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22, it is easily seen that da,b(O, p) -- (Xp -- yp)a + ypb K;- ...................... /(b) t Ye ~. e. for all p such that Xp - K and 0 < yp < K). 23. e. a, b) met by the line ( x - K) are indicated between brackets. 23 (b) f f (E~,b)'= 0 K The graph of Ea, b in the first octant CHAPTER 1: DIGITAL TOPOLOGY 29 Since Ea,b is a convex function on [0, K], its local extreme can be obtained atpsuchthat OE~b o y' (O , p) -- O. Now, for a l l p s u c h t h a t x p I OEa,b ( O , p ) -- K and O _< yp < _ K, - ( ( b _ a ) _ ( ( K - yp)a + ypb)yp ) 1" " In t h a t case ~ Ea,b(O,p) -- v/a2+(b-a)2 e m a x - max{[Ea,b (O,p)l.

28(A), R - 1 0 a v / 2 - 30x/~ and q = ql. By definition of R, AE(p, R) is the minimal Euclidean disc that contains all the points in Aa,b(p,D) (Aa,b(p,D) C AE(p,R)). e. A - A E ( p , R ) - Aa,b(p,D) # 0). 28). 27 where D - 120, R - 1 0 x / ~ and r - (40, 10). After rotation, dE(p, q') may not (and generally does not) represent the radius R' of the minimal Euclidean disc that contains the discrete disc Aa,b(P, D') with D' - da,b(P,q'). 28(C). 01. The point r' = (33,25) is no longer included in A' - AE(p, R') - Aa,b(p,D') and it is easy to verify that d3,4(p, q') > d3,4(p, r') and dE(p, q') > dE(p, r').

15 is straightforward. Now, given p, q and r, three points on the triangular lattice, if q is included in the CHAPTER 1" DIGITAL TOPOLOGY 15 shortest 6-arc from p to r, then, d6(p, r) = d6(p, q ) + d6(q, r). Assume that point q is not included in the shortest 6-arc from p to r. If d6(p, q)+ d6(q, r) < d6(p, r) then, there exists a 6-arc which contains q and such that its length is smaller than d6(p, r). By the definition of d6(p, r), this is impossible. Therefore, d6 satisfies the triangular inequality.