Estudio de confiabilidad del sistema

δ p q = q −p + q −p + q −p ²

ˆ_i

(V.2)

The specification of this Euclidean distance function on a Cartesian product n-space provides the structuring of the familiar Euclidean descriptions of elemental elements such as lines and points. Since this function is true also of simple permutations of real numbers, Euclidean space and Cartesian product n-spaces of independent or orthogonal variables are often used interchangeably. In the following discussion, we will encounter spaces representing parameters on objects for which this simple metric function no longer holds.

B. VECTOR SPACES

Spaces of numbers are expanded by considering spaces of vectors. The notion of a vector may be considered initially to be a primitive object, characterizing the spatial concept of directedness in space. For each variable, (x¹, x², . . . , xⁿ ), we may assign a vector in space, and assign the corresponding coordinate as representing a multiple of this vector. A set of n linearly independent vectors serve as a basis for the space; by linearly independence we can simply state that no vector of the set may be expressed as a combination of any others of the set. In R³ this simply means that the 3 basis vectors may not be co-planar.

Euclidean spaces may be characterized as vector spaces, where an obvious set of basis vectors exists: the unit vectors e , representing a unit length in each of the directions of the axes. It is important to keep this dual nature of Euclidean n space firmly in mind, as an ordered set of n scalar values, and as a vector space characterized by the unit vectors e . In Euclidean spaces, the basis vectors are orthogonal and invariant, in that their magnitude and direction remain constant over the space. The notion of a point in space is equivalent to that of a vector in space, since

ˆ_i ˆ_i e

Figure V-2: Euclidean space basis vectors

we can describe the point p = (x₁, x₂, x₃) as a vector from the origin, defined as the sum of i vectors of length x_i in the direction of : eˆ_i

iˆi

=

∑

p e (V.3)

In representing curved objects, we must draw on spatial constructs whose characteristics are substantially more complicated than those of Euclidean spaces. By way of departure, we may consider affine vector spaces (Figure V-3), characterized by n basis vectors, whose directions are not necessarily orthogonal, and whose magnitude is not necessarily unit.

Euclidean space may be considered to be a special case of affine space.

1. Vector Fields

The notion of a vector field may now be introduced. Consider the application of a

tangent vector v_p at a point p in Rⁿ space: the tail of the vector begins at the point p, while the direction of the vector is described by n ordered coordinates, indicating the direction and distance described by the vector. A tangent vector v_p is thus described by two ordered sets of n variables: the point of application p, and the vector part v.

Figure V-3: Description of a point in an affine vector space

The set of possible vectors with application at p thus may be characterized as a Euclidean space in its own right. We may consider each such vector to be equivalent or dual to a point in this vector space, associated with the point of application p. By extension, we may consider the combination of all points and all tangent vectors to these points in Rⁿ to be Cartesian product space of character Rⁿ × Rⁿ = R²ⁿ , a six dimensional Cartesian product space for tangent vectors in R³. Often, however, in considering deformations of spatial objects, we will be concerned with multiple vectors emanating from a single point of consideration. We may thus consider these vectors from the perspective of a coordinate system with its origin at the common application point of the vectors p. In doing so, we

diverge from the simple framework of a single coordinate system defined over the entire space.

A vector field is a function F: p → v(p)| p ∈ M, that assigns a tangent vector to each point p of some region M of Rⁿ. Specifying a region M allows the construct of a general vector field to be applied only to domains of Rⁿ of interest to a given application, such as curves or surfaces, where the vector field may be undefined for regions of space outside of the domain.

2. Coordinate Fields

With the construct of vector fields in place, the basis is established for defining non-rectilinear coordinate systems in Rⁿ. Vector fields may be combined to construct general coordinate systems on Rⁿ, by establishing an n –part function:

φ i : p → vi (p) | i ∈ (1. . . n ), p ∈ M (V.4)

The vectors vi form a basis of M provided that v_i are linearly independent for each point p inM. Thus any vector whose point of application is p may be resolved into components in terms of the basis vectors v_i. We will limit ourselves to consideration of vector basis functions that are differentiable over M, allowing assumptions such as

continuity and smoothness to be assumed. Figure V-4: a vector field in R³

The notion of a coordinate system varying at every point of application may cause some initial concern, since it seems incongruous with initial notions derived from the usual conventions of coordinate systems on Euclidean n space. For example, we can not directly determine the coordinates of a point p₂ from the coordinate field at p₁ by mapping the vector p₂ – p₁ onto the basis vectors at p₁, the way one could in the natural coordinate system.

However, again, in considering curved objects in space, we will principally be concerned with the local character of the object, “near” a point under consideration. The localized coordinate field will provide a structure for considering this localized character of the object.

3. Frame Fields

The previous section established the notion of a coordinate system defined by basis vectors, continuously varying over a region of space. This coordinate system was established in terms of a set of n linearly independent basis vectors, whose direction and magnitude was dependent on the given point in space. While this construct provides broad generality for coordinate systems, this generality comes at some computational complexity. Even a simple operation as resolving a vector into its components in a general basis requires solution of a set of inter-dependent, linear equations.

With one important constraint on the basis vectors, the complexity of the coordinate system can be dramatically reduced. A frame field is a set of mutually orthogonal, unit length vectors e , applied to every point in a region of Rⁿ. Since these vectors are orthogonal and unit length, the dot product between any pair of these vectors, e • , is 0 unless i=j, in which case the dot product is equal to 1. We may introduce the important short hand notation of the Kroneker delta δ:

ˆ_i

ˆe ˆ

Figure V-5: Expansion of a vector by orthonormal basis vectors.

The components of a vector v in the coordinate system defined by basis vectors are simply the dot product of the vector with the corresponding basis vector: The coordinate of v in the terms of the basis vector e is simply the real value: is then the sum of these n components:

ˆ_i

e eˆ_i

( ˆ ˆ_i)

=

∑

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.