arXiv:math/0506386v1_[math.GM] 20 Jun 2005

Parallel Bundles in Planar Map Geometries

Linfan Mao

Institute of Systems Science of Academy of Mathematics and Systems Chinese Academy of Sciences, Beijing 100080, P.R.China

E-mail: maolipfan@163.com

Abstract: Parallel lines are very important objects in Euclid plane geometry and its behaviors can be gotten by one’s intuition. But in a planar map geometry, a kind of the Smarandache geometries, the situation is complex since it may contains elliptic or hyperbolic points. This paper concentrates on the behaviors of parallel bundles in planar map geometries, a generalization of parallel lines in plane geometry and obtains characteristics for parallel bundles. Key Words: parallel bundle, planar map, Smarandache geometry, map geometry, classification.

AMS(2000): 05C15, 20H15, 51D99, 51M05

1 Introduction

A map is a connected topological graph cellularly embedded in a surface. On the past century, many works are concentrated on to find the combinatorial properties of maps, such as to determine whether exists a particularly embedding on a surface ([7|[11]) or to enumerate a family of maps ([6]). All these works are on the side of algebra, not the object itself, i.e., geometry. For the later, more attentions are given to its element’s behaviors, such as, the line, angle, area, curvature, ---, see also [12] and [14]. For returning to its original face, the conception of map geometries is introduced in [10]. It is proved in [10] that the map geometries are nice model of the Smarandache geometries. They are also a new kind of intrinsic geometry of surfaces ([1]). The main purpose of this paper is to determine the behaviors of parallel bundles in planar geometries, a generalization of parallel lines in the Euclid plane geometry.

An axiom is said Smarandachely denied if the axiom behaves in at least two different ways within the same space, i.e., validated and invalided, or only invalided but in multiple distinct ways.

A Smarandache geometry is a geometry which has at least one Smarandachely denied axiom(1969)([5][13]).

In [3][4], Iseri presented a nice model of the Smarandache geometries, called s-manifolds by using equilateral triangles, which is defined as follows([3][5][9]):

An s-manifold is any collection C(T, n) of these equilateral triangular disks T;, 1 < i < n satisfying the following conditions:

(i) Each edge e is the identification of at most two edges e;,e; in two distinct triangular disks T;, Tj}, 1 < i,j < n andi Æj;

(ii) Each vertex v is the identification of one vertex in each of five, six or seven distinct triangular disks. The conception of map geometries without boundary is defined as follows ({10]).

Definition 1.1 For a given combinatorial map M, associates a real number pu(u),0 < ulu) < r, to each vertex u,u E€ V(M). Call (M, u) a map geometry without bound- ary, (u) the angle factor of the vertex u and to be orientablle or non-orientable if M is orientable or not.

In [10], it has proved that map geometries are the Smarandache geometries. The realization of each vertex u,u € V(M) in R? space is shown in the Fig.1 for each case of p(u)u(u) > 2r, = 2r or < 2r, call elliptic point, euclidean point and hyperbolic point, respectively.

plu)u(u) < 2r plu)u(u)} = 27 pluju(u) > 27 Fig.1

Therefore, a line passes through an elliptic vertex, an euclidean vertex or a hyperbolic vertex u has angle paeta at the vertex u. It is not 180° if the vertex u is elliptic or hyperbolic. Then what is the angle of a line passes through a point on an edge of a map? It is 180°? Since we wish the change of angles on an edge is smooth, the answer is not. For the Smarandache geometries, the parallel lines in them are need to be given more attention. We have the following definition.

Definition 1.2 A family L of infinite lines not intersecting each other in a planar geometry is called a parallel bundle.

In the Fig.2, we present all cases of parallel bundles passing through an edge in planar geometries, where, (a) is the case of points u, v are same type with p(w) u(u) = p(v)u(v), (b) and (c) the cases of same types with p(u)u(u) > p(v)u(v) and (d) the case of u is elliptic and v hyperbolic.

