An n-form f on A can be evaluated, not only at points in A, but at any anchor over A. As a consequence, f can also be differentiated, not only in the directions ofvectors over A, but in the direction ofany anchor over A. Evaluating a form at arbitrary anchors doesn’t lead to confusion. But differentiating a form in the directions of anchors that aren’t vectors leads to a subtlety that is worth discussing.
By the way, these generalized flavors ofevaluation and differentiation became available to us as soon as we homogenized. We started out, in the nested-spaces framework, with a polynomial function f: A → R ofdegree at most n. In converting to the homogenized approach, we linearized the domain space A into ˆA and we homogenized f into the n-form f: ˆA → R. Already at this point, it started making sense to use arbitrary anchors in evaluation, and hence also in differentiation. Thus, the subtlety that this section discusses has nothing to do with the algebra ofsites.
The subtlety involves the naive concept of“all possible derivatives”. In some cases, what this turns out to mean, precisely, is the derivatives in all possible directions that are vectors — but not the derivatives in directions that are anchors ofnonzero weight.
For example, consider the notion of“agreement to kth order”. Two smooth, real-valued functions f and g defined on A are said to agree to
7.11. AGREEMENT TO KTH ORDER 101
kth order at a point P in A when
Dπ1· · ·Dπjf(P) =Dπ1· · ·Dπjg(P),
for allj in [0. .k] and all vectorsπ1 throughπj over A. That is, all derivatives
of f and g oforder at most k agree at P.
Suppose now that f is given by a polynomial ofdegree at most n, and let that same symbol f denote the resulting n-form; and similarly for g, an m-form. The identity above then reduces to the identity
f, π1· · ·πjP(n−j)/(n−j)!=g , π1· · ·πjP(m−j)/(m−j)!.
Might this identity hold with the vectorsπ1throughπj generalized to become
arbitrary anchors?
If n =m, then that generalized identity does hold. A more concise way to phrase the situation is as follows: Two n-forms f and g agree tokth order
at P just when f, s=g , s for all n-sites s that are multiples of Pn−k, as
we essentially saw in Proposition 6.7-2.
If k = 0 and hence j = 0, the generalized identity holds trivially, since there are no parameters πi to remove restrictions from.
But, if n and m are distinct and k ≥ 1, there is no hope. Substitut- ing π1 := · · · := πj := P, we find that we must have f, Pn/(n −j)! =
g , Pm/(m−j)!for allj from 0 tok, and that is possible only ifbothf and
g are zero to kth order at P. Thus, when we require two forms of differing degrees to agree to some order at some point, we must restrict the directions ofdifferentiation (πi) to be vectors.
Suppose that f is a fixed n-form and that we want to determine g to be the unique k-form that agrees with f tokth order atP. How do we construct
that unique g via the paired algebras? We must arrange that
f, π1· · ·πjP(n−j)/(n−j)!=g , π1· · ·πjP(k−j)/(k−j)!,
(7.11-1)
for allj in [0. . k] and all vectorsπ1 throughπj over A. Let (P, ϕ1, . . . , ϕd) be
some Cartesian reference frame for the affined-spaceAthat uses the point P as the center ofits coordinate system. Every k-site over A can be expanded as a linear combination ofmonomials ofthe formPk−jϕα, where j is in [0. .k]
and α = (α1, . . . , αd) is a multi-index with |α|:=α1+· · ·+αd=j. For any
monomial k-site ofthe form Pk−jϕα, Equation 7.11-1 tells us the value that we must assign to g , Pk−jϕα. Assigning arbitrary values to those pairings
determines a unique k-form g, since those monomials form a basis for the space Symk( ˆA) of k-sites. And the k-form g that is so determined will, in fact, satisfy Equation 7.11-1 for all vectors π1 through πj, since each πi is a
linear combination of(ϕ1, . . . , ϕd).
In Section 8.4.2, we shall analyze a differencing algorithm for computing nth derivatives ofan n-form, when that n-form is given to us by its values
102 CHAPTER 7. THE PAIRED-ALGEBRAS FRAMEWORK
at the points ofan evenly n-divided d-simplex. That algorithm is another example where we must restrict the directions ofdifferentiation to be vectors.
Chapter 8
Exploiting the Pairing
We have built the algebra ofsites, in parallel with the algebra offorms; and we have chosen, for each n, a pairing map between n-sites and n-forms. So each dual functional on n-forms is now represented, for us, by an n-site. Symmetrically, each dual functional on n-sites is represented by an n-form. In this chapter, we study several ways in which those representations clarify and simplify CAGD.
