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Topology and differential geometry

Miklós Hoffmann

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Chapter 9. Foundations of differential geometry of surfaces

Chapter 9. Foundations of differential geometry of surfaces

Similarly to the discussion of curves we shall begin our study of surfaces by the definition of what we mean by surface in terms of differential geometry. This definition is - analogously to the definition of curves - a kind of restriction of the everyday concept of surface, but even this notion can describe most of the important surfaces, including those we use in geometric modeling.

Definition 9.1. By regular surface we mean a vector function defined over a simply connected (open) parameter domain, where the endpoints of the representatives of vectors starting from the origin form the surface in , if a) the function defines a topological mapping b) is continuously differentiable in both parameters c) vectors and are not parallel in any point .

Figure 9.1. The definition of regular surfaces

The function uniquely defines the geometric surface but not vice versa - a surface can have several representations, or even parametric representations which do not fulfill the above mentioned criteria. Those representations, which fulfills the criteria mentioned in the definition, are called regular representations. A technical note: partial derivatives of a vector valued function is computed analogously to the one-parameter case, by separately differentiating the coordinate-functions with respect to the actual parameter.

The topological mapping over the parameter domain can be performed in the simplest way by an orthogonal projection. This way we obtain a domain in the parameter plane . Consider a topological mapping of this domain to another domain in this plane. The relation between the surface and cannot be simply described by a projection any more of course, however this can also yield a regular representation of the surface.

Regular surfaces form such a small subset of surfaces that well-known surfaces such as sphere or torus are not in this subset, because cannot be mapped onto a simply connected open subset of the plane. These surfaces however, can be constructed by a union of a finite number of regular surfaces, e.g. the sphere can be a union of two, sufficiently large hemisphere with the two poles as centers, overshooting the equator a bit. Analogous solution can be found for other, non-regular surfaces as well. Thus we can define the notion of surface as union of finite number of regular surfaces and for any point of the surface has a sufficiently small neighborhood, which is a regular surface piece on the surface. A surface is connected if any two points of the surface can be connected by a regular curve on the surface.

Various algebraic representations of surfaces

  1. Explicit representation. Consider a Cartesian coordinate-system in and an explicit function with two variables: . Those points the coordinates of which are form a surface. This representation is also called Euler-Monge-type form.

  2. Implicit representation.Consider again a Cartesian coordinate-system in and an implicit function with three variables: . Those points the coordinates of which satisfy the equation for some constant, form a surface called slice surface. Typically, we are dealing with surfaces with equation .

  3. Parametric representation.This form is the one we defined at the beginning of the section as a regular surface representation: the function , which can be described for practical computations by three coordinate functions:

    This form is also called as Gauss-type representation.

Now we will focus on the possibility of transformation of the surface from one representation type to another one.

1)2)

in this case one can easily transform the equation to implicit representation.

2)1)

in this case the following theorem shows the possibilities. Suppose that holds in a point , and is defined on a small neighborhood of and holds. Under these conditions there exists a sufficiently small neighborhood of the point where there is one and only one function which satisfies the equation and for which holds.

3)1)

in this case we show, that for any point of the surface there is a sufficiently small neighborhood, in which the surface can be represented in one of the forms , or . Based on the assumptions in the definition, the vectors and cannot be parallel, or equivalently

Due to the rank of the matrix, in the point the matrix has a non-vanishing minor matrix. For example let this matrix be

Due to the continuity of partial derivatives is non-vanishing in a neighborhood of , let this neighborhood be denoted by . Thus the system of functions which maps to another neighborhood is as follows

and in it has an inverse system

Substituting these functions to we have

where the right side of the equation depends only on and . Let us denote it by . This function generates the same points over the domain as the function over the domain .

There are several different possibilities to express the surface in Gauss-type representation, that is a representation uniquely determines the surface but not vice versa. Let be a surface over the domain and consider a pair of continuously differentiable functions over

which generates a 1-1 mapping between domains and and where

over the whole domain . Then the above described pair of functions has an inverse system

which is continuously differentiable as well. Substituting it to , the function generates the same points as . This change of parameters is called admissible change of parametrization or simply admissible reparametrization.

Example 9.2. Consider the plane passing through the point which is determined by the linearly independent (i.e. non parallel) vectors and . The representation of this plane, based on the relation is as follows

Example 9.3. The implicit equation of the sphere with radius and origin as center is:

The same sphere can be represented in explicit form by a pair of equations:

where the first equation describes the hemisphere above the plane while the second one describes the hemisphere below the plane . For parametric equation let us choose a point and the surface is generated by endpoints of the representations of vectors starting from the origin. Project the point onto the plane , let the image point be . Let the parameter be the angle of the axis and the vector , while the parameter be the angle of the axis and the vector . This way, the parametric coordinate functions are as

The Gauss-type representation is

If and , then the representation describes the whole sphere, while if for example and , then we get the lune between the positive axes.

Example 9.4. The Gauss-type representation of a hyperbolic paraboloid can be formulated by the coordinate functions as

Thus and parameters can be chosen from the entire parameter plane.