for all a, b,

ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria I. PRACE MATEMATYCZNE XXVI (1986)

An d r z e j Se u z a l e c (Czçstochowa)

Projection in an affine space described in terms of lattices

Introduction. The paper analyses different kinds of projections in an affine space, which has arbitrary dimension n, and are constructed over arbitrary field, not necessarily over the field of real numbers. Since every projection space can be treated as a lattice, it is convenient to describe projections in terms of the lattice theory. The elements of the lattice are then interpreted as linear manifolds, i.e., points, lines, planes, and so on, whilst the lattice operations are seen as intersection and join of these manifolds. In this paper we shall analyse projection systems, where both the projection centre and the projection plane are of arbitrary dimension. Next we shall describe projection and reconstruction of manifolds with respect to these systems. The lattice theory seems most convenient in defining the view and the trace, because in classical geometry the view is the result of joining of a manifold with the projection centre and then intersecting the join with the projection plane.

«-space as lattices. Now we shall define geometric spaces as lattice structures. It is worth noting that this approach is well known, but this point presents a systematic treatment helpful in solving specific problems, i.e., problems of projection. Papers [1], [2], [3] are used as points of reference in defining the notions. We begin with the definition of a lattice.

1. De f in i t i o n. A structure < i f , vj, n , в, 1) is called a lattice if i f is a set, 0, l e i f , and u , n are operations i f 2 -*• i f satisfying the axioms:

a u b = b u a, (a u b)и c = a u ( b u c), a и (a n b) = a,

а и в = a,

a n b — b n a, (ia n b ) n c = a n (bn c), a n (a и b) — a,

a n 1 = a

for all a, b,

In a lattice < if, u , n , 0, 1) the relations c , < are defined as follows:

for every a, be i f

a cr b if and only if a n b — a, or equivalently a cr b if and only if a u b = b, and

a < b if and only if a <=. b and аФ b.

We write b zz> a instead of acz b, and b > a instead of a < b.

2. Definition. A lattice < if, u , n , 0, 1) is called a modular lattice if acz c implies a ^ j( b n c ) = {a u b) n c for every a, b, ce

3. Definition. By n-dimensional metric lattice will be meant a structure

< if, u , n , 0, 1, dim), where < if, u , n , 0, 1> is a lattice and dim is a function dim: { — 1, 0, 1, ..., n} satisfying the following conditions:

dim(0) = — 1, dim(l) = n,

dim (a<u b) + dim (a n b ) = dim (a) + dim (b) for every a, b e i f , if a < b, then dim(a) < dim(b) for every a, be if.

It is well known that

4. Theorem. For every metric lattice (Jr£ , u , n , 0, 1, dim) rbe structure

< if, u , n , 0, 1) is a modular lattice.

Other properties of the lattices can be found, for instance, in [1]. A metric lattice may be considered as an n-dimensional projective space, therefore we can admit

5. Definition. By n-dimensional projective space we mean n-dimensional metric lattice.

The elements of a projective space will be called the manifolds. The function dim assigns to every element its dimension, therefore we shall use the following terminology:

the empty manifold — the element of dimension —1, points — elements of dimension 0,

lines — elements of dimension 1, planes — elements of dimension 2,

hyperplanes — elements of dimension fce{0, 1, ..., n — 1}, the space — the element of dimension n.

This terminology agrees with that used in [1], [2], [3], [5].

In classical geometry one can obtain affine space by choosing one hyperplane in projective space. Now, this may be done in the same manner.

6. Definition. By n-dimensional affine space will be meant the structure

< if, J , u , n , 0, ¹, h, dim), where <Jzf, u , n , 0, ¹, dim) is n-dimen­

sional projective space, be if , dim(b) = n —1, J = { x e S f: x c h], t?

— {xe if : ~ x c: h}.

The element h will be called the improper (n -l)-hy per plane; J is the set o f improper manifolds, & the set of proper manifolds.

By the definitions of a projective space and affine space, all theorems on projective spaces are valid for affine spaces.

7. De f i n i t i o n. The manifolds a,a, . . . , h in a projective space are u -

independent if and only if

dim( (J â ) = £ dim(û) + p — 1.

i = 1 /=1

We give some properties of u-independence of manifolds.

8. Th e o r e m (proof, see [5]). In a projective space the manifolds

a , a , . . . , h are и -independent if and only if for every f , l2, ..., lg

e [ 1 , 2 , . . . , p] such that {lu l2, ..., lk} n {lk + u ..., lg) = 0 M l2

(a и a и ⁴ + 1

u û) n ( a и . . . u a ) = 0.

