Spaces in Mathematical Forms

CAD systems will have an understanding of the conventional representations of primitive objects, as being defined in terms of coordinates. The notion of a point, residing in 3D space, as being defined by an ordered set of number p = (x,y,z) should be readily apparent. The description of more complex objects in terms of basic functions on coordinates should also be readily apparent.

We may describe a line segment in terms of an ordered pair of points, (p,q), which are may in turn be described by their coordinates as above. It should be recognized that this basic description is not the totality of the behavior of a line segment in space, rather, the end point coordinates form the basis for a function, which describes a set of points on the interior of the line segment (Figure V-1A). The end points of the line segment form one representation of these conditions; there are

alternative descriptions of the conditioning of points on the line segment as well. For example (Figure V-1B), we may equivalently describe this same line segment through a starting point

A

B

Figure V-1: A line as a function on end points

(p), a vector describing a direction in space (q - p), and a range of distances along this vector from the initial point which represent the range of the line segment (0 ≤ t ≤ | q – p|).

These examples rely on a common, unifying coordinate representation of 3-space, where a point in the space may be uniquely described by its ordered triple of coordinates. Two points in this space are equivalent if and only if their coordinates are equivalent:

p = q iff (xp = xq, yp = yq, zp = zq)

While this representation of space suffices for basic Euclidean primitives, such as points, lines, and planes, this representation of a space and the elements that reside in it becomes problematic for more complicated spatial elements. For example, a curve in space may be construed as being comprised of a set of points in space similar to that of a line. However, the description of the curve through an analogous function on its interior points will be more difficult.

Additionally, it is often useful to consider objects from the perspective of other spatial constructs than a homogenous 3-space representation. For example, architectural drawings typically consider the objects of their inquiry in terms of two dimensional representations such as plans and sections. These representations are inherently two dimensional, and may be read as such. The observer of these configurations may vary her frame of reference on these objects, alternating between considering the represented objects as two dimensional objects in the space of the drawing, as well as “reading through” to the three dimensional counterparts represented by these artifacts. A rigorous, formal description of spatial representations which allows translation between these interpretations is of benefit, rather than considering the two dimensional form of the drawing as some sort of emaciated abstraction relative to the “true” nature of the objects in some preferred, integrative 3-D space.

To circumvent the limitations of descriptive capabilities of a common 3-D descriptive space, we will need to turn to more general representations of spaces, which allow multiple spatial representations to co-exist. Such a system should support the development of spatial organizations as independent constructs – allowing the objects within a given spatial representation to be considered, while additionally allowing opportunities for associations

between spatial organizations to be supported. Such a structure will allow individual objects such as curves and surfaces to be considered from a spatial perspective that highlights the invariant qualities of the object, while in turn allowing its consideration relative to other objects that may in turn have spatial structuring in their own terms.

To expand on this notion of mathematical spaces, we must initially disregard some of the innate notions of physical space, and rather consider spaces whose character is simply based on permutations of numerical values. A space, in mathematical terms, is simply an ordered set of variables, e.g. (x₁, x₂, . . . , x_n). A point in this space is equivalent to a specific instantiation of values for these variables, such as (x1= c1, x2= c2, . . . , xn= cn). The specific values as (c₁, c², . . . , cⁿ) are termed the coordinates of the point. A space with this structure is termed a Cartesian product space of n real variables or Rⁿ.

Such mathematical spaces, developed simply as a description of ordered sets of variables, may be naturally combined into higher order constructs. Our definition of a line as defined by two points may be constructed as an ordered set of two R³ Cartesian product spaces – one for each set of the possible coordinates of each end point. The set of all possible line segments is defined in the (R³ × R³) space, instantiated by the specification of six parameters: (x1p= c¹, x2p= c², x3p= c³, x1q= c⁴, x2q= c⁵, x3q= c⁶).

Cartesian product spaces provide much of the structuring required for the description of geometric objects. However, geometric spatial constructs often require the definition of an additional metric on the relationship between locations defined in the space. In and of themselves, Cartesian product spaces do not establish any notion of proximity or nearness between elements defined in the space. While it seems obvious that the point (0,0,0) is nearer to the point (1,0,0) than it is to the point (10,0,0), this is not a property of the simplistic notion of ordered sets of numbers. The establishment of a metric or distance function on members of a Cartesian product space allows notions of nearness, neighborhoods and continuity of spatial elements between coordinates in space to be defined. A metric function is a (symmetric, positive, nondegenerative) function of the form

: ^{n n} R

δ R ×R → (V.1)

that takes two locations (equivalently, points) in the space and returns a real valued distance between them. One obvious such metric function is the Euclidean distance function:

(

) (

) (

)

( , )

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.