Fig.2 Here, we assume the angle at the intersection point is in clockwise, that is, a line passing through an elliptic point will bend up and a hyperbolic point will bend down, such as the cases (b),(c) in the Fig.2. For a vector O on the Euclid plane, call it an orientation. We classify parallel bundles in planar map geometries along an orientation O.

2. A condition for parallel bundles

We investigate the behaviors of parallel bundles in the planar map geometries. For this object, we define a function f(x) of angles on an edge of a planar map as follows.

Definition 2.1 Denote by f(x) the angle function of a line L passing through an edge uv at the point of distance x to u on the edge uv.

Then we get the following result.

Proposition 2.1 A family L of parallel lines passing through an edge uv is a parallel bundle aff

F) 59.

dx|\, ~

Proof If £ is a parallel bundle, then any two lines Lı, Lo will not intersect after them passing through the edge uv. Therefore, if 01,02 are the angles of Lı, Lo at the intersect points of Lı, Lo with wv and L is far from u than Lz, then we know that 02 > 01. Whence, for any point with x distance from u and Az > 0, we have that

f(a+ Az) — f(x) > 0. Therefore, we get that P| _ m fet Ae) —~ Ke), 9 dz], Ar—+0 Az

As the cases in the Fig.1.

Now if £ y > 0, then f(y) > f(x) if y > zx. Since £ is a family of parallel lines before meeting uv, whence, any two lines in £ will not intersect each other after them passing through uv. Therefore, £ is a parallel bundle. h

A general condition for a family of parallel lines passing through a cut of a planar map being a parallel bundle is the following.

Proposition 2.2 Let (M, p) be a planar map geometry, C = {u101, Uzv,- +, utr} a cut of the map M with order uiv, Ugv2,--+, uv, from the left to the right, l > 1 and the angle functions on them are fı, fo,---, fi, respectively, also see the Fig.3.

Fig.3 Then a family L of parallel lines passing through C is a parallel bundle iff for any z,x > 0,

fi(x) = 0 fi, (2) + fo, (2) = 0 fiz(2) + fo4(@) + fa, (2) 2 0

Filipe falt) +t f(a) 0,

Proof According to the Proposition 2.1, see the following Fig.4,

Fig.4

we know that any lines will not intersect after them passing through u,v, and ugv2 iff for VAx >Oandx>0,

fo(a + Ax) + fi, (x)Ax > falx). That is,

fi, (2) + f8,.(2) > 0.

Similarly, any lines will not intersect after them passing through w,v1, u2v2 and u3zv3 iff for VAx > 0 and z > 0,

fa(a + Ax) + fi (£)Ac + fi (£) Ac > f(z). That is,

FEF E f34(z) > 0.

Generally, any lines will not intersect after them passing through u,v, U2ve, +- , Wj—1Uj-1 and uv, iff for VAr > 0 and x > 0,

fila+ Ax) + filt) Ar +: fi, @)Az > filz). Whence, we get that

fOr hia ter+ 7@) 2 0.

Therefore, a family £ of parallel lines passing through C is a parallel bundle iff for any x,x > 0, we have that

filz) = 0 AAO) + fil) 20 fis (2) + far (2) + fa4(a) = 0

fix (@) + far (2) + +++ + fix (@) 2 0. This completes the proof. h.

Corollary 2.1 Let (M, u) be a planar map geometry, C = {uiv1, U2V2,:--, ur} a cut of the map M with order uiv, U2v2,--+, uu from the left to the right, | > 1 and the angle functions on them are fi, fo,---, fi. Then a family L of parallel lines passing through C is still parallel lines after them leaving C iff for any x, x > 0,

>0 fig) h(a) 20 fiz(2) + fo4(2) + f34(x) = 0 fir (2) + fa @) +--+ fiapla) 20

fi la) + fo, (a) +++ + fi, (2) = 0.

Proof According to the Proposition 2.2, we know the condition is a necessary and sufficient condition for £ being a parallel bundle. Now since lines in £ are parallel lines after them leaving C iff for any x > 0 and Az > 0, there must be that

fila + Ax) + fiala) Ar +--+ fi, (~)Ag = file). Therefore, we get that

ha) + falt) + ie) =0 '

When do the parallel lines parallel the initial parallel lines after them passing through a cut C in a planar map geometry? The answer is in the following result.