It is not difficult to prove

9. Th e o r e m. In a projective space, if a ' a 1 for every ie {1, 2, . . . , p — 1},

then d im (a) ^ d im (â )- b p — 1.

1 0. T h e o r e m (proof, see [6 ]). I f a is a proper manifold in an affine space J , и , n , 9, 1, h, dim), then

dim (an h) = dim(a)— 1.

It is not difficult to prove

11. Th e o r e m. In an affine space if a, a, , . . , h are improper and u -

independent manifolds and b is a proper manifold, then

p

dim (h)> £ dim (frnà) + p — 1.

i = 1

Projection. Let us consider two elements s and r of a projective space.

Element s will be called a projection centre and element r a projection plane.

These elements will be called the projection system. We denote the projection system by F = (s, r).

12. De f i n i t i o n. A projection system F = (s, r) fulfilling the condition s u r = 1

w ill be c a lle d semiregular.

13. De f i n i t i o n. A p ro je ctio n system F = (s, r) fu lfillin g the co n d itio n s

s n r = 0, s u r = l

w ill be ca lle d regular.

The definitions above imply that a regular projection is a semiregular one.

We will now give the definition of view of the manifold.

1 4. De f i n i t i o n. The manifold ( a u s ) n r will be called the view of the element a (with respect to F = (s, r)). We denote it by aF. The function g : S?

-> i f defined by the formula g (a) = (a u s)n r is called the projection in F.

1 5. De f i n i t i o n. The manifold a n r will be called the trace of the element a (with respect to F — (s, r ) . We denote it by aF.

1 6. Th e o r e m (proof, see [5]). I f the projection system F = (s, r) is semiregular, then a u s = aF и s.

If we have in n-space several projection systems Fx = (s, r), F2

= (s, r), ..., Fp = (I, f), then the view of a manifold a in FÉ= (s, r), ie (1, 2, ..., p} is denoted by a' and the trace is denoted by at.

Now we give the definition of parallel projection.

1 7. De f i n i t i o n. In an affine space a projection in projection system F = (s, r) will be called a parallel projection if F is semiregular, s is an improper manifold and r is a proper manifold.

This notation of parallel projection is more general than that adopted in elementary geometry.

Reconstruction of manifolds. The purpose of this section is to analyse methods of reconstruction of manifolds from its views. We know that projection from one centre onto one plane is never sufficient. Thus every projection method will be characterized by the system of projection apparatus, i.e., by the system of projection centres and corresponding projection planes. Suppose that in a projective space we have p projection systems F, = (s, r) (i = 1, 2, ...» p). If only p views a1 {i — 1 ,2 , . . . , p) of the manifold a are known, then we only know that the manifold a is included in the manifolds a‘ u s (/ = 1, 2, ..., p). We say that the manifold a is

p j

reconstructed from the views a 1 , a 2, ..., ap w henever a = f] (a1 us).

i= 1

1 8. Th e o r e m. Suppose that in a projective space и , n , 0, 1, dim)

11 2 2 1 2

we have two projection systems F x — (s, r), F 2 = (s, r). I f a n (s и s) = в and

1 2 2 ^.

s n s = 0, then a — f) (flos).

» = l

P ro o f. By Definition 3, we have

(1) dim ((a u s) n (a и s)) + dim((a и s) u (a и I)) = dim(a и s) + dim(u и s).

We know that

1 2 1 2

dim ((a u s ) и (a и s)) = dim ( a u ( s u s))

= dim (a) + dim (s u s ) — dim (a n ( s u s)),

hence

(2) dim ((a u s ) u ( a u s)) = dim (a) + dim (s) + dim (s) 4-1 + 1.

Since a n s <=. a n (s \js) = 0, we have a n s — в and similarly a n s = 0.

Next, by Definition 3, we obtain

(3) dim (a u s) = dim(a) + dim(s)-f 1

and

(4) dim (a u s) = dim (a) 4-dim (I) +1.

2

By (1), (2), (3), and (4) we obtain dim ( 0 (a us)) = dim (a). Hence a = 0 (a u s). ■

i = 1

Theorem 16 implies

19. Theorem. Suppose that in a projective space (<£, u , п , в, 1, dim)

1 1 2 2

we have two semiregular projection systems F l =(s,r), F 2 = (s, r). I f

1 2 1 2

a n ( s u s ) — в and s n s — 0, then the manifold a can be reconstructed from its views a 1, a2.

