Asymptotic Nets and Discrete Affine Surfaces with Indefinite Metric

# Asymptotic Nets and Discrete Affine Surfaces with Indefinite Metric

Marcos Craizer
###### Abstract

Asymptotic net is an important concept in discrete differential geometry. In this paper, we show that we can associate affine discrete geometric concepts to an arbitrary non-degenerate asymptotic net. These concepts include discrete affine area, mean curvature, normal and co-normal vector fields and cubic form, and they are related by structural and compatibility equations. We consider also the particular cases of affine minimal surfaces and affine spheres.
Keywords: Asymptotic nets, Discrete affine surfaces, Discrete affine indefinite metric.

Pontifícia Universidade Católica - Rio de Janeiro - Brazil.
craizer@mat.puc-rio.br

## 1 Introduction

The expansion of computer graphics and applications in mathematical physics have recently given a great impulse to discrete differential geometry. In this discrete context, surfaces with indefinite metric are generally modelled as asymptotic nets. ([2],[1]). In particular, there are several types of discrete affine surfaces with indefinite metric modelled as asymptotic nets: Affine spheres ([3]), improper affine spheres ([7], [6]) and minimal surfaces ([5]). In this work we define a general class of discrete affine surfaces that includes these types as particular cases.

Beginning with an arbitrary asymptotic net, with the mild hypothesis of non-degeneracy, we define a discrete affine invariant structure on it. The discrete geometric concepts defined are affine metric, normal vector field, co-normal vector field, cubic form and mean curvature, and they are related by equations comparable with the correspondent smooth equations. The asymptotic net is called minimal when its mean curvature is zero, while the asymptotic net is an called an affine sphere when certain discrete derivatives described in section 4 vanishes. Asymptotic discretizations of the one-sheet hyperboloid are basic examples of discrete surfaces that satisfy our definition of affine spheres.

The main characteristic of our definition of the discrete affine geometric concepts is that they are related by discrete equations that closely resembles the smooth equations of affine differential geometry of surfaces. The simplicity of these equations are somewhat surprising, since the construction is very general. Structural equations describe the discrete immersion in terms of the affine metric, cubic form and mean curvature, and these concepts must satisfy compatibility equations. When the discrete affine metric, cubic form and mean curvature satisfy the compatibility equations, one can define an asymptotic net, unique up to eqüi-affine transformations of , that satisfies the structural equations.

The paper is organized as follows: Section 2 reviews the basic facts of smooth affine differential geometry with asymptotic parameters. Section 3 contains the main results: It shows how you can define the affine metric, normal and co-normal vector fields, cubic form and mean curvature from a given asymptotic net. It also relate this work with the discrete affine minimal surfaces of [5] and give an example of a ”non-minimal” asymptotic net. Section 4 describes the structural equations and propose a new definition of affine spheres that is more general than the one proposed in [3]. Section 5 describes the compatibility equations and shows that an affine surface is defined, up to eqüi-affine transformations of , by its metric, cubic form and mean curvature.

Acknowledgements. The author want to thank CNPq for financial support during the preparation of this paper.

## 2 Smooth affine surface with indefinite metric

In this section we review the structure of smooth surfaces with indefinite Berwald-Blashcke metric. The concepts and equations of this section are the inspiration for the discrete concepts and equations of the rest of the paper. For details and proofs of this section, see [4].

### 2.1 Concepts and equations

Notation. Given two vectors , we denote by the cross product and by the dot product between them. Given three vectors , we denote by their determinant.

Consider a parameterized smooth surface , where is an open subset of the plane and denote

 L(u,v) = [qu,qv,quu], M(u,v) = [qu,qv,quv], N(u,v) = [qu,qv,qvv].

The surface is non-degenerate if , and, in this case, the Berwald-Blaschke metric is defined by

 ds2=1|LN−M2|1/4(Ldu2+2Mdudv+Ndv2).

If , the metric is definite while if , the metric is indefinite. In this paper, we shall restrict ourselves to surfaces with indefinite metric. We say that the coordinates are asymptotic if . In this case, the metric takes the form , where .

The vector field is called the co-normal vector field. It satisfies Lelieuvre’s equations

 qu = ν×νu, qv = −ν×νv.

And the vector field is called the affine normal vector field. One can easily verify that .

