Diseño de la tarjeta de control, procesamiento y transmisión de datos

v v e e_i (V.7)

The process of resolving a vector into its components in terms of the frame vectors is termed orthonormal expansion.

Having previously established the notion of a general coordinate field in Section 6, the frame field represents a “step backward” to a less general construct. Note that the “natural”

coordinate system (x,y,z), is a special, additionally constrained frame field whose basis vectors are invariant over the space. Frame fields enjoy some of the simplicity of Euclidean space, but only at local points of application. In investigating the shape of curved spatial objects, we will be establishing appropriate frame fields on these objects that highlight their localized characteristics.

C. MAPPINGS

We will next consider mappings between spatial representations - the means by which disparate spaces and objects expressed in these spaces may be integrated. A mapping may be defined between pairs of spatial representations. A mapping from a space N = (x¹, x²,. . . , xⁿ), to a space M = (y¹, y²,. . . , y^m) is a set of m functions, that expresses the relationships between the coordinates of the two spaces:

(y¹, y²,. . . , y^m) =

(

)

We may consider this mapping both in terms of the individual coordinate mapping functions:

φⁱ ∈ (φ¹. . . φ^m),

or simplify our notation in considering the set of functions to be a single multi-part function, of the form:

φ: N → M (read: φ maps N to M )

Similarly, we should expect that if such a mapping function exists, then an inverse function exists, mapping from the space M back into the space N, of the form:

φ^-1 : M → N (read: the inverse of φ maps M to N )

For the purposes of exploring curved spatial objects, we will presume that these mapping functions are differentiable to the degree appropriate for the required application, i.e. that rates of change in the coordinate variables of the space M can be determined by considering the rates of change in the variables of the space N. A mapping function is infinitely differentiable if each of the coordinate mapping functions φⁱ possesses valid derivates up to infinite order. Typically we will be interested in mappings that present derivatives at least up to some finite order.

In considering the forms of curved spatial objects, the structure presented by these differentials will often have greater significance than the actual mapping functions themselves. While we may of course be interested in the actual placement of objects in space, the geometric nature of these objects are frequently invariant, should the object be subject to some translation in the space. In such an event, the localized rates of change of the mapping function between the parametric and embedding space will be preserved, while the specific functions mapping coordinates between these spatial representations may not.

We will from time to time consider the structure of curved spatial objects in terms of these derivative functions, without explicitly stating the characteristics of the mapping functions themselves.

Mapping between frame fields is a straight forward computation. Given a vector v with components vi = ci , we wish to determine its components vj = dj in some second coordinate system, whose basis are the orthogonal, unit length vectors . The transformation is found by resolving each basis vector into its components in the frame . These components are found by conducting an orthonormal expansion of each vector :

ˆ_i

The j^th component of v in the coordinate frame can then be found by summing the contribution of each of its components in the directions , as they are resolved into the direction off :

The above discussion has used the term space in a fairly unformulated manner. We have seen the term applied to ordered sets of numbers, and alternatively have discussed the concept of vectors spaces. The concept of a manifold allows a rigorous definition of geometric objects and their occupancy in space through mappings between a local coordinate system, intrinsically defined on spatial object and Euclidean in nature, and some extrinsic, containing space. We will refer loosely to the intrinsic space of the object as the parametric or embedded space of the object, and the space into which this object is mapped as the containing or embedding space. The topological characteristics of the object are determined by the orders of each of these spaces. In turn, the shape of the object

in the containing space is largely determined by the mapping function.

Figure V-6: A manifold, defined as a mapping between parametric and containing spaces

A manifold is properly defined as a topological space in which some neighborhood of each point admits a coordinate system. The passage between coordinate systems at neighboring points is smoothly continuous, allowing notions of differentiability of the space and bodies

described in the space. If X is a topological space, a chart at p ∈ X is a function µ : U → R^d, where U is an open set containing p. The concept of a manifold allows notions of elements of a space to be constructed independently of an particular coordinate system, and provides the basis for establishing mapping between coordinate systems. Additional structuring appropriate for modeling of physical systems, such as distance and other spatial metrics may be overlaid on the structure of manifolds, whereas notions of continuity, smoothness, and differentiability are part of the structure of manifolds themselves.

2. Submanifolds and Imbeddings

With the structure of vector spaces and mapping functions between these spaces in place, we have the formal basis necessary to discuss the means by which spaces – and the objects defined in these spaces – can be integrated.

In contrast to the “conventional” spatial structure of simple CAD systems, where the structuring of objects occurs through a single, principal, 3-D Euclidean space, we instead consider a notion of space constructed from multiple manifolds of differing dimension, with equal weight in the total description of spatial objects, integrated by mapping functions between these objects. Given a manifold N of dimension n, we will consider the means by which other spatial constructs may be embedded within this space.

A manifold M is imbedded in N if there is a invertible, differentiable map Φ: M → N, which supports a invertible coordinate function for every point Φ(m) in N. A proper imbedding of M into N requires that the image of M in the space of N not self intersect, since this would prohibit the unambiguous mapping backward of certain points in N. This limitation can be addressed by considering the imbedding to be valid on regions of M that are not mapped to points of overlap in the image.

This section has presented some of the mathematical foundations necessary for describing curved objects in space. In particular, we have moved away from space as being defined by a global coordinate system, and objects in this space as being merely functions on points in this space. Instead, both space and spatial objects are resolved through the concept of manifolds. Manifolds are not simply coordinate functions, but rather objects and sets of objects, spatial, numerical or other. Their structure and relationships are determined by the ability to map these objects to others, through mapping functions on their ranges, and their

variation in these ranges. The imbedding of objects in spaces, or spaces into other spaces, is achieved strictly through notions of imbedding.

The subsequent sections will describe general applications of these principals to two important classes of deformable objects: curves and surfaces.