It is not difficult to prove

20. Theorem. Suppose that in a projective space we have p projection systems Ff = (s, r), i = l , 2 , ...,p . I f there exist projection centres

* 1 * 2 fjij 1 2 p ^ i ^ ;

s , s , ..., s e {s, s, ..., s} such that a — f) (a u s)> then a = f) (a u s)-

i = 11 i = l

If projection systems are semiregular, then we have

21. Theorem. Suppose that in a projective space we have p semiregular projection systems F, = (s, r), i = l , 2 , ...,p . I f a manifold a can be reconstructed from views d y, d z, . . . , a k for some к < p, then it can be reconstructed from a1, a2, ..., ap.

In an affine space we are interested mainly in the reconstruction of proper manifolds.

22. Theorem. Suppose that in an ajfine space we have p projection systems Fi = (s, r), i = 1, 2, ..., p. I f s, s, ..., I are improper and kj-independent manifolds, then for every proper manifold a such that dim (a) ^ p — 2 there exist i, j e {1, 2, ..., p], i Ф j such that a — (a u s) n(a u i).

P ro o f. Assume that

(1) for every i , j e { l , 2, ..., p} if i ф j, then ~ a = ( a u s ) n ( a u i ) .

By the assumption and by Theorem 8:

for every i , j e { 1, 2, ..., p}, if i Ф j, then s n i = в.

Hence, by Theorem 18, and (1) we get

(2) for every i, j e {1, 2, ..., p} if i # j, then a n ( s u è ) Ф в.

Let C„ = U ( 0 a n (s u s)).

i = 1 k( = i + 1

Suppose that there exists me {3,4, ...,p ] such that

(3) Cm- j = Cm

and

(4) m is the greatest number satisfying (3).

Hence we have

i = 1 k, = i + l / = 1 / = 1 kj = i + 1

Hence we get

m- 1 m- 2 m— 1

(5) (J (an (sи s)) c (J ( (J a n (su s')).

i = 1 i = l l; = i + l

Since a r \s a a n (s<j s) for every ie { 1, 2, ..., m — 1}, we obtain

m — 1

(6) a n s cz (J (a n (s u s)).

i= i

Since a n (s u s) c (s u s) for every ie {1, 2, ..., m— 1}, we have

m — 2 m — 1 iç. m — 1

(7)

U ( U

i = 1 lt, = i+l i = 1

m — 1 .

By (5), (6), and (7), we get a n s e (J s, and hence (a n s )n ( (J s) = a n s.

i = 1 Hence it follows that

m - 1 ,

:u '

(8)

m - 1 .

m / i i m

a n s n ( (J s) = a n s.

i = 1

We know that s n ( (J s) = 0; hence, by (8), we get a n 6 = a n s , which i= 1

gives

(9) a n s = в.

By (5) and (7), we obtain

m - 1 .

an(s<js)cz (J s for every k e { l , 2, m — 1}.

i = 1

к m m ~ 1 i k m

Hence we obtain a n ( s u ? ) = ( (J s ) n ( a n ( s u s)), and we get

; = i

. m - 1 .

(10) ü n ( s u s ) = ( 1J s ) n ( s u s ) n a .

i = 1 By (3), we see that

m —1 . m — 1 . m — 1 .

dim(( (J s)n (s u s)) == dim( (J s)+ d im (su s) — dim( (J s),

i = 1 i = 1 i=l

hence it follows that

m — 1 . m — 1 . m

dim(( (J s)n (s u s)) = dim( (J s) — dim( (J s) + dim(s) + d im (s)+ 1

i = 1 1=1 i = 1

= — dim(s) —1+dim(s) + d in i(s)+ 1 = dim(s), but

i m - 1 .

s <= ( (J i ) n ( i u s),

i = 1 so we get

m -1 ,

(11) ( y s ) n ( s u s ) for every ке (1, 2, m — 1}.

i= 1

By (10) and (11), we obtain

(12) a n (s u s) ~ s u a.

Hence, by (2), we have

(13) a n s ^ O for every ke (1, 2, ..., m — 1}.

