Dupin cyclides osculating surfaces
Paweł G. Walczak
Geometry and Symbolic Computation
Haifa, May 16, 2013
E-mail: [email protected]
Paweł G. Walczak Geometry and Symbolic Computation Haifa, May 16, 2013 E-mail: [email protected]
Dupin cyclides osculating surfaces
Goal
Our goal here is to present after
A. Bartoszek, P. Walczak, Sz. Walczak, Dupin cyclides
osculating surfaces, preprint 2012
the answer to the following:
Question
What is the best order of tangency of a Dupin cyclid with a generic
surface and how to find such osculating cyclid?
Paweł G. Walczak Geometry and Symbolic Computation Haifa, May 16, 2013 E-mail: [email protected]
Dupin cyclides osculating surfaces
General setting
Here:
Σ and S – two hypersurfaces of RN given resp. by:
the equation F = 0, F : RN → Rn being a submersion,
a parametrization φ, S = φ(Rk ), φ : Rk → RN being an immersion.
f = F ◦ φ and p = φ(0) ∈ S ∩ Σ.
Paweł G. Walczak Geometry and Symbolic Computation Haifa, May 16, 2013 E-mail: [email protected]
Dupin cyclides osculating surfaces
Tangency type
Next:
V = (V1 , V2 , V3 , . . .) – a sequence of linear subspaces Vj of Rk
such that Vj+1 ⊂ Vj for all j ∈ N.
Definition
Σ and S are tangent along V at p whenever f (0) = 0 and
df (0)|V1 = 0 and d j f (0)| j−1 Vj−1 ⊗ Vj = 0,
j > 1,
(1)
If r = max{j ∈ N; Vj 6= {0}}, then the rank of the tangency equals
r . If d = (d1 , d2 , d3 , . . .) and dj = dim Vj , then Σ and S are
tangent of type d.
Paweł G. Walczak Geometry and Symbolic Computation Haifa, May 16, 2013 E-mail: [email protected]
Dupin cyclides osculating surfaces
Osculation
Let S = φ(Rk ) be as before while S = {Σλ , λ ∈ Λ}, Λ being an
open subset of Rm , be a smooth family of hypersurfaces given by
Fλ = 0, Fλ : RN → Rn being submersions.
Fix a non-increasing sequence d = (d1 , d2 , d3 , . . .) of nonnegative
integers and denote by rλ , λ ∈ Λ, the rank of tangency at p of type
d of Σλ and S.
Definition
An element Σλ0 (λ0 ∈ Λ) of S with
rλ0 = max{rλ ; λ ∈ Λ}
(2)
is said to be d-osculating S at p.
Paweł G. Walczak Geometry and Symbolic Computation Haifa, May 16, 2013 E-mail: [email protected]
Dupin cyclides osculating surfaces
Simple examples for curves
Generically:
Given a curve Γ : s 7→ γ(s) in RN , the affine r -dimensional,
r < N, hyperplane P through γ(s) spanned by the vectors
γ 0 (s), . . . , γ (r ) (s) is tangent to Γ of order r and there is no
r -hyperplane tangent to Γ of higher order, so P is
(1, 1, . . .)-osculating to Γ.
For all N, circles osculating curves in RN are tangent of
order 2. The same holds for helices, however, one may expect
a 1-parameter family of osculating helices and, in fact, this is
true and was known already to T. Olivier in the first half of
19th century.
Paweł G. Walczak Geometry and Symbolic Computation Haifa, May 16, 2013 E-mail: [email protected]
Dupin cyclides osculating surfaces
Simple examples for surfaces
Let d(r1 , r2 ) = (d1 , d2 , . . .), where dj = 2 for 1 ¬ j ¬ r1 , dj = 1 for
r1 < dj ¬ r1 + r2 and dj = 0 for all j > r1 + r2 . Generically:
The affine hyperplane H through p = φ(u) spanned by all the
derivatives (∂ i+j φ/(∂ i x1 ∂ j x2 ))(u) with i + j ¬ r1 and, for
example, (∂ k φ/∂ k x1 )(u), r1 < k ¬ r1 + r2 is tangent to S of
type d(r1 , r2 ) to S ∈ RN at p, therefore, one has a
1-dimensional family of (r2 + (r12 + 3r1 )/2)-dimensional
hyperplanes osculating S in this type. For example, if N > 6,
then one can find such a family of 6-dimensional hyperplanes
osculating S of types either d(1, 4) or d(2, 1).
