Mathematical Representation

Mathematical 3D surface representations play a significant role in the context of indus- trial applications where they are typically used, in connection with CAD software, to build 3D models of some object. Mathematical representations also often provide the foundation for other representations. With respect to the work described in this thesis mathematical representations are utilised in Chapters 3 and 5. In Chapter 3 it was adopted for the purpose of springback calculation (given a before and an after surface), whilst in Chapter 5 it was adopted for the purpose of identifying “critical features” (corners and edges) within a given 3D surface. More detail concerning the particular mathematical representation process used is presented in Section 2.4.4. Mathematical 3D representations can be further divided into two categories: (i) Parametric and (ii) Implicit. Each is discussed in the following two subsections.

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Range data is 2D data where each “pixel” value describes the distance between the points in a 3D scene (object) to a specific point such as a camera. It is sometimes considered to be a special form of 3D data and referred to as “2.5D” data

Parametric Representations

A parametric representation is a mapping from R2(u, v) → R3(x, y, z) where a vector

valued functionf(u, v) comprised of two variablesu and v is applied as follows:

     x y z      =      fx(u, v) fy(u, v) fz(u, v)      (2.10)

whereumin≤u≤umax and vmin≤v≤vmax are surface parameters and x, y, z are the

coordinates of a point located on the surface as shown in Figure 2.12. The domain can be defined more generally when u and v are normalised as (0≤u, v≤1) [30].

The parametric surface representations tend to be simple representations. In [176] it is noted that “parametric surfaces are generally easier [than implicit surfaces] to draw, tessellate, subdivide, and bound, or to perform any operation on that requires a knowledge of where on the surface”. In [133] their simplicity with respect to the modification of object shapes is noted. Due to this simplicity parametric representations have been widely utilised: (i) as a surface construction technique [113] where by curve fitting and smoothing processes are applied to support visualisation [99] and (ii) for inferring new ranges of data points for regions that are either sparsely represented or do not contain any data points.

Given the normal vector~n=ha, b, ci of a pointp0 = (x0, y0, z0), then the line along

the normal n can be defined using a parametric equation which can then be used for further processing (as in the case of the work described in Chapters 3 and 5). However, it is usually not feasible to describe an entire 3D object using parametric mathematical representations because [60, 133]: (i) it is difficult to determine whether the position of a given point is inside or outside a given volume, and (ii) it is difficult to combine multiple “patches” to form a complex shape due to smoothness problems that may arise at patch boundaries. So we can say that the parametric representations are best suited to representing interesting geometrical features (such as lines) but not an entire 3D surface.

Implicit Representations

An implicit 3D surface representation is defined as a mapping R3(x, y, z)→Rgiven by

the following function of three variables:

f(x, y, z) = 0 (2.11)

Given a point (x0, y0, z0), with a normal~n=< a b c >, located on a plane, the implicit

representation of the plane is given by:

f(x, y, z) = 0

a(x−x0) +b(y−y0) +c(z−z0) = 0 ax−ax0+by−by0+cz−cz0 = 0 ax+by+cz=ax0+by0+cz0

Now, letd=ax0+by0+cz0 and thus we have:

ax+by+cz=d

ax+by+cz−d= 0 (2.12)

Equation 2.12 should be satisfied by each point (x, y, z) located on this plane. The coefficients of the implicit surface (a,b,c,d) indicate that the value of function f for the point (x, y, z) gets close to zero. These coefficients can be estimated [198]. The main advantage of implicit surface representations is the ease with which they can be used to indicate the location of a given point with respect to a 3D surface as shown in Equation 2.13. More specifically they are well suited to describing topological information (infor- mation about connected components). This property is used, with respect to the work described in this thesis, to determine springback magnitude (as described in Chapter 3).

(

f(x, y, z)≥0 if the point is on/outside the surface.

f(x, y, z)<0 if the point is inside the surface. (2.13) However, implicit mathematical 3D representations are not well suited to identify- ing points on a surface, although they are well suited to identifying whether particular locations are “inside” (below) or “outside” (above) the surface (this is why the repre- sentation is referred to as being implicit). Implicit representations are thus well suited toraytracing (for more details about raytracing see [71]).

One of the most popular examples of an implicit mathematical representation is the Isosurface representation [143]. An Isosurface (isos in Greek translates to “the same”) is defined as the surface that contains all points that feature the same scalar value in the context of volumetric 3D data [200] (thus a contour in 2D). The Isosurface representation

is employed, for example, in the medical field to represent a region (object) of interest in medical images.

The main disadvantage of implicit mathematical representations is that there is a “trade off” between efficiency (speed) and effectiveness (quality); this is particularly significant in the context of visualisation. The solution is to transform it into some other format that does support visualisation (for example a mesh representation) as described in the following section. In the context of the work described in this thesis an implicit representation is used to identify the point where the extension of a normal ~

n to a given pointp0 = (x0, y0, z0) on an input surface intersects the output surface as

will become clear in Chapter 3.