By (12), (13) and Theorem 11, we have

m -1 m -1

(14) dim(Cm) ^ dim( y a n (s u s)) = dim( y a n s ) ^ m — 2.

i=l i — i

Now we shall analyse all pairs Cn and C„_x where n e { 3, 4 , . . . , p}. We know that C„_i c C„. Hence we have either C„_! = C„ or C„-1 < C„.

By (3) and (4), we have Cm = Cm_ l5 where m e {3,4, ..., p}. Then we have Cm_ j = Cm < ... < Cp_ j < Cp.

Hence, by (14), we have

dim(Cp) ^ dim(Cm) + p — m > m — 2 + p — m — p —2.

(15)

We know that

(16) Cp cza and Cp c (J s .

i = 1

Since s c h for every ie {1, 2, . . p}, we obtain

(17) U*<=ft.

i= 1 By (16) and (17), we get Cp <= a n h .

Hence, by the condition that a is a proper manifold, we obtain a n h

< a, and by (15) we see that dim (a)> p — 2, contrary to the assumption.

If does not exist m fullfilling (3), then

(18) C2 < C 3 < . . . < C p_ l < C p.

Since C2 = a n (s u I), by (2), dim(C2) ^ 0; then (18) implies dim(Cp) ^ p — 2.

Hence, by the condition Cp < a, we get dim (a) > P~ 2, and again we have a contradiction. This completes the proof of the theorem. ■

23. Th e o r e m. Suppose that in an affine space we have p projection systems

F( = (s, r), i = 1, 2, ..., p, p = (“ ), q > 0, s = U s wh^re l/? J f . •••.;?} for i = 1, 2, ..., p are distinct combinations of the set {1, 2, ..., u], s, s, ..., s (!)

1 2 u

are improper and и -independent, and и ^ q+ 1. 77ien /or euery proper manifold

p i

a such that 0 < dim (a) < (u — 2)/q we have a — f) (a vs).

i — 1

P ro o f. We shall prove this theorem by induction on q. By Theorem 22, Theorem 23 holds for q — 1. Let te {1, 2, ..., p) . Assume that Theorem 23 holds for t, i.e.:

If in an affine space we have p projection systems Ft = (s, r), i

= 1, 2, ..., a p = s = U s , where for i = 1, 2, ..., p are distinct combinations of the set {1, 2, ..., u), s, s, ..., s are improper and

1 2 и

u-independent, and u ^ t + 1 , then for every proper manifold a such that 0 ^ dim (a) ^ (u — 2)/t

a = П (a v s).

i= 1

Now we shall prove the theorem for r + 1.

(*) The elements у, s (with subscripts below) do not denote the traces; they denote the manifolds generating projection centres (i.e., elements s, i = 1 , 2 , . . . , p).

(19) Suppose that in an affine space we have p projection systems Ft

= (s, r), i = 1, 2, p, p = ( “ ), s = U s , where ■ ■■,fi + l ) for i = 1, 2, p are distinct combinations of the set (1, 2, и}, s, s, s are improper and u-independent, м ^ f + 2, a is a proper

1 2 и

manifold, and 0 < dim (я) ^ (u — 2)/(t+ 1).

Since 0 ^ dim (a) < (u — 2)/(t + 1), we have either 0 ^ (u —2)/(t+ 1) < 1 or (u - 2 )/(t+ l)> 1. Let (« —2)/(f +1) < 1. Hence

(20)

Using (20), we obtain u —(f +1) —2

и ^ f + 3.

и — t +1 - 1 = и — + 1 + - - ^ - - 1

t 4-1 t 1

ui i — L ) + _ i ^ ( t+ 3 ) ( i _ _ L ) + . 2 t+ 1 7 f+1

= t + 3 - ^ 2 + - ? ~ = t + 3 - 1 t +1 t *+-1

t - f - 1 / t -{-1

2 ■+— r = t + 2.2 t Ч- 1 t -f-1

Hence we have (21)

We know that >

M ; u —(f-t-1) — 2 _ _ _ _ _ _ _ ^m—(r-f-1) —2

Hence we have

N u —{t+ l) — 2

u - { t + l ) - 2 ^ ^ и --- ---1 > t + 2.

t +1

t 4-1 + f0, where 0 < t0 < 1 (2).

(22)

t+ 1

w—(t + 1) —2 _

— 1 = и --- .---1 4- to ^ t 4- 2.

t+ 1

Let s = u - ( u - ( f + l)-2 )/(f4 -1)-1 . Assume that (23) there exist j 1, j 2, ..., j s e {1, 2, ..., и} such that

# 0 j j 2, - j ‘s) ( 3) and a n ( U s ) = 0.

«= i Ji

(2) [x ] is the greatest integer function.

