Parallel Covering And Packing Of Plane Polygons With Squares

Let P,Pn be plane convex polygons.We say that the sequence {Pn} can be packed into P if∪ UPn?C P and for arbitrary i,j∈{1,2,...},i≠j,int(Pi)∩int(Pj)=?.We say that {Pn} permits a covering of P if P?∪Pn.In particular,a packing or covering of P with {Pn} is called parallel if there is a side of P such that each Pn has a side parallel to this side of P.The area of a polygon P is denoted by A(P).Let C be a convex polygon and let{Dn} be a collection of the homothetic copies of a convex polygon D.Let f(C,D)be the smallest number such that any collection {Dn} with the total area not less than f(C,D)·A(C)permits a parallel covering of C.Let P(C,D)be the greatest number such that any collection {Dn} of the total area not greater than P(C,D)·A(C)can be parallel packed into C.In Chapter 1,we introduce the research background and the main results of this thesis.Let T(α,β)be an obtuse triangle with base length 1 and with base angles measuringα and β(where β>90°).Denote by the height of T(α,β)by h.Clearly,h=1/cot α+cot β·Let f(T(α,β),S)be f(T(α,β)).In Chapter 2 we consider parallel covering obtuse triangles with squares and obtain the following results:If 0<tan α<(?),then f(T(α,β))=2hcot2α=2cot2α/cot α+cot β:If(?)≤tan α<1 and h≤1,then f(T(α,β))=4h=4/cot α+cot β;If(?)≤tan α<1 and(?)<tan(π-β)≤(?)+1,then f(T(α,β))=2h cot2 α=2 cot2α/cot α+cot β;If(?)≤tan α<1,tan(π-β)>(?)+1,and h>1,then f(T(α,β))=4h cot2α/(1-cotβ)2=4 cot2α/(cot α+cot β)(1-cot β)2;If 1 ≤tan α<(?)and 1<tan(π-β)≤(?)+1,then f(T(α,β)=2h=2/cotα+cotβ;If 1 ≤tan α<(?)and tan(π-β)>(?)+1,then f(T(α,β))=max{2h,4h cot2α/(1-cotβ)2}=max{2/cotα+cotβ,4cot2α/(cot α+cot β)(1-cot β)2};If tan α≥(?),then f(T(α,β))=2h=2/cot α+cot β.In Chapter 3,we consider the parallel covering and packing of parallelogram with squares.