The classical 2-dimensional spheres osculating surfaces in R3
are tangent to them of type d(1, 1).
Paweł G. Walczak Geometry and Symbolic Computation Haifa, May 16, 2013 E-mail: [email protected]
Dupin cyclides osculating surfaces
Local conformal invariants 1
A. Fialkov, Conformal differential geometry of a subspace,
Trans. Amer. Math. Soc. 56 (1944), 309 – 433.
G. Cairns, R. W. Sharpe and L. Webb. Conformal invariants for
curves in three dimensional space forms, Rocky Mountain J.
Math. 24 (1994), 933 – 959.
Theorem
For a non-umbilic point p of a surface S ⊂ R3 there exists a unique
Möbius map g such that g (p) = (0, 0, 0) and g (S) is given by
z
=
+
1
1 2
(x − y 2 ) + (θ1 x 3 + θ2 y 3 )
2
6
1
(ax 4 + bx 3 y + Ψx 2 y 2 + cxy 3 + dy 4 ) + H.O.T .
24
Paweł G. Walczak Geometry and Symbolic Computation Haifa, May 16, 2013 E-mail: [email protected]
Dupin cyclides osculating surfaces
Local conformal invariants 2
The blue quantities in the above together with two vector fields ξi
given by
1
· Xi , i = 1, 2,
(3)
ξi =
k1 − k2
where ki ’s are principal curvatures and Xi are unit principal vectors,
constitute the complete set of local conformal invariants
determining a surface up to a Möbius transformation.
Paweł G. Walczak Geometry and Symbolic Computation Haifa, May 16, 2013 E-mail: [email protected]
Dupin cyclides osculating surfaces
Peculiar surfaces
Examples
Canal surfaces = envelopes of 1-parameter familes of spheres
= surfaces with θi ≡ 0 for some i = 1, 2.
Special canal surfaces (according to Bartoszek, Langevin and
W.) = canal surfaces with, say, θ1 ≡ 0 and ξ1 (θ2 ) ≡ 0
= surfaces of revolution, cones or cyliners over planar (or,
spherical) curves.
Dupin cyclides = canal surfaces in two ways = surfaces with
both θi ’s identically zero = conformal images of tori, cylinders
or cones of revolution = (by Banchoff) surfaces with STTP
(spherical two piece property).
Paweł G. Walczak Geometry and Symbolic Computation Haifa, May 16, 2013 E-mail: [email protected]
Dupin cyclides osculating surfaces
Tangency of Dupin cyclides
Canonical equations of a surface S
z
=
+
1
1 2
(x − y 2 ) + (θ1 x 3 + θ2 y 3 )
2
6
1
4
3
(ax + bx y + Ψx 2 y 2 + cxy 3 + dy 4 ) + H.O.T .
24
and of a Dupin cyclid
1
1
z = (x 2 − y 2 ) + + (3x 4 + Ψx 2 y 2 − 3y 4 ) + H.O.T .
2
24
show that generically they are tangent of p
order 3 in the Dupin
direction determined by y = tx with t = 3 −θ1 /θ2 .
Paweł G. Walczak Geometry and Symbolic Computation Haifa, May 16, 2013 E-mail: [email protected]
Dupin cyclides osculating surfaces
Osculating Dupin cyclides
Generically, one can choose Ψ for which all the terms of order 4
coincide in the Dupin direction. In our terminology, the Dupin cyclid
with this value of Ψ osculates S in type d = (2, 2, 1, 1, 0, 0, . . .).
Paweł G. Walczak Geometry and Symbolic Computation Haifa, May 16, 2013 E-mail: [email protected]
Dupin cyclides osculating surfaces
A striking example
Example. Helcats are minimal surfaces Sα which form a
1-parameter family connecting the helicoid and catenoid:
x1 = cosα · sinh s · sin t + sin α · cosh s cos t,
x2 = − cos α · sinh s · cos t + sin α · cosh s · sin t,
x3 = sin α · s + cos α · t,
For these surfaces
θ1 /θ2 =
cos α
= const.,
1 + sin α
that is the Dupin direction forms constant angles with principal
directions.
Paweł G. Walczak Geometry and Symbolic Computation Haifa, May 16, 2013 E-mail: [email protected]
Dupin cyclides osculating surfaces
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Paweł G. Walczak Geometry and Symbolic Computation Haifa, May 16, 2013 E-mail: [email protected]
Dupin cyclides osculating surfaces