Proposition 2.3 Let (M, pn) be a planar map geometry, C = {u101, Uzv,- +, uv} a cut of the map M with order uiv, ugu,- -, uj from the left to the right, l > 1 and the angle functions on them are fı, fo,---, ft. Then the parallel lines parallel the initial parallel lines after them passing through C iff for Vx > 0,

>0 fir (a) + fa (x) = 0 fx @)+ ii @)+ hye) > 0

Jiz) + fo, (2) ae te fil-4 (2) > 0

and

fila) + fo(x) +--+ 4+ filz) = lr.

Proof According to the Proposition 2.2 and Corollary 2.1, we know the parallel lines passing through C is a parallel bundle.

We calculate the angle a(i, x) of a line L passing through an edge u;v;,1 <i <1 with the line before it meeting C at the intersection of L with the edge u;v;, where x is the distance of the intersection point to u; on uiv, see also the Fig.4. By the

definition, we know the angle a(1,z) = f(x) and a(2,x) = falx) — (m — fi(x)) = f(a) + f(a) -r

Now if a(i, x) = fi(x) + falx) +---+ fi(x) — (i — 1)r, then similar to the case i = 2, we know that a(i + 1,x) = fipı(£) — (m — ali, x)) = fils) + ali, 2) — r. Whence, we get that

ali +1, x) = filx) + fala) +--+ firilx) — ir.

Notice that a line L parallel the initial parallel line after it passing through C iff a(l, £) = T, i.e.,

fila) + fle) +--+ + fi(x) = lr. This completes the proof. 4 3. Linear condition and combinatorial realization for parallel bundles

For the simplicity, we can assume the function f(z) is linear and denoted it by fi(a). We can calculate fi(x) as follows.

Proposition 3.1 The angle function fi(x) of a line L passing through an edge uv at the point with distance x to u is

aiea A) he ER) fitz) = d(uv) ) 2 d(uv) 2 ?’ where, d(uv) is the length of the edge w.

Proof Since fi(x) is linear, we know that fiı(x) satisfies the following equation.

fila) — pluulw o ax puw) _ puelu) d(uv)’ 2 2

Calculation shows that

Ata yee x plv)u(v)

Corollary 3.1 Under the linear assumption, a family L of parallel lines passing through an edge uv is a parallel bundle iff

) ~ plu)’ Proof According to the Proposition 2.1, a family of parallel lines passing through an edge uv is a parallel bundle iff for Yx, x > 0, f'(x) > 0, i.e.,

Z e E

ployer) _ pluu) S 2d(uv) 2d(uv) 7

Therefore, a family £ of parallel lines passing through an edge uv is a parallel bundle iff

Whence,

p(v) ~ ulu) For a family of parallel lines pass through a cut, we have the following condition for it being a parallel bundle.

plu) — Ble) ;

Proposition 3.2 Let (M, p) be a planar map geometry, C = {u101, Ugve,- ++, ur} a cut of the map M with order uv, U2v2,---, uy from the left to the right, | > 1. Then under the linear assumption, a family L of parallel lines passing through C is a parallel bundle iff the angle factor u satisfies the following linear inequality system

p(v1)u(v1) > pur) u(ur)

(vi) u(e1) plva)ulva) oles )u(e) plu2)u(u2)

d(uvı) d(ugv2) d(uv) d(u2V2) p(vi) He) plvz)ulva) 4, elven) d(u,v1) d(uzv2) d( uv) > Pualu) | plua)ulua) plu) u(u) d(u1, v1) d(ug, v2) d(u, v) `

Proof Under the linear assumption, for any integer i, i > 1, we know that ier p(vi) evi) = plus) ului)

2d(uj;v;) by the Proposition 3.1. Whence, according to the Proposition 2.2, we get that a

family L of parallel lines passing through C is a parallel bundle iff the angle factor u satisfies the following linear inequality system

p(v1)u(v1) > pur) u(ur)

(v1) e(v1) plv2)e(v2) pur) Hu) p(uz) (Ua) d(u4v1) d(ugv2) E d(uv) d(ugv2)

pwi) ulv) plva)u(va) | plu) eer)

d(u,v1) d(uzv2) d( uv) p(ui)u(u:) k p(uz)u(uz2) dining? pur) e (a) d(u4, U1) d( ug, v2) d(uz, vi) This completes the proof. h

For planar maps underlying a regular graph, we have the following interesting results for parallel bundles.