The functions

 A(u,v) = [qu,quu,ξ], B(u,v) = [qv,qvv,ξ]

are the coefficients of the affine cubic form (see [8]). We can write

 quu = 1Ω(Ωuqu+Aqv), (1) qvv = 1Ω(Bqu+Ωvqv). (2)

The function

 H(u,v)=ΩuΩv−ΩΩuv−ABΩ3

is called the affine mean curvature. One can write

 ξu = −Hqu+AvΩ2qv, (3) ξv = BuΩ2qu−Hqv. (4)

Equations (1), (2), (3) and (4) are the structural equations of the surface. For a given surface, the quadratic form , the cubic form and the affine mean curvature should satisfy the following compatibility equations:

 Hu = ABuΩ3−1Ω(AvΩ)v, (5) Hv = BAvΩ3−1Ω(BuΩ)u. (6)

Conversely, given and satisfying equations (5) and (6), there exists a parameterization of a surface with quadratic form , cubic form and affine mean curvature .

### 2.2 Example

###### Example 1

Consider the asymptotic parameterization of the one-sheet hyperboloid ,

 q(u,v)=csinh(u+v)(−cosh(u−v),−sinh(u−v),cosh(u+v)).

Taking derivatives

 qu = csinh2(u+v)(cosh(2v),−sinh(2v),−1), qv = csinh2(u+v)(cosh(2u),sinh(2u),−1).

Straightforward calculations shows that

 Ω=2c3/2sinh2(u+v),
 ν(u,v)=c1/21sinh(u+v)(cosh(u−v),sinh(v−u),cosh(u+v))

and

 ξ(u,v)=c−1/21sinh(u+v)(−cosh(u−v),sinh(v−u),cosh(u+v)).

One can also verify that and that is the affine mean curvature.

## 3 Geometric concepts from an asymptotic net

In this section, we define the following quantities associated with a given non-degenerate asymptotic net: Affine metric, normal and co-normal vector fields, mean curvature and cubic form.

### 3.1 Discrete affine concepts

Notation. For a discrete real or vector function defined on a domain , we denote the discrete partial derivatives with respect to or by

 f1(u+12,v) = f(u+1,v)−f(u,v) f2(u,v+12) = f(u,v+1)−f(u,v).

The second order partial derivatives are defined by

 f11(u,v) = f(u+1,v)−2f(u,v)+f(u−1,v) f22(u,v) = f(u,v+1)−2f(u,v)+f(u,v−1) f12(u,v) = f(u+1,v+1)+f(u,v)−f(u+1,v)−f(u,v+1).

#### Non-degenerate asymptotic nets

The asymptotic net can be described by a vector function , called the affine immersion, such that , , and are co-planar. For each quadrangle, let

 M(u+12,v+12)=[q1(u+12,v),q2(u,v+12),q2(u+1,v+12)].

We say that the asymptotic net is non-degenerate if does not change sign. We shall assume throughout the paper that , for any .

#### Affine metric

The affine metric at a quadrangle is defined as

 Ω(u+12,v+12)=√M(u+12,v+12).

#### Co-normal vector field

The co-normal is defined at each vertex and is orthogonal to the plane containing , , and . The length of is defined by the following lemma:

###### Proposition 2

Consider an initial quadrangle and fix . Then there exist unique scalars such that the following formulas for coincide:

 ν(u,v) = γ−1(u+12,v+12)Ω(u+12,v+12)(q1(u+12,v)×q2(u,v+12)), ν(u,v) = γ(u−12,v+12)Ω(u−12,v+12)(q1(u−12,v)×q2(u,v+12)), ν(u,v) = γ−1(u−12,v−12)Ω(u−12,v−12)(q1(u−12,v)×q2(u,v−12)), ν(u,v) = γ(u+12,v−12)Ω(u+12,v−12)(q1(u+12,v)×q2(u,v−12)).

Moreover, the co-normals satisfy the discrete Lelieuvre’s equations

 ν(u,v)×ν(u+1,v) = q1(u+12,v), ν(u,v)×ν(u,v+1) = −q2(u,v+12).

The products and do not depend on the choice of .

• We begin by fixing an initial value . The coincidence of at determines . Observe that

 ν(u,v)×ν(u+1,v) = 1Ω2(u+12,v+12)(q1(u+12,v)×q2(u,v+12))×(q1(u+12,v)×q2(u+1,v+12)) = q1(u+12,v),

and so coincidences also at . Similarly, define by coincidence of at and formula

 ν(u,v)×ν(u,v+1)=−q2(u,v+12)

implies coincidence of at . Finally define by coincidence of at and the above formulas guarantee coincidence of at and . By repeating this procedure, we can define in all domain satisfying the above formulas.

