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Mathematical Representations: A Closer Look
Where do the functions – implicit or parametric – come from?
In many simple cases, we just write them down from geometric descriptions. Conic sections, for example, are easy. Given the center and radius of a circle, we can just write down the implicit and parametric representations of the circle. The same is true of simple quadric surfaces like spheres, cylinders, and cones. In a like manner, lines and planes are easily handled.
Even some more complex curves and surfaces – e.g., the torus whose implicit surface equation is degree four and whose parametric representation is rational biquadratic – can also just be "written down" given simple geometric descriptions. Although the effort is somewhat more involved.
Geometric Representations
We somewhat loosely and informally say such curves and surfaces as these have another type of representation: a geometric representation. Implicit and parametric representations of curves and surfaces with well-known geometric representations are easily constructed from these geometric descriptions as illustrated by the examples in the table below.
| curve or surface type |
geometric representation | implicit representation | parametric representation |
| line | B, w | for distinct vectors u and v,
each perpendicular to w: (X - B)·u = 0 and (X - B)·v = 0 |
Q(t) = B + tw; -∞ ≤ t ≤ ∞ |
| circle | C, w, r | (X - C)·w = 0 and (X - C)·(X - C) - r2 = 0 | for mutually perpendicular unit vectors u and v, each also
perpendicular to w: Q(θ) = C + r * [cos(θ)u + sin(θ)v]; -π ≤ θ ≤ π |
| plane | B, w | (X - B)·w = 0 | for distinct vectors u and v, each perpendicular to w: Q(s, t) = B + su + tv; -∞ ≤ s ≤ ∞; -∞ ≤ t ≤ ∞ n(s, t) = w |
| (right circular) cylinder | C, w, r | (X - C)·(X - C) - ((X - C)·w)2 - r2 = 0 | for mutually perpendicular unit vectors u and v,
each also perpendicular to w: Q(θ, t) = C + r * [cos(θ)u + sin(θ)v] + tw; -π ≤ θ ≤ π; -∞ ≤ t ≤ ∞ n(θ, t) = cos(θ)u + sin(θ)v |
Geometric representations are an informal, but convenient, "intermediate representation". As implied in the table above, there is no formal fixed mathematical representational form. Rather they are generally represented as some encoding of: (type, {points...}, {vectors...}, {scalars...}).
Returning to the more formally defined implicit and parametric representations, we are left wondering:
The answer to the first questions is: no. While the details are beyond the scope of the notes here, it turns out that the set of 2D curves (3D surfaces) that have parametric representations is a strict subset of the set of 2D curves (3D surfaces) that have implicit representations.
Moving on to the second question, how do I create representations of so-called freeform curves and surfaces, i.e., those whose shapes are more complicated and not describable using common terms like "circular" or "parabolic"? Should I use parametric forms or should I explore the richer set of curves and surfaces that only have implicit representations? While the set of implicit forms is clearly richer, it turns out that most curve and surface design and representation strategies have been developed for parametric forms. To explore how those representations are created, we need to understand the typical operations designers use because common creation algorithms were developed to directly satisfy those operational needs.
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