# Download e-book for kindle: Axiomatic characterization of physical geometry by Heinz-Jurgen Schmidt

By Heinz-Jurgen Schmidt

ISBN-10: 3540097198

ISBN-13: 9783540097198

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Discrete differential geometry is an lively mathematical terrain the place differential geometry and discrete geometry meet and have interaction. It offers discrete equivalents of the geometric notions and strategies of differential geometry, similar to notions of curvature and integrability for polyhedral surfaces.

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3. In t h e ease p E k 2 we hence p ~ mI. c. a substitution from p A k I that is p r o v e d p = ~p A ~ k I = m I, for minimal (~,k1,~1,~,~,mT). 8k 2 = ek2 (= m 2 ) . 2. = pos(llm) £ ~1" The latter by definition By virtue of since ~m 6 R, am = Bm = m. implies (2210) (2216) and pos(~,ml) (iv). By V n 6 M(ml) , pos(l,n) (2212), there exists (T,k2,~2,B,l,m2) , which 2. ~2 ~ <2 is analogously (2223) Lemma: = pos(llml) completes assume = pos(l~m) ~k 2 = Bk 2 follows. a substitution the proof of <2 ~ ~2" proved.

Be satisfied: => p ~ m 2. mI = ~ kI E ~ k 2 = m 2. p is m i n i m a l , p C ~ and p A kI p ~ m I. 1. Consider the case p A k 2. 2. 2. 3. In t h e ease p E k 2 we hence p ~ mI. c. a substitution from p A k I that is p r o v e d p = ~p A ~ k I = m I, for minimal (~,k1,~1,~,~,mT). 8k 2 = ek2 (= m 2 ) . 2. = pos(llm) £ ~1" The latter by definition By virtue of since ~m 6 R, am = Bm = m. implies (2210) (2216) and pos(~,ml) (iv). By V n 6 M(ml) , pos(l,n) (2212), there exists (T,k2,~2,B,l,m2) , which 2.

1. 1. We assume m 6 C(111,ki) and will Clearly, am 6 C(112). Consider the case m ~ k I. H e n c e showam 6 C(112,ki). 2. In 6 C(112,ki). case m A k I assume: 3 h r-am, a k I. It h F" k I, k I A n 2 follows: (see I. 2. that ak I k I in c o n t r a d i c t i o n am A mk I = k I and We~assume First s E C ( 1 1 2 , n 2) that s = at. 1. 2. If s A k I, a s s u m e h 6 C(111) (iii) ~ak and s = mr. s 6 C ( 1 1 2 , k I) 3h such (see r 6 C(111,ki). 6[D I] = D 2 a n d We have a) a n d b) Let to to show are m 6 C(111) pos(111,m) I.