Corollary 3.2 Let (M, u) be a planar map geometry with M underlying a regular graph, C = {u101, Ugv2,-++, ww} a cut of the map M with order u,v, uzwa,- , ww] from the left to the right, l > 1. Then under the linear assumption, a family L of parallel lines passing through C is a parallel bundle iff the angle factor u satisfies the following linear inequality system

(v) > ulu)

p(v:) [u(v2) > phu) p(u2) d(uv) d(u2V2) E d(u v1) d(u2V2)

ulv) wie) elm) WO) a), el) d(uyv1) dluzvə) d(uwi) 7 d(uyv1) ~~ d(ugv2) dfu)

and particularly, if assume that all the lengths of edges in C' are the same, then

5 £ IV IV 5 =

(ur) + u(u2)

H (ug) + +++ + p(w).

vV A =

a

(vi) + pve) +++ + ului)

Certainly, by choosing different angle factors, we can also get combinatorial conditions for existing parallel bundles under the linear assumption.

Proposition 3.3 Let (M, p) be a planar map geometry, C = {u101, uate, +++, utr} a cut of the map M with order uiv, uguo,- --, uyw from the left to the right, | > 1. If for any integer 1,1 > 1,

Ne)

then under the linear assumption, a family L of parallel lines passing through C is a parallel bundle.

Proof Notice that under the linear assumption, for any integer 7,7 > 1, we know that

/ _ piyul) — p(us)u(us)

by the Proposition 3.1. Whence, fj (x) > 0 for i = 1,2,---,1. Therefore, we get that

filz) = 0 filt) + fi, (x) = 0 hee) + fe) ile) 20

fiala) + falt) + file) 2 O.

By the Proposition 2.2, we know that a family L of parallel lines passing through C is a parallel bundle. 4

4. Classification of parallel bundles

For a cut C in a planar map geometry and e € C, denote by f.(x) the angle function on the edge e, f(C,x) = X f.(x). If f(C,x) is independent on z, then eC

we abbreviate it to f(C). According to the results in the Section 2 and 3, we can —

classify the parallel bundles with a given orientation O in planar map geometries

into the following 15 classes, where, each class is labelled by a 4-tuple 0,1 code.

Classification of parallel bundles

(1) Ciooo: for any cut C along O: F(C) = |C|r;

(2) Coo: for any cut C along Ø; f(C) < |Clr;

(3) Coo1o: for any cut C along O, f(C) > Clr ;

(4) Cooo1: for any cut C along O, fi(C,x) > 0 for Yx, x > 0;

(5) Ci1o0: There exist cuts C1, C2 along O, such that f(Ci) = |Cilr and f(C2) = c < |Colr;

|C2|7;

(7) Cioo1: there exist cuts C1, C2 along O, such that f(C1) = |Ci|a and fi (Co,2) > 0 for Vx,x > 0;

(8) Cono: there exist cuts Cy,C2 along O, such that f(Ci) < |Cilr and f(C2) > |Calr;

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>

(9) Coor: there exist cuts C1, Co along O, such that f(C,) < |Cilr and fi(C2,£) > 0 forYz,z > 0;

(10) Coo: there exist cuts C1, C2 along O, such that f(Ci) > |Cilr and fi(C2,£) > 0 for Yz, x > 0;

(11) Cimo: there exist cuts Ci, C2 and C3 along O, such that f(C1) = |Cilz, f(C2) < |Calmr and f(C3) > |Cs|z;

(12) Cy1o1: there exist cuts C1, C2 and C3 along O, such that f(C1) = |Cilr, f(C2) < |Coļr and f! (C3,x£) > 0 forYx,x > 0;