(3) The symbol ¥^{jij2, ■■■Js) means that . . . J s are pairwise distinct.

10 — Prace Matematyczne 26.2

By (22) and (23), we have

(24) there exist j u j 2t ..., j ti j rx, j r2 such that Ф (Д, j 2, ..., Л, Л ,, Л2) and

(25) Let

я n ( (J s u s u s ) — 0.

i=lJi Jrj Jr 2

(26) j i J i i ' - ' J l i ï j ' 2* such that # (/•J ? , j? 1,. /<2), where г = 1, 2, ..., у and у = ^ satisfy (25).

By (3), we have

(27) dim (( (J ^ и s u a) n ( (J ^ u s u a)) = dim( IJ s__ u s u a) + m= 1 Ji л m = l j i ‘ Jl/2 m = 1 .™ Ji

j i

+ dim( (J smu s u a) — dim ( (J ^ u s u s u a)

m=lji Jl/2 m= 1Л jjl j'2

г t t - l

= d im (lj sm u s ) + dim (a)-dim (( (J s u s ) n a) + dim( (J s u s ) +

m= 1 J i J, i . 1 j r l i m = 1 Ji” J,.r 2

t t

+ dim(a) —dim(( [j s u s ) n a ) — dim ( (J s u s u s ) — dim(u) +

p m= 1 / f f . l p

+ dim(( U s u s и s ) n a) for every i e {1, 2, ..., у}.

* = 1 J ? f i 1 J Ï 2

By (19), (23), (24) and (25), we get dim(( !J s и s u a ) n ( [j s u s и a))

“ " * - i jT f t 2

'= '/? fi1

= У dim( s ) + dim( s ) + t + 1 4- У d im (s)-f

_ 1 -m

m = l j , m ~ 1

+ dim( s ) + f+ dim(a) + 1 - ( £ dim( sj) + dim( s ) + dim( s ) + f + 1)— 1

Ji- ^{m = 1 J i} П /1

Ji Ji

~ £ dim( si) + t + dim(a) = £ dim( s^ f t — 1 + dim(a) + 1

m = 1 j i

m- 1 Ji t

dim( U S ) + dim (u)+ 1 = dim( U s u a )

m=i jJ" »=i jT

for every ie { l, 2 , y}. Hence we have (28)

for every ie { l, 2, y}.

t t t

( U smu s u a ) n ( (J ^{Sm u s} u a ) = ( \J ^{sm u} a)

m — 1 Ji

j't -U ! / 2

tfi² f i 2 ^{p .}

By (19), there exist s , s e{s, s, $}, where m = 1, 2, y, such that

(29) s = (J s u s , s = (J s u s . fc= 1 / •/m /1 1 / «/m Jm/2 By (28), we get

гй i /Й2 1

(30) ( a u s ) n ( a u s ) = (J s u a for every me {1, 2, ..., y).

k=l J»

By (19), we get

(31) p m p , + 1

0 ( a u « ) = P (a u (J s ), m= 1 - 1

where + 1}, i — 1, 2, ..., p are distinct combinations of the set (1, 2, m} and p =

By (28), (29), (30) and (31), we get (32)

We have

(33)

By (33), we obtain и —(t + 1) — 2

0 ( a u s ) = 0 (flu U s ) n ...

m= 1 r= 1 n= 1 /

n - ( t + l ) - 2 W---—:--- 1 - 2

t H- 1 m 2

t t + i

и — f + 1

1 w - ( t + 1 ) — 2 ! 0

— 1 — 2 и--- .--- hto- 1—2 t+ 1 ____________

~~ t

u — (t+1) — 2 u --- l — z

t+ 1 u — 2

^ --- -- --- - where 0 ^ f0 < 1.

t t+ 1

Since (и — 2)/(f + 1) ^ dim (a), we get и — (t + 1) — 2 и —

dim (a) ^ t+ 1 ^{- 1 - 2 5} — 2

If we denote (J s by s for every ie { l, 2, y}, where y = ( ),

m= 1 ч /

{ j i i j f t for i = l , 2 , y are distinct combinations of the set {1,2, and s , s , . . . , s are u-independent, then by the inductive

1 2 s

assumption for t we have

a = fl (e u s ).

m — 1

P i

Hence, by (31), we obtain a = f) (au s).

