Jeannin [21] proposed the term “massic vector”.
But none ofthose phrases is adequate. We should dignify the elements ofthe linearization by giving them a single-word name. Here is my proposal: Given any affine space A, let’s refer to an element of its linearization ˆA as an anchor over A. So a point in A is an anchor over A ofweight 1, while a vector over A is an anchor over A ofweight 0. Every anchor is either a vector or a scalar multiple ofa point.
(In defense of the word “anchor”, it connotes a fixed point and something weighty, both ofwhich are appropriate. Indeed, control points in computer drawing systems are sometimes called “anchors”. Also, there is no estab- lished mathematical meaning of“anchor” with which this new sense might be confused. Finally, it is quite convenient that the noun “anchor” has two syllables and ends in “-or”, like “vector” and “tensor”.)
Consider the domain plane A ofa cubic B´ezier triangle F: A → O, as in our recurring example. Every anchor p over the plane A can be written uniquely as a linear combination
p=w(p)C+u(p)ϕ+v(p)ψ, (4.4-1)
where we no longer place any constraint on the weight w(p). Equivalently, every anchor pcan be written uniquely as a linear combination
p=q(p)Q+r(p)R+s(p)S, (4.4-2)
with no constraint on the sum q(p) +r(p) +s(p).
4.5
Coanchors
Now that an element ofthe linearized space ˆAis an anchor overA, an element ofthe dual space ˆA∗ — that is, a linear functional on anchors — is acoanchor onA(nothing to do with a co-anchor ofa television newscast). In particular, the linear functionals q, r, s, w, u, and v are coanchors on A. The weight coanchor is the coanchorw=q+r+sthat satisfiesA=w−1(1). A Cartesian
coordinate system for A, such as (w, u, v), is a basis of ˆA∗ that contains the weight coanchor as one basis element. A barycentric coordinate system, such as (q, r, s), is a basis of ˆA∗ whose coanchors sum to the weight.
Using Cartesian coordinates, every coanchor h on the plane A can be written uniquely as a linear combination ofthe coanchors w, u, andv:
h=C(h)w+ϕ(h)u+ψ(h)v =h(C)w+h(ϕ)u+h(ψ)v =h, Cw+h, ϕu+h, ψv.
38 CHAPTER 4. THE HOMOGENIZED FRAMEWORK
We wrote the right-hand side ofthat equation three times because there is an issue about how to write it. On the first line, we wrote the exact dual of Equation 4.4-1. The coefficients on that first line look strange because we aren’t used to treating an anchor as a function that gets applied to a coanchor as its input datum. We typically prefer to break the symmetry in the opposite direction, treating the coanchor as the function and the anchor as the datum, as on the second line. Ofcourse, the underlying reality is symmetric, as we discussed in Section 2.3: We are really pairing the coanchor with the anchor, however we choose to write it.
In barycentric coordinates, the story is much the same. We can write any coanchorhon the planeA uniquely as a linear combination ofthe coanchors q, r, and s:
h=h, Qq+h, Rr+h, Ss.
The dual ofa coordinate system is a reference frame. For example, the reference frame for the plane A that is dual to the Cartesian coordinate system (w, u, v) consists ofthe center pointC and the unit vectors ϕ and ψ. The three anchors (C, ϕ, ψ) form a basis for the linear space ˆA ofanchors over A, and we have the duality constraints
wu v C ϕ ψ= 1 0 00 1 0 0 0 1 .
In general, a Cartesian reference frame for an affine space is a basis for its linearization that is all vectors, except for a single point.
The reference frame that is dual to the barycentric coordinate system (q, r, s) consists ofthe three points Q, R, and S. Those three points also form a basis for the linearization ˆA, and they satisfy the duality constraints
qr s Q R S= 1 0 00 1 0 0 0 1 .
In general, a barycentric reference frame for an affine space is a basis for its linearization that consists entirely ofpoints.
A comment about notation: We are denoting the fundamental pairing between the linear space ˆA ofanchors and the linear space ˆA∗ ofcoanchors as a function , : ˆA∗ ×Aˆ → R. In particular, given an anchor p and a coanchor h, we shall pair them by writing h, p, with h on the left and p on the right. We adopt that convention for two related reasons. First, when breaking the symmetry, people more often think of the coanchor h as the function and the anchorp as its input datum, and it is convenient to end up with the function on the left. Second, people typically represent an anchor
4.6. THE BENEFITS OF LINEARIZATION 39 (or vector) in coordinates as a column ofnumbers, while they represent a coanchor (or covector) as a row ofnumbers; it is convenient to end up with the row to the left of the column, so that the dot product that effects the pairing follows the standard rules for matrix multiplication. Hence, we prefer to write our pairings with their arguments in the order dual,primal.