Definitions: Flock, Herd Space

Proposition 8.1.1: The herd space of a flock F is determined by any three of its functions whose associated points are not collinear.

Pf: Let fP, fQ, and fR be three functions of the herd space of F where P =(xP,yP,zP,0), Q = (xQ,yQ,zQ,0) and R = (xR,yR,zR,0) are non-collinear points of W = 0. For each value of the parameter t (indexing the planes of F), the points (xP,yP,zP,-fP(t)), (xQ,yQ,zQ,-fQ(t)), (xR,yR,zR,-fR(t)) are non-collinear points in by the definition of the herd space functions. Thus, , and hence all the planes of F are determined by these three functions. Given the indexed planes, the functions at all the other points of W = 0 are determined.

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