«=1

Assume that (23) is not true. Then we have

(34) for every j\ , j 2, ..., j se {1, 2, ..., u}, if # {jx, j 2, • • •, À),

S

then a n ( (J s) Ф в.

i = 1 J'i

Let gae = (J Ъ where e = 1 , 2 , (U) and \ j f j f , for i

m= 1 yf ' S/

= 1, 2, ..., L J are distinct combinations of the set {1, 2 , _, m}. Let (в)

C ?= ^{m= 1}

Ù

Now we shall consider all pairs and where u — k —n

^ 2 , s — k ^ 1, u — к — 1 — n ^ s. Since cz we obtain either

Си - к - н 1 к h n r . г ' и - к - н ^ / ^ u - l - k - n

s —k '-'s—k ОГ C's—if C j- j,

Assume that

(35) there exist ne{0, 1, u — s} and k e { 0, 1, . s — 1} such that

f i n u — k — n /~>и— l - k - n

'-'s — k — s - k

Let n*, k* satisfy (35)„jfe and s — 1— к* ^ 1. Let g{e~k*~n**

= (J s u s , where e = l , 2 , ...,x , x = ( J, and f°r * = 1, 2, . . . , x are distinct combinations of the set {1, 2, ..., u —1—^{/с* —}и*}. Let = U (a

We know that *= QZj:-"*. Hence by (35),

we obtain C“l l l k*Jun*k*-n*) = C%l£*~k*~"*. Hence we have

Cu - l - k * —n* Г ' и - 1 - k * ~ n * s — 1 — k*,(u — k*— n*) ' - ’s — k*

We get

и- 1 — k *~ n*

U s

m= 1 m

~ л ( и - к * -и * ) /">u- 1 ^{- k * - n*} /^ii- 1 - k * - n * _

a r ^ Ç f i CZ C s _ j _ * * ( „ _ * * _ „ * ) Cl L s _ ^ * C

for every ie Jl, 2, x}, hence it follows that

Hence we obtain 1 -k*

ni = 1 w

и - 1 — k*~ n* s— 1 — k*

(36) a n ( (J s u s ) n ( (J s) = a n ( И s u s )

m = 1 f t « - к * " - * ' V m = 1 V ж = 1 « - к * - » * '

for every ie { l, 2, x}. By (3), we get

s - 1 -Л* u — 1 — Ic*— n*

dim(( U s u s ) n ( (J £»

m= 1 Jj u - k * —n* m= 1

s- 1 -k* и - 1 - k * - n * u — k*— n%

= dim( U « ù s )+dim ( (J s )-d im ( (J s)

m= 1 Jj u — k*—n* m= 1 m= 1

s~ 1 —k*

= dim( (J s u s ) —dim( s ) —1 m= 1 jj u — k * —n* u — k*— n*

s - 1 - k *

= dim( (J s)+ d im ( s ) + l — dim( s ) — 1

m — 1 j j и — к*— n* и - к * — n*

К —— | — tr ♦

= dim( U m = 1 Jj&)■

Hence by (36), we get

s_ i —fc* s — l —k*

(37) a n ( |J s) = a n ( U ^ )

m — 1 j j m= 1 j j u — k * ~ n*

for every j'J j? . j r l ~k*e {1, 2 , u - 1 - k * - n * } and ie { l, 2 ,..., x}.

It is obvious that

(38) for every manifold a, for every k e { 0, 1 ,__ s — 1} and ne [0, 1, ...

..., и — 2} such that u - n ^ s , if i = 1, 2, ...,

are distinct combinations of the set {1, 2, u — k) and a n ( ( J s ) # 0 , then a n ( ( J s ) # 0, where ... J?"*},

m= 1 jj

/ = 1 , 2 , . . . , ^{u — k — n}

s — k

m= 1 Jj

are distinct combinations of the set

» 1, 2, ..., u - k - n \ ,

(39) if a n b = a n (b и c) and a n ( b u c ) Ф 9, then a n b Ф в for all a, 6, c e ^ .

Let n0 and £ be the least numbers satisfying (35)„>fc. Let £ ф 0. Hence we get C“ > C“_1 > ... > C“. Hence, by Theorem 9, we obtain

(40) dim (C“) ^ dim (C*) + и - s.

We have (41) (42)

С" c a, C" c (J s cz h .

f = l ,- By (41) and (42), we get

(43) C“ cz a n h .

