4.9 Linearization revisited
5.1.2 Fixing a basis
Given a linear space X ofdimension k, the simplest way to construct the symmetric algebra Sym(X) is to fix a basis (ξ1, . . . , ξk) f or X and then to
construct Sym(X) as the algebra R[ξ1, . . . , ξk] ofall polynomials in those k
basis elements, treated as variables. Using this approach, we can construct the algebra offorms Sym( ˆA∗) asR[w, u, v], and we can construct the algebra ofsites Sym( ˆA) as R[C, ϕ, ψ]. While this approach is delightfully simple, it might seem to unfairly favor the fixed basis.
5.1.3
Defining an equivalence relation
We would prefer to use different bases at different times and, even better, to use multiple bases simultaneously. For example, we would like any coanchor polynomial to denote a form, even if its variables don’t all come from any single basis for the space ˆA∗ ofcoanchors. Once we allow coanchors that are linearly dependent, however, distinct polynomials may denote the same form. For example, the linear dependenceq+r+s=wtells us that the two polynomials q+r+s and w denote the same 1-form — to wit, the weight coanchor. It follows that the quadratic polynomialsqu+ru+su= (q+r+s)u and wu must denote the same 2-form. Thus, we can think of a form as an
5.1. THE ALGEBRA OF SITES 53 space of n-sites space of n-forms
Symn( ˆA) Symn( ˆA∗) Rn[C, ϕ, ψ] Rn[w, u, v] Rn[ ˆA]/≈Aˆ Rn[ ˆA∗]/≈Aˆ∗ Polyn( ˆA∗,R) Polyn( ˆA,R)
Table 5.2: Formulas for the spaces of n-sites and n-forms
equivalence class ofcoanchor polynomials.
Given two coanchor polynomials, elements ofthe huge algebra R[ ˆA∗], how do we test whether they are equivalent? Answer: We rewrite all ofthe coanchors in both ofthem as linear combinations ofw, u, and v and check whether the rewritten polynomials coincide. Ofcourse, there is nothing special about the basis (w, u, v); we can adopt any basis for ˆA∗ in performing this equivalence test without affecting the result.
The same goes for sites. We would like any anchor polynomial inR[ ˆA] to denote a site, even if its anchors don’t all come from any single basis for ˆA. But once we allow anchors that are linearly dependent, distinct polynomials may denote the same site. For example, let E := (Q +R +S)/3 be the centroid ofthe reference triangle QRS. The linear polynomials 3E and Q+R+S denote the same 1-site — an anchor ofweight 3. It follows that the quadratic polynomials 3Eψ andQψ+Rψ+Sψ = (Q+R+S)ψ denote the same 2-site. We can think ofa site as an equivalence class ofanchor polynomials, where two such polynomials are equivalent when rewriting all ofthe anchors in both ofthem as linear combinations ofthe anchors in a common basis for ˆA would make them coincide; and which common basis we adopt in this test ofequivalence doesn’t matter.
5.1.4
Exploiting duality
One thing that you can do with a coanchor polynomial is to define a real- valued function on anchors; for example, the coanchor polynomial wu de- fines the function that takes an anchor p to the real number w(p)u(p) =
w, pu, p. As it happens, we are quite interested in real-valued functions on anchors, since we intend to use three ofthem to define thex,y, andz coor- dinates ofour B´ezier triangle. Note that the coanchor polynomialqu+ru+su defines the same real-valued function as doeswu; so each real-valued function actually arises from some equivalence class of coanchor polynomials. In fact, these equivalence classes are the same ones that we introduced in the third
54 CHAPTER 5. THE SEPARATE-ALGEBRAS FRAMEWORK
row ofTable 5.1. Thus, we can think ofa form either as an equivalence class ofcoanchor polynomials or as the common real-valued function on anchors that any one ofthose equivalent polynomials defines.
The same goes for sites, except that CAGD doesn’t give us any particu- lar reason to be interested in the resulting real-valued functions. An anchor polynomial defines a real-valued function on coanchors; for example, the an- chor polynomial 3Eψ defines the function that takes a coanchorhto the real number 3E(h)ψ(h) = 3h(E)h(ψ) = 3h, Eh, ψ. Two anchor polynomials define the same real-valued function just when they are equivalent, in the sense ofthe third row ofTable 5.1. For example, the equivalent polynomials 3Eψ and Qψ+Rψ+Sψ define the same real-valued function. Thus, we can think ofa site either as an equivalence class ofanchor polynomials or as the common real-valued function on coanchors that any one of those equivalent polynomials defines.
Mathematically, forms and sites are completely symmetric; but their ap- plications to CAGD are not. Since CAGD gives us good uses for real-valued functions on anchors, the definition of forms in the fourth row seems more attractive than the one in the third row. Indeed, when we first defined forms in Chapter 4, we talked only about real-valued functions on anchors, leaving implicit the equivalence relation on coanchor polynomials. But CAGD does not give us similarly good uses for real-valued functions on coanchors. Hence, in defining sites, the third row seems more attractive than the fourth. Math remark: Why does it work to algebrize by exploiting duality, that is, to construct Sym(X) as Poly(X∗,R)? It works because, over the real numbers, the coefficients ofa polynomial are uniquely determined by that polynomial’s values. The same works over any infinite field, even infinite fields ofprime characteristic. But not over finite fields. Let p be a prime. Over the Galois field oforder pk, the two polynomialsξpk and ξ have all ofthe same values;
thus, two distinct elements ofSym(X) would be indistinguishable as elements ofPoly(X∗,R). For more on this, see Section C.2.3.