Continuous Maps on Digital Simple Closed Curves

Applied Mathematics, 2010, 1, 377-386 doi:10.4236/am.2010.15050 Published Online November 2010 (http://www.SciRP.org/journal/am)

Laurence Boxer1,2 1Department of Computer and Information Sciences, Niagara University, New York, USA 2Department of Computer Science and Engineering, State University of New York at Buffalo, Buffalo, USA E-mail: [email protected] Received August 2, 2010; revised September 14, 2010; accepted September 16, 2010

Abstract

We give digital analogues of classical theorems of topology for continuous functions defined on spheres, for digital simple closed curves. In particular, we show the following: 1) A digital simple closed curve S of more than 4 points is not contractible, i.e., its identity map is not nullhomotopic in S ; 2) Let X and Y be digital simple closed curves, each symmetric with respect to the origin, such that | Y |> 5 (where | Y | is the number of points in Y ). Let f : X  Y be a digitally continuous antipodal map. Then f is not nullhomotopic in Y ; 3) Let S be a digital simple closed curve that is symmetric with respect to the origin. Let f : S  Z be a digitally continuous map. Then there is a pair of antipodes {x,x}  S such that | f (x)  f (x) | 1.

Keywords: Digital Image, Digital Topology, Homotopy, Antipodal Point

1. Introduction Euclidean space; from this viewpoint, it is of interest to study conditions under which this digitization process A digital image is a set X of lattice points that model a preserves topological (or other geometric) properties” “continuous object” Y , where Y is a subset of a Eucli- [3]. dean space. Digital topology is concerned with deve- In this spirit, we obtain in this paper several properties loping a mathematical theory of such discrete objects so of continuous maps on digital simple closed curves, that, as much as possible, digital images have topological inspired by analogs for Euclidean simple closed curves. properties that mirror those of the Euclidean objects they In particular, we show that digital simple closed curves model; however, in digital topology we view a digital of more than 4 points are not contractible, and we obtain image as a graph, rather than, e.g., a metric space, as the several results for continuous maps and antipodal points latter would, for a finite digital image, result in a discrete on digital simple closed curves. topological space. Therefore, the reader is reminded that in digital topology, our “nearness” notion is the graphical 2. Preliminaries notion of adjacency, rather than a neighborhood system as in classical topology; usually, we use one of the na- Let Z be the set of integers. Then Z d is the set of tural cl -adjacencies (see Section 2). Early papers in the lattice points in d -dimensional Euclidean space. Let field, e.g., [1-8], noted that this notion of nearness allows X  Z d and let  be some adjacency relation for the us to express notions borrowed from classical topology, members of X . Then the pair (X ,) is said to be a e.g., connectedness, continuous function, homotopy, and (binary) digital image. A variety of adjacency relations fundamental group, such that these often mirror their are used in the study of digital images. Well known analogs with respect to Euclidean objects modeled by adjacencies include the following. respective digital images. Applications of digital topo- For a positive integer l with 1 ld and two distinct d logy have been found shape description and in image points ppppqqqq=(12 , , ,dd ), =( 12 , , , ) Z , p processing operations such as thinning and skeletoni- and q are cl -adjacent [10] if zation [9]. • there are at most l indices i such that

A. Rosenfeld wrote the following: “The discrete grid | pi  qi |= 1 , and (of pixels or voxels) used in digital topology can be • for all other indices j such that | p j  q j | 1 , regarded as a ‘digitization’ of (two or three-dimensional) p j =q j . See Figure 1.

morphism). By a digital  -path from x to y in a digital image X , we mean a (2,) -continuous function

f :[0,m]Z  X such that f (0) = x and f (m) = y . We say m is the length of this path. A simple closed  -curve of m  4 points (for some adjacencies, the minimal value of m may be greater than 4; see below) in a digital image X is a sequence {(0),(1),,ff 2 Figure 1. In Z , each of the points p1 = (1,0) and fm(1)} of images of the  -path fm:[0, 1]Z

p2 = (0,1) is both c1 -adjacent and c2 -adjacent to  X such that f (i) and f ( j) are  -adjacent if m1 p0 = (0,0) , since both p1 and p2 differ from p0 by 1 in and only if j = (i 1) mod m . If S = {xi }i=0 where exactly one coordinate and coincide in the other coordinate; xi = f (i) for all i [0,m 1]Z , we say the points of S

p3 =(-1,1) is c2 -adjacent, but not c1 -adjacent, to p0 , are circularly ordered. Digital simple closed curves are often examples of since p0 and p3 differ by 1 in both coordinates. digital images X  Z d for which it is desirable to