8.1The duals of popular monomial bases
Several ofthe most popular bases for the linear space Symn( ˆA∗) of n-forms onA are monomial bases. For example, a power basis is the monomial basis associated with a Cartesian reference frame for A, while a Bernstein basis is a rescaling ofthe monomial basis associated with a barycentric reference frame. In the paired-algebras framework, Proposition 7.4-2 tells us that the duals ofthese popular bases are also rescalings ofmonomial bases.8.1.1
Power-basis forms and Taylor-basis sites
Let A be an affine space offinite dimension d, and let (C, ϕ1, . . . , ϕd) be
a Cartesian reference frame for A. The point C together with the vectors ϕ1 through ϕd form a basis (C, ϕ1, . . . , ϕd) for the linearized space ˆA. Let
(w, u1, . . . , ud) be the dual basis for ˆA∗. The monomials oftotal degree n in
the variables (w, u1, . . . , ud) f orm thepower basis forn-forms onAassociated
with this reference frame. To denote those monomials, let α := (α0, . . . , αd)
be a multi-index with |α|=n, and let α+ denote the dehomogenized multi-
index α+ := (α1, . . . , αd), so that α0 +|α+| = |α| = n. The power basis
consists ofthe n-forms (wα0uα+)
|α|=n.
We now apply Proposition 7.4-2. Since we have adopted the summed pairing, we conclude that the dual basis for n-sites is the rescaled monomial
104 CHAPTER 8. EXPLOITING THE PAIRING
basis (Cα0ϕα+/α! )
|α|=n. We shall refer to this basis as the Taylor basis
for n-sites associated with the reference frame (C, ϕ1, . . . , ϕd), since pairing
an n-form with the n-sites in this Taylor basis (Cα0 ϕα+/α! )|
α|=n precisely
corresponds to expanding that n-form in a Taylor series around C.
Take the case d = 2 and n = 3, for a concrete example. Here, listed on successive lines, are the power-basis cubic forms, the Taylor-basis cubic sites, and the results ofpairing those cubic sites with an arbitrary cubic form f:
w3 w2u w2v wu2 wuv wv2 u3 u2v uv2 v3 C3 6 C2ϕ 2 C2ψ 2 Cϕ2 2 Cϕψ Cψ2 2 ϕ3 6 ϕ2ψ 2 ϕψ2 2 ψ3 6 f(C)Dϕf(C)Dψf(C) (Dϕ)2f(C) 2 DϕDψf(C) (Dψ)2f(C) 2 (Dϕ)3f 6 (Dϕ)2Dψf 2 Dϕ(Dψ)2f 2 (Dψ)3f 6
Note that all three ofthe corner sites in this example represent evaluations. The siteC3/6 =+
C represents evaluation at the center point C, clearly. But
the site ϕ3/6 = +
ϕ also represents evaluation — evaluation at the vector ϕ;
we have f(ϕ) =f, ϕ3/6= (D
ϕ)3f /6.
8.1.2
Bernstein-basis forms and B´ezier-basis sites
Let’s consider Bernstein bases for n-forms next. Let (R0, . . . , Rd) be abarycentric reference frame for the d-dimensional affine space A. And let (r0, . . . , rd) be the basis for ˆA∗ that is dual to the basis (R0, . . . , Rd) f or ˆA.
The Bernstein basis for n-forms on A associated with the reference frame (R0, . . . , Rd) (or with the reference d-simplex [R0, . . . , Rd]) consists ofthe
n-forms nαrα
|α|=n. The multinomial scaling factor
n
α
makes the Bern- stein n-forms a partition of unity; that is, we have
|α|=n n α rα(P) = |α|=n n α r0(P)α0· · ·rd(P)αd = (r0(P) +· · ·+rd(P))n = 1n= 1,
for all points P in A.
What basis is dual to the Bernstein basis? In traditional approaches to CAGD, that dual basis consisted ofcertain dual functionals (ρα)|α|=n. The d+ 1 functionals at the corners, from ρ(n,0,...,0) through ρ(0,...,0,n), were
recognized as being the point evaluations+R0 through+Rd. But the remaining dual functionals were not typically viewed as having any simple form.
The paired-algebras framework lets us represent every one of those dual functionals quite simply, as a monomial in the points (R0, . . . , Rd). By Propo-
sition 7.4-2, the basis dual to the Bernstein basis consists ofthe n-sites (Rα/n! )
8.2. THE DE CASTELJAU ALGORITHM 105