EL SUELO URBANO

in some plane in our object space O, and suppose we want to use the arc F([¯a . .¯b]). No matter what design methodology we employ, the parabolaF will be an affine transform of the particular parabola κ2. That is, there will

exist an instancing transformation, an affine map f: Sym₂( ˆL)↓ → O with F(P) =f(κ2(P)) =f(P2), for all pointsP onL. In this way, the particular

parabola κ2 can serve as a prototype for all parabolas.

To add the parabolic arc F([¯a . .¯b]) to our design, it suffices to specify the instancing transformation f. And one simple way to specify the map f is to specify the images under f ofthe three sites ¯a2_{, ¯a}¯_{b, and ¯b}2_{; note that,}

whenever the real numbers a and b are distinct, those three sites constitute an affine frame for the plane Sym₂( ˆL)↓ ofFigure 6.1. The images ofthose three sites under f are, ofcourse, the three B´ezier points ofthe parabolic segment F([¯a . .¯b]). To see this algebraically, suppose that the point P onL is located t ofthe way from ¯a to ¯b, so that P = (1−t)¯a+t¯b. We then have

F(P) = f(κ2(P)) =f(P2)

=f((1−t)¯a+t¯b)2

=f(1−t)2¯a2+ 2t(1−t) ¯a¯b+t2¯b2 = (1−t)2f(¯a2) + 2t(1−t)f(¯a¯b) +t2f(¯b2).

In degenerate cases, we might choose the three B´ezier pointsf(¯a2),f(¯a¯b), andf(¯b2_{) to be collinear, or even choose all three to coincide. The instancing}

transformationf would then collapse the plane ofFigure 6.1 down either to a line or to a point. But this collapsing happens only to our parabolic instance F, not to the prototypical parabola κ2.

6.3

The moment curves

In a similar way, the nth_{-power mapκ}

n: L→Symn( ˆL)↓ defined by κn(P) :=

Pn provides a prototype for all polynomial curves of degree at most n. For example, when n= 3, we have ¯t3 _{= (C+tϕ)}3 _=C3_{+3t C}2_ϕ+3t2_Cϕ2_+t3_ϕ3_,

so our prototypical cubic is the twisted cubic curve (x, y, z) := (3t,3t2_{, t}3_),

sitting in the affine 3-space

Sym₃( ˆL)↓ =R3[C, ϕ]↓ ={C3+x C2ϕ+y Cϕ2+z ϕ3 |x, y, z∈R}.

In projective geometry, the curve that results from the analogous construction is called the rational normal curve of degree n. Since we are working in an affine space, instead ofin its projective closure, we’ll refer to κ_n by its other name: the moment curve of degree n.

The tangent lines, osculating planes, and so forth of the moment curve κ_n are related to the multiplication in the algebra ofsites as follows.

62 CHAPTER 6. THE VERONESE PROTOTYPES

Proposition 6.3-1 Let κ_n: L→Sym_n( ˆL)↓ be the moment curve of degree

n given by κ_n(P) :=Pn_{, for all points P on the affine line L. A unit-weight}

n-sites overLlies in the affine k-flat that osculates the curve κ_ntokthorder atPn _{just when the (n−k)-site P}n−k _{divides s.}

Proof As the real number h tends to 0, we have t+hn= (¯t+hϕ)n= 0≤i≤n n i ¯ tn−ihiϕi = 0≤i≤k n i ¯ tn−ihiϕi+O(hk+1) = ¯tn−k 0≤i≤k n i ¯ tk−ihiϕi+O(hk+1).

Thus, the moment curveκ_nis approximated to kth order, near then-site ¯tn, by thek-flat that consists ofall multiples of¯tn−k_.

For example, consider the 3-site ¯a¯b2_{. The one factor of ¯a puts us in the}

osculating plane to the twisted cubic κ3 at ¯a3, while the two factors of ¯b put

us in the osculating plane at ¯b3 twice, that is, on the tangent line at ¯b3. So the 3-site ¯a¯b2 _{sits where that tangent line cuts that osculating plane.}

The moment curve κ_n can serve as a prototype for all n-ic polynomial parametric curves. Given any such curve F: L→ O, sitting in some object spaceO, there exists a unique affine map f: Sym_n( ˆL)↓ →O that realizes F as an instance ofthe prototypeκ_n, that is, that satisfies F(P) = f(κ_n(P)) = f(Pn), for all pointsP onL. Given some parameter interval [¯a . .¯b] onL, one convenient way to determine whichn-ic segment F([¯a . .¯b]) we want in some design of ours is to specify the instancing transformationf by specifying the images under f ofthe n-sites ¯an, ¯an−1¯b, through ¯bn, those images being the B´ezier points ofthe segment F([¯a . .¯b]).

Note that the instancing transformation f may well fail to be injective. Indeed, the prototypical cubic κ3 is twisted, spanning the affine 3-space

Sym₃( ˆL)↓. So, the instancing transformation for any planar cubic segment will definitely fail to be injective; its four B´ezier points will be coplanar. When the instancing transformation f fails to be injective in this way, the differential geometry in the object space gets affected. For example, all of the osculating planes ofa planar cubic coincide, so we can’t construct the point f(¯a¯b¯c) geometrically by intersecting the osculating planes to F at the parameter values ¯a, ¯b, and ¯c. But the differential geometry ofthe prototype is not affected. We can still intersect the osculating planes to κ3 at ¯a3, ¯b3,

and ¯c3 to find the 3-site ¯a¯b¯cand then apply the instancing transformationf. The resulting pointf(¯a¯b¯c) is the blossom value ˜F(¯a,¯b,c), as we discuss next.¯