Definitions: Flock, Herd Space, Star Flock

Theorem 8.1.1: If two functions in the herd space of a flock F are scalar multiples, then F is a star flock.

Pf: Suppose that fP' = kfP for distinct points P and P'. We can assume that these are not constant functions (k0) and that any functions which appear in this proof are not constant by Proposition 8.1.3 . Let T be any point of W = 0 which is not on the line PP'. Choose points Q on PT, and R on P'T which are distinct from P, P' or T (this is always possible since q2). If any of fQ, fR or fT is a scalar multiple of fP, then F is a linear flock by Proposition 8.1.6 . By Proposition 8.1.7 , there exist scalars a and b so that fT = afP + bfQ, and scalars c and d so that fT = cfP' + dfR = ckfP + dfR. Since fT is not the constant function, nor a scalar multiple of fP, b and d are not zero. Equating the two expressions for fT gives, (a-ck)fP + bfQ - dfR = 0, and so, fP, fQ and fR are linearly dependent functions associated to non-collinear points. That F is a star flock now follows immediately from Proposition 8.1.5 .