(13) Cion: there exist cuts C1, C and C3 along O, such that f(C1) = |C,|z, f(C2) > |Cola and fi. (Ci,x2) > 0 for Vz,x > 0;

(14) Cor: there exist cuts C1, Co and C3 along O, such that f(C1) < |Cilr, f(C2) > |Cala and f! (Ci,2) > 0 for Vz,x > 0;

(15) Cii: there exist cuts C1, C2, C3 and C4 along O, such that f(C1) = |Cilr, f(C2) < |Colr, f(Cs) > |Cs|r and f} (C4, x£) > 0 for Vz,x > 0.

Notice that only the first three classes may be parallel lines after them passing through the cut C. All of the other classes are only parallel bundles, not parallel lines in the usual meaning.

Proposition 4.1 For an orientation O, the 15 classes Ciooo ~ Cı111 are all the parallel bundles in planar map geometries.

Proof Not loss of generality, we assume C1, C2,- -, Cm, M 21, are all the cuts along O in a planar map geometry (M, u) from the upon side of O to its down side. We find their structural characters for each case in the following discussion.

Cioo0: By the Proposition 2.3, a family £ of parallel lines parallel their initial lines before meeting M after the passing through M.

Co1oo: By the definition, a family £ of parallel lines is a parallel bundle along O only if

f(Ci) < f(C2) <- < f(Cm) <T.

Otherwise, some lines in £ will intersect. According to the Corollary 2.1, they parallel each other after they passing through M only if

f(Ci) = f(C2) =+ = f(Cm) <T.

Coo10; Similar to the case Co10o0, a family £ of parallel lines is a parallel bundle along O only if

m< f(Ci) S$ E S F(Cm) and parallel each other after they passing through M only if

ms f(Ci = (C2) = Sf(Gm)-

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Cooo1: Notice that by the proof of the Proposition 2.3, a line has angle f(C, x) — (|C| —1)a after it passing through C with the initial line before meeting C. In this case, a family £ of parallel lines is a parallel bundle along O only if for Vx;, 2; > 0,1<i<m,

f(Ci, £1) < (Gs, £2) < -+ < f(Cm, Em).

Otherwise, they will intersect. = C1100: In this case, a family £ of parallel lines is a parallel bundle along O only if there is an integer k,2 < k < m, such that

F(C) < f(C2) S++ < F(Cr-1) < f(Ck) = f (Ceri) = +--+ = f(Cn) = T.

Otherwise, they will intersect. Cio10: Similar to the case C1100, in this case, a family £ of parallel lines is a parallel bundle along O only if there is an integer k,2 < k < m, such that

m= f(Ci1) = f(C2) =--- = a S +++ S F(Cm).

Otherwise, they will intersect. =

Cioo1: In this case, a family £ of parallel lines is a parallel bundle along O only if there is an integer k,l, 1 <k < l< m, such that for Vz;, x; > 0,1 <i< kor l<i<m,

F(Ci,zi) < F(C, £2) <--> < f(Ch te) < FCCk) = f (Chae) = = f (Ci) = 7 < fC ti) <- < f(Cm, m).

Otherwise, they will intersect. = Co110: In this case, a family £ of parallel lines is a parallel bundle along O only if there is integers k,1 <k < m, such that

(CQ) Sf (Ca) <c < f(Ck) < m < f( Cen) SS f(Cm).

Otherwise, they will intersect. Bp Co101: In this case, a family £ of parallel lines is a parallel bundle along O only if there is integers k,1 < k < m, such that for Vx; x; > 0,1 <i <m,

F(Ci, £1) < f(C2, £2) < t < f(Ch Ek) <T < F Cki; Eeti) < L f (Cm, Em),

and there must be a constant in f(C1, x1), f(C2, £2), -, f(Ck, £k). Coo11: In this case, the situation is similar to the case Cojo; and there must be a constant in F(Ck+1, Tk+1), F(Ck+2, Tk+2), Sa f(Cn, Lm).