It is interesting to observe that the above co-normal vector field defines a Moutard net, i.e.,

 γ2(u+12,v+12)(ν(u,v)+ν(u+1,v+1))=ν(u,v+1)+ν(u+1,v)

(see [2]). And, in terms of the co-normals, the affine metric is given as

 Ω(u+12,v+12)=γ−1(u+12,v+12)[ν(u,v),ν(u,v+1),ν(u+1,v)].

#### The normal vector field

The affine normal vector field is defined at each quadrangle by

 ξ(u+12,v+12)=q12(u+12,v+12)Ω(u+12,v+12).

It satisfies the following equations:

 ν(u,v)⋅ξ(u+12,v+12) = γ−1(u+12,v+12), ν(u+1,v)⋅ξ(u+12,v+12) = γ(u+12,v+12), ν(u,v+1)⋅ξ(u+12,v+12) = γ(u+12,v+12), ν(u+1,v+1)⋅ξ(u+12,v+12) = γ−1(u+12,v+12).

#### The cubic form

Define the discrete cubic form as , where

 A(u,v) = [q1(u−12,v),q1(u+12,v),γ(u+12,v+12)ξ(u+12,v+12)], = [q1(u−12,v),q1(u+12,v),γ−1(u+12,v−12)ξ(u+12,v−12)], = [q1(u−12,v),q1(u+12,v),γ−1(u−12,v+12)ξ(u−12,v+12)], = [q1(u−12,v),q1(u+12,v),γ(u−12,v−12)ξ(u−12,v−12)]

and

 B(u,v) = [q2(u,v−12),q2(u,v+12),γ(u+12,v+12)ξ(u+12,v+12)], = [q2(u,v−12),q2(u,v+12),γ−1(u+12,v−12)ξ(u+12,v−12)], = [q2(u,v−12),q2(u,v+12),γ−1(u−12,v+12)ξ(u−12,v+12)], = [q2(u,v−12),q2(u,v+12),γ(u−12,v−12)ξ(u−12,v−12)].

Since we are interested only in the coefficients and of the discrete cubic form, we shall not discuss in this paper the meaning of the symbols and .

#### Mean curvature

We shall define mean curvature at each edge of the asymptotic net. At edges and define

 h(u,v+12) = p(u,v+12)−p−1(u,v+12), h(u+12,v) = p(u+12,v)−p−1(u+12,v).

The mean curvature and are defined as

 H(u,v+12) = h(u,v+12)√Ω(u−12,v+12)Ω(u+12,v+12), H(u+12,v) = h(u+12,v)√Ω(u+12,v−12)Ω(u+12,v+12).

The reason for this definition will be clear later in subsection 4.2, when we obtain the equations for the derivatives of .

### 3.2 The case of minimal surfaces

Examples of asymptotic nets can be obtained as follows: Begin with a smooth -dimensional vector field satisfying and consider a sampling . The discrete immersion is then obtained by integrating the discrete Lelieuvre’s equations.

This type of asymptotic net can be characterized by the constancy of the parameter . More precisely, for these nets we can choose , for any , which implies that the discrete affine mean curvature defined above is zero. This nets were studied in [5], where they were called discrete affine minimal surfaces.

### 3.3 An example with non-vanishing mean curvature

###### Example 3

Since the asymptotic curves the hyperboloid of example 1 are straight lines, sampling it in the domain of asymptotic parameters generate an asymptotic net. Denote by and the distance between samples in and directions, respectively.

Integrating and one can show that

 q1(u+Δu2,v) = csinh(Δu)sinh(u+v)sinh(u+v+Δu)(cosh(2v),−sinh(2v),−1), q2(u,v+Δv2) = csinh(Δv)sinh(u+v)sinh(u+v+Δv)(cosh(2u),sinh(2u),−1).

Denoting by the co-normal vector of the smooth hyperboloid, one can verify that

 ν(u,v)×ν(u+Δu,v) = q1(u+Δu2,v), ν(u,v)×ν(u+Δv,v) = q2(u,v+Δv2).