We see that

(44) и — s = lu и — (f +1) — 2

—

( u - ( t + 1 )-2 f+ 1

—1 1 = U—U+ u — (t +1) — 2

f+1 + 1

w—( f + 1 ) —2

— f0 + 1 = ---- — :--- HI — f0), where 0 ^ f0 < I-

Since Css, by (39), (40), (43), (44) and Theorem 10, we obtain ,. . . и —(f + 1) —2

dim (a) — 1 $5---— --- + ( l - f 0).

Hence, dim (a) > (w — 2)/(f +1), contrary to the assumption. Let к = 0, hence we have

(45) с “ > . . . > с ! Яо = с! "° \

Assume that

(46) there exist l e N a t u JO] such that for every me{0, ..., /} if there exists nme N a t u J 0 ] such that

f ' u ~ n ~ k m - i - m „ u - n - k m _ j - m - 1

s -*m-l~ m

for every nejn ,,,-!, ..., nm- l -{-nm}1l then there exists fcmeN atu { 0 } such that

( 4 7 ) Г ' и ~ пт ~ к _ r u ~ n m - k - 1

s - k — ^ s - k

for every k e \ k m- 1+m, km- l +m + km], and l is the greatest number satisfying (47).

By (45), k - 1 = 0, n - 1 = 0. By (46) we get

(48) c s“ >

c r 1 > c r 2

с“~”°

— — r^u ~ nQ ~ 2 f u ~ n0 ~ k0 Г ’и ~ к0 ~ п0 ~ 1

! - ' - s - l => •• • =3 = C s _ fcQ

n0 k0 1 ^ /^<u ~ n0k 0 ~ 2 / ^ “ — n0 ~ K0 — ” 1 / ^ u — " 0 * 0

'-s -fc o -l -> , -'s-k0-2 > •.. > l - ' s - k Q -~<u~ n0~k0~ nl 1 — Cs-k0- liu~ n0k0~ ”1 ~ 1

u - r , 0 - k 0 - n i - 1 _ г и - п 0к 0 ~ г ц - 2

s — кq — 2 ^S- kQ ~~ 2

«-( I »m + k mi

c mss°1

s - l - ( I k m) m - 0

1 1 1

«“ ( I > + y - l »-( I ("m + ^rn))-1 «-( I (»m + km))-2

= C ",= 01 1 =) C M=0, >

c m=z°

s - l - ( I Mm=0 s 2 (m£ 0fcm)

s - 2 - ( I i k m)

m= 0

> ...

1

“ < I " m + * W ~ 1

. . . > C W=° ! s- l - (

m

1 1

«-( I (nm + kmï ï ~ n2 ~ 1 «-( I + 1

C W,= 01 ₁ D C

a - l - ( X Л„,)-2

m= 0

m = 0

* -( I *m>~3m= 01

“ ~( I "m + km ) ~ n2 ~ 2

C ⁿⁱ⁼ 0 s~( I *m»~3m= 01

2 2

u-( I («m + *m» «"< I ("m + km))-l

З . . . Г Э С mj° = C mJ°

«-( S k m > ~ 2 *-( I k m ) - 2

m= 0 m= 0

2 2

« - ( I ( " n t + /cm ) ) _ 1 “ - ( I ( " m + k m ) ) “ 2

d c m:° > c > . . .

s-( I fcm)-3m= 0 *-< I fc»)-3_{m = 0}

2 / - 1 ^- i- 1

"~( X (”m + /cm»“ «3 «-< I (nm + k m>i~nl X + 1

> c m:° > . . . > C t ,0 = c MSS°

s~( £ k^j) 3 m= 0

l - 1 s~( X km>~1

m= 0

l- 1

*-( X *m)- 1

m = 0

l-l l-l

X ("m + k m)i~ nl ~ 1 X (nm + km » ~ nl ~ 2

q m= 0 = C m~ 0

s-( I km)-(H-l) m= 0

s-( X km)-(J+l)/- 1

m= 0

...d C w7°

« “ ( X ( n m + k m))

*-( X km) - im= 0

/ /

u ~ ( X (nm k mii ~ 1 u_ ( X ^

C w 7° =>C w7°

»-( X km) - / * -( X кда>-</+1>

m = О m =0

> c

l l

X (nm + fcm»-2 s~< X m= 0 i < > . . . > C ^ Г' m =i 0

s-( I k m) - ( l + 1) s-( I кш)-(/+1)

m =0 m=0

IIM

Since

( н- ( м- и0))+((м- П о - 7 г о « о - f c o - ” i)) + (“ - £ {nm + km) - l ) ~ m= O

( « - Z (nm + km) - n 2)+ ... + ( u - £ (nw + fem) - l ) - ( s - ( /cm) - ( /+ l) )

m= О m=O m=О

I /

= Z nm- l + u - s - £ nm +1 = u - s ,

m= О m= О

by (48) and by Theorem 9, we obtain

« * - ( I km) - ( / + l )

(49) dim (C“) ^ dim С , T 0 + u - s .