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C1110: In this case, a family £ of parallel lines is a parallel bundle along oO only if there is an integer k,l, 1 <k < L< m, such that

PCs F(C2) <- < f(Ck = = f(O) =

Otherwise, they will intersect.

C1101: In this case, a family £ of parallel lines is a parallel bundle along O only if there is an integer k,l, 1 <k < l< m, such that for Vz;,2; > 0,1 < i< kor Lt oy,

(Gi, 41) Sf (Cy £2) <- < F(Ck £k) < J (Char) = -= f(Ci) =m < f(C, a) <---< f(Cn, tm)

and there must be a constant in f(C), 21), f (C2, £2), -+ , f(Cr, £k). Otherwise, they will intersect.

Cio11: In this case, a family £ of parallel lines is a parallel bundle along oO only if there is an integer k,l, 1 <k < l< m, such that for Vz; xi > 0,1 <i< kor STs

f(Ci z1) < f(Co,22) <- < f(Ch, £k) < f(Cr4s) = o= f(O) =T < FCn ati) < < f(Cns tm)

and there must be a constant in f(C), 21), f(Cip1, £i+1); t, f(Cm, £m). Otherwise, they will intersect.

Co111: In this case, a family £ of parallel lines is a parallel bundle along O only if there is an integer k,1 < k < m, such that for Vz;, xi > 0,

F(Ci, £1) < f(C2, £2) < -++ < f(Ce, £k) < m < fC 1) <- < f(Cm, tm)

and there must be a constant in f(C1, 21), f(C2, £2), =- -, f(Ck, £k) and a constant in f (Cir, £1), f (Cipi, t1), f(Cm;&m). Otherwise, they will intersect.

C1111: In this case, a family £ of parallel lines is a parallel bundle along © only if there is an integer k,l, 1 <k < l< m, such that for Vx;, x; > 0,1 <i < kor Li,

f(Ci z1) < f(Co,22) <- < f(Ch, £k) < f(Cr4i) = o= f(O) =n < FCn ati) <- < f(Cns tm)

and there must be a constant in f(C,, 21), f(C2,22),---, f(Ck, £k) and a constant in f(Ci, £1), f (Cipi, £1), f(Cm, €& m). Otherwise, they will intersect.

Following the structural characters of the classes Ciooo ~ C1111, by the Proposition 2.2, 2.3 and Proposition 3.1, we know that any parallel bundle is in one of the classes Ciooo ~ C1111 and each class in Ciooo ~ C1111 is non-empty. This completes the proof. '

A example of parallel bundle in a planar map geometry is shown in the Fig.5, in where the number on a vertex u denotes the number p(u)u(u).

Fig.5

5. Generalization

All the planar map geometries considered in this paper are without boundary. For planar map geometries with boundary, i.e., some faces are deleted ([10]), which are correspondence with the maps with boundary ({2]). We know that they are the Smarandache non-geometries, satisfying one or more of the following conditions:

(Al )It is not always possible to draw a line from an arbitrary point to another arbitrary point.

(A2~)It is not always possible to extend by continuity a finite line to an infinite line.

(A37)It is not always possible to draw a circle from an arbitrary point and of an arbitrary interval.

(A4~)not all the right angles are congruent.

(A5~ )if a line, cutting two other lines, forms the interior angles of the same side

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of it strictly less than two right angle, then not always the two lines extended towards infinite cut each other in the side where the angles are strictly less than two right angle.

Notice that for an one face planar map geometry (M, )~' with boundary, if we choose all points being euclidean, then (M, )~+ is just the Poincaré’s model for the hyperbolic geometry.

Using the neutrosophic logic idea, we can also define the conception of neutro- sophic surface as follow, comparing also with the surfaces in [8] and [14].

Definition 5.1 A neutrosophic surface is a Hausdorff, connected, topological space S such that every point v is elleptic, euclidean, or hyperbolic.

For this kind of surface, we present the following problem for the further re- searching.

Problem 5.1 To determine the behaviors of elements, such as, the line, angle, area, --, in neutrosophic surfaces.

Notice that results in this paper are just the behaviors of line bundles in a neutrosophic plane.

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