Asymptotic Nets and Discrete Affine Surfaces with Indefinite Metric
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
nondegenerate asymptotic net. These concepts include discrete
affine area, mean curvature, normal and conormal 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.pucrio.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 nondegeneracy, we define a discrete affine invariant structure on it. The discrete geometric concepts defined are affine metric, normal vector field, conormal 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 onesheet 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 eqüiaffine 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 conormal 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 ”nonminimal” 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 eqüiaffine 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 BerwaldBlashcke 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
The surface is nondegenerate if , and, in this case, the BerwaldBlaschke metric is defined by
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 conormal vector field. It satisfies Lelieuvre’s equations
And the vector field is called the affine normal vector field. One can easily verify that .
The functions
are the coefficients of the affine cubic form (see [8]). We can write
(1)  
(2) 
The function
is called the affine mean curvature. One can write
(3)  
(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:
2.2 Example
Example 1
Consider the asymptotic parameterization of the onesheet hyperboloid ,
Taking derivatives
Straightforward calculations shows that
and
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 nondegenerate asymptotic net: Affine metric, normal and conormal 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
The second order partial derivatives are defined by
Nondegenerate asymptotic nets
The asymptotic net can be described by a vector function , called the affine immersion, such that , , and are coplanar. For each quadrangle, let
We say that the asymptotic net is nondegenerate 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
Conormal vector field
The conormal 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:
Moreover, the conormals satisfy the discrete Lelieuvre’s equations
The products and do not depend on the choice of .

We begin by fixing an initial value . The coincidence of at determines . Observe that
and so coincidences also at . Similarly, define by coincidence of at and formula
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 conormal vector field defines a Moutard net, i.e.,
(see [2]). And, in terms of the conormals, the affine metric is given as
The normal vector field
The affine normal vector field is defined at each quadrangle by
It satisfies the following equations:
The cubic form
Define the discrete cubic form as , where
and
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
The mean curvature and are defined as
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 nonvanishing 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
Denoting by the conormal vector of the smooth hyperboloid, one can verify that
Thus, since Lelieuvre’s formulas hold, we can consider as the conormal of the discrete surface as well. Straightforward calculations shows that
and
The affine normal is given by
Also
and
The affine mean curvature is thus
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 BerwaldBlaschke metric :
Proposition 4
The second derivatives of the affine immersion are given by
and