» - ( I kw) - ( / + l )

m= 0

By (37), (38), (39), (43), (44), (49) and by Theorem 10, we obtain dim (a)

> (u — 2)/(f+l), contrary to the assumption. If (46) is not true, then we have:

for every /e N atu { 0 } , there exist me{0, 1, . . . , / } and umeN atu { 0 } such that

„ ч - п - к т -1 -m r u - n - k m _ 1 - m- 1

s ~ k m - l ~ m > Lys - k m _ 1 - m

for every ne[nm- u ..., пт^ {+пт], and for every &meN atu { 0 }, there exists k e \km— ! m, ..., km- i + m + km] such that

r^u ~ nm ~ k ^ /~<u ~ nm ~ k ~ *

^ s - k '-'s — k

For 1 = 0, km = 0, and we obtain C“~"° > C“- ”0_1 contrary to (45).

If (35) is not true, then С“Г*- " > C"!*1 -fc~" for every ne {0, 1, . . . , и — s} and к e |0, 1, ..., s -1 } . Hence we get C“ > C“~1 > ... > C*. Hence, by (39), (40), (43) and (44), we get dim (a) > (u — 2)/(f+1), contrary to the assumption.

Let 0 < (u — 2)/(t +1) < 1. Hence dim (a) = 0. Since a is a proper manifold and s cz к for every ie { 1, 2, ..., u}, we get

I

(50) a n { \ J s ) = 9.

i = 1 i

Since и ^ f + 2, by (50), we have

for every j l , j 2, . . . , j t, f 1, f 2e{ 1, 2, ..., u}.

Let (j l , j f , »Ji}* i = 1, 2, ..., p be distinct combinations of the set

'Since (и — 2)/(r + l) ^ (u — 2)/t, dim (а) < (u — 2)/t. By (51), we can use (27), (28), (29), (30), (31) and (32), and by the truth of the theorem for r, we have a

= ^П {avs).

i = 1

In such a way we have proved the theorem for every te{ 1, 2, p}, in particular for t = q. ■

If projection systems in Theorem 23 will be systems of parallel projections, then by Theorem 16, Definition 17 and by the condition of reconstruction of manifolds we have the following theorem:

24. Th e o r e m. Suppose that in an affine space we have p systems of parallel

projections Ft = (s, r), i = l, 2, p, p = ( “), q > 0, s = [j s , where {j\ ■> j f •>'••■> j\) f or / = 1,2, . . . , p are distinct combinations of the set {1, 2, ..., u}, s, s, ..., s are u -independent and и ^ q+l . Then every proper

1 2 и

manifold a such that 0 ^ dim (a) < (w — 2)/q can be reconstructed from views a 1, a2, ..., ap.

Let us consider two-dimensional affine space constructed over arbitrary field. As the projection systems we choose: two proper lines as the projection planes and two improper points as the projection centres. Of course, projection centres must be u-independent. In this case q = 1, и = 2 and by Theorem 24 we get that every proper point of this space can be reconstructed from views.

References

[1] G. B ir k h o ff, Lattice theory, Colloq. Publ. 25, Amer. Math. Soc. Prov. (1967).

[2] M. L. D u b r e il- J a c o t in , L. L e sie u r , R. C r o is o t , Leçons sur la théorie des treillis des structures algébriques ordonnées et des treillis géométriques, Paris 1953.

[3] M. H a ll, Combinational theory, Baisdall Publ. Comp., Waltham, Massachusetts 1967.

[4] E. O t to , Descriptive geometry (in Polish), PWN, Warszawa 1963.

[5] A. S lu z a le c , Some remarks on projection methods, Demonstratio Math. 2 (1984).

[6] —, On perspective in n-space, ZN Geometria (Poznan) 13 (1983), 27-33.