Thus, since Lelieuvre’s formulas hold, we can consider as the co-normal of the discrete surface as well. Straightforward calculations shows that

 Ω(u+Δu2,v+Δv2)=2c3/2sinh(Δu)sinh(Δv)√sinh(u+v+Δu+Δv)sinh(u+v+Δu)sinh(u+v+Δv)sinh(u+v)

and

 γ(u+Δu2,v+Δv2)=√sinh(u+v+Δu+Δv)sinh(u+v)sinh(u+v+Δu)sinh(u+v+Δv).

The affine normal is given by

 2c1/2ξx = −cosh(Δu)sinh(2v+Δv)−cosh(Δv)sinh(2u+Δu)√sinh(u+v+Δv)sinh(u+v+Δu+Δv)sinh(u+v)sinh(u+v+Δu), 2c1/2ξy = cosh(Δu)cosh(2v+Δv)−cosh(Δv)cosh(2u+Δu)√sinh(u+v+Δv)sinh(u+v+Δu+Δv)sinh(u+v)sinh(u+v+Δu), 2c1/2ξz = sinh(2u+2v+Δu+Δv)√sinh(u+v+Δv)sinh(u+v+Δu+Δv)sinh(u+v)sinh(u+v+Δu).

Also

 p(u,v+Δv2)=√sinh(u+v−Δu)sinh(u+v+Δu+Δv)sinh(u+v−Δu+Δv)sinh(u+v+Δu)

and

 h(u,v+Δv2)=−2sinh(Δv)sinh(Δu)cosh(Δu)√sinh(u+v−Δu)sinh(u+v+Δu)sinh(u+v−Δu+Δv)sinh(u+v+Δu+Δv).

The affine mean curvature is thus

 H(u,v+Δv2)=c−3/2cosh(Δu)(sinh(u+v+Δv)sinh(u+v))1/2(sinh(u+v−Δu)sinh(u+v+Δu)sinh(u+v−Δu+Δv)sinh(u+v+Δu+Δv))1/4.

## 4 Structural Equations

In this section we show the equations relating the affine immersion with the geometric concepts defined in section 3.1, that we shall call structural equations.

### 4.1 Equations for the second derivatives of the parameterization

To obtain equations for the second derivative of the affine immersion, we begin by defining derivatives of the area element of the Berwald-Blaschke metric :

 Ω−1(u,v+12) = p(u,v+12)Ω(u+12,v+12)−Ω(u−12,v+12), Ω+1(u,v+12) = Ω(u+12,v+12)−p(u,v+12)Ω(u−12,v+12), Ω−2(u+12,v) = p(u+12,v)Ω(u+12,v+12)−Ω(u+12,v−12), Ω+2(u+12,v) = Ω(u+12,v+12)−p(u+12,v)Ω(u+12,v−12).
###### Proposition 4

The second derivatives of the affine immersion are given by

 p(u,v+12)q11(u,v) = Ω−1(u,v+12)Ω(u+12,v+12)q1(u+12,v)+γ(u−12,v+12)A(u,v)Ω(u+12,v+12)q2(u,v+12), q11(u,v) = Ω+1(u,v−12)Ω(u+12,v−12)q1(u+12,v)+γ(u+12,v−12)A(u,v)Ω1(u,v−12)q2(u,v−12), q11(u,v) = Ω−1(u,v+12)Ω(u−12,v+12)q1(u−12,v)+γ(u−12,v+12)A(u,v)Ω(u−12,v+12)q2(u,v+12), p(u,v−12)q11(u,v) = Ω+1(u,v−12)Ω(u−12,v−12)q1(u−12,v)+γ(u+12,v−12)A(u,v)Ω(u−12,v−12)q2(u,v−12)

and

 p(u+12,v)q22(u,v) = γ(u+12,v−12)B(u,v)Ω(u+12,v+12)q1(u+12,v)+Ω−2(u+12,v)Ω(u+12,v+12)q2(u,v+12), q22(u,v) = γ(u+12,v−12)B(u,v)Ω(u+12,v−12)q1(u+12,v)+Ω−2(u−12,v)Ω(u+12,v−12)q2(u,v−12), q22(u,v) = γ(u−12,v+12)B(u,v)Ω(u−12,v+12)q1(u−12,v)+Ω+2(u+12,v)Ω(u−12,v+12)q2(u,v+12), p(u−12,v)q22(u,v) = γ(u−12,v+12)B(u,v)Ω(u−12,v−12)q1(u−12,