

3/4 1/4 3/8 1/2 2/3

2 1 3 6 5 4

Cross-ratio Motivation in R3 Surfaces in R4 Cross-ratio for surfaces in R4

Theorem The configuration of the curves ∆ and S2 at P3(c)-point.

3 2

1 5

4

a b cP = 0 cS = 0

3/4

• 1/6

1/24

• 1/4

2/3

2 1 3 6 5 4

6

Cross-ratio Motivation in R3 Surfaces in R4 Cross-ratio for surfaces in R4

Theorem The configuration of the curves ∆ and flecnodal at P3(c)-point.

3 2

1 5

4

a b cP = 0 cF = 0

3/4

• 1/6

2/3

2 1 3

6

• 1/6

7

6 5 4 7

Cross-ratio Motivation in R3 Surfaces in R4 Cross-ratio for surfaces in R4

Theorem At P3(c)-point the asymptotic curves have a folded singularity if (−9β2 − 6α + (15/2)β) = 0, 1

16.

• (−9β2 − 6α + (15/2)β < 0 ⇐

⇒ folded saddle;

• 0 < (−9β2 − 6α + (15/2)β) < 1

16 ⇐

⇒ folded node;

• (−9β2 − 6α + (15/2)β) > 1

16 ⇐

⇒ folded focus.

Figure: A folded saddle (left), node (centre) and focus (right).

Cross-ratio Motivation in R3 Surfaces in R4 Cross-ratio for surfaces in R4

References

• J. W. Bruce and A. C. Nogueira, Surfaces in R4 and duality.
• Quart. J. Math. Oxford Ser. Ser. (2), 49 (1998), 433–443.
• J. W. Bruce and F. Tari, Families of surfaces in R4. Proc. Edinb.
• Math. Soc. 45 (2002), 181–203.
• C. G. Gibson, C. A. Hobbs, W. L. Marar, On Versal Unfoldings of

Singularities for General Two-Dimensional Spatial Motions. Acta Applicandae Mathematicae 47: 221242, 1997.

• D. K. H. Mochida, M. C. Romero-Fuster and M. A. S. Ruas, The

geometry of surfaces in 4-space from a contact viewpoint. Geometria Dedicata 54 (1995), 323–332.

Cross-ratio Motivation in R3 Surfaces in R4 Cross-ratio for surfaces in R4

• D. M. Q. Mond, On the classification of germs of maps from R2 to
• R3. Proc. London Math. Soc. 50 (1985), 333–369.
• O. A. Platonova, Projections of smooth surfaces, J. Soviet Math.

35 no. 6 (1986), 2796-2808 [Tr. Sem. I. G. Petvoskii 10 (1984), 135-149 in Russian].

• R. Uribe-Vargas, A projective invariant for swallowtails and

godrons, and global theorems on the flecnodal curve. Mosc. Math.

• J. 6 (2006), 731–768, 772.