consider Z d \ X as a digital image with some adja- The notation cl is sometimes also understood as the d cency (not necessarily the same adjacency as used by number of points q  Z that are cl -adjacent to a d X ). For example, by analogy with Euclidean topology, given point p  Z . Thus, in Z we have c1 = 2 ; in 2 2 3 it is desirable that a digital simple closed curve X  Z Z we have c = 4 and c = 8 ; in Z we have 1 2 satisfy the “Jordan curve property” of separating Z 2 c = 6, c = 18, and c = 26 . 1 2 3 into two connected components (one “inside” and the More general adjacency relations are studied in [11]. d other “outside” X ). An example that fails to satisfy this Let  be an adjacency relation defined on Z . A  - property [10] if we allow | X |= 4 is given by (X ,c ) , neighbor of a lattice point p is  -adjacent to p . A 1 d where digital image X  Z is  -connected [11] if and only if for every pair of different points x, y  X , there is a X = {(0,0),(1,0),(1,1),(0,1)} set {x , x ,, x } of points of a digital image X such 2 0 1 r is circularly ordered, and (Z \ X ,c ) is c -connected. that x = x , y = x and x and x are  -neighbors 2 2 0 r i i1 However, this anomaly is essentially due to the “small- where i {0,1,,r 1}. ness” of X as a simple closed curve; it is known Let a,b  Z with a < b . A digital interval [7] is a [1,15,16] that for m = 8 and m = 4 , if X  Z 2 is a set of the form 1 2 digital simple closed ci -curve such that | X | mi , then 2 [a,b]Z = {z  Z|a  z  b}. Z \ X has exactly 2 c3i -connected components,

d d i {1,2}. Thus, it is customary to require that a digital Let X  Z 0 and Y  Z 1 be digital images with simple closed curve X  Z 2 satisfy | X | 8 when c -  -adjacency and  -adjacency respectively. A 1 0 1 adjacency is used; | X | 4 when c -adjacency is used. function f : X  Y is said to be ( , ) -continuous 2 0 1 Let X  Z d0 and Y  Z d1 be digital images with [2,8], if for every  -connected subset U of X , 0  -adjacency and  -adjacency respectively. Two f (U ) is a  -connected subset of Y . We say that 0 1 1 ( , ) -continuous functions f , g : X  Y are said to such a function is digitally continuous. 0 1 d d be digitally ( , ) -homotopic in Y [8] if there is a Proposition 2.1 [2,8] Let X  Z 0 and Y  Z 1 be 0 1 positive integer m and a function H : X [0,m]  Y digital images with  -adjacency and  -adjacency Z 0 1 such that respectively. Then the function f : X  Y is ( , ) - 0 1 • for all x  X , H (x,0) = f (x) and H (x,m) = g(x) ; continuous if and only if for every  -adjacent points 0 • for all x  X , the induced function H :[0,m ] {x , x } of X , either f (x ) = f (x ) or f (x ) and x Z 0 1 0 1 0  Y defined by f (x1 ) are 1 -adjacent in Y .

This characterization of continuity is what is called an H x (t) = H (x,t) for all t [0,m]Z , immersion, gradually varied operator, or gradually is (2,1 ) -continuous; and varied mapping in [12,13]. • for all t [0,m] , the induced function H : X  Y Given digital images (X , ) , i {0,1} , suppose Z t i i defined by there is a ( 0 ,1 ) -continuous bijection f : X 0  X 1 1 H t (x) = H (x,t) for all x  X , such that f : X 1  X 0 is (1 , 0 ) -continuous. We say X 0 and X 1 are ( 0 ,1 ) -isomorphic [14] (this is ( 0 ,1 ) -continuous. was called ( 0 ,1 ) -homeomorphic in [7]) and f is a We say that the function H is a digital ( 0 ,1 ) -

( 0 ,1 ) -isomorphism (respectively, a ( 0 ,1 )-homeo- homotopy between f and g .

Additional terminology associated with homotopic Theorem 3.2 Let S be a simple closed  -curve maps includes the following. and let H : S [0,m]Z  S be a (,) -homotopy

• If g is a constant map, we say f is ( 0 ,1 ) - between an isomorphism H 0 and Hm =,f where nullhomotopic [8]; if, further, (X , 0 ) = (Y,1 ) and f (S)  S . Then | S |= 4 . n1 f = 1X (the identity map on X ), we say X is  0 - Proof: Let S = {xi }i=0 , where the points of S are contractible [7,17]. circularly ordered.

• If H (x0 ,t) = f (x0 ) for some x0  X and all There exists w[1,m]Z such that t [0,m]Z , we say H is a ( 0 ,1 ) -pointed homotopy w = min{t [0,m]Z |H t (S)  S}. [18]. Without loss of generality, x1  H w (S) . Then the • If H :[0,m0 ]Z [0,m1 ]Z  Y is a (c1 ,)-homotopy induced function H w1 is a bijection, so there exists between (c1 ,)-continuous functions xu  S such that H (xu , w 1) = x1 . By Proposition 3.1, f , g :[0,m0 ]Z  Y such that for all t [0,m1 ]Z , H w1 ({x(u1) mod n , x(u1) mod n }) = {x0 , x2 } , and the continuity H (0,t) = f (0) = g(0) and H (m0 ,t) = f (m0 ) = g(m0 ) , property of homotopy implies H (xu , w) {x0 , x2 } . we say H holds the endpoints fixed [18]. Without loss of generality,

Proposition 2.2 [8] Suppose f 0 , f1 : X  Y are H  xwx(1)modun ,1=  0 (1) (,) -continuous and (,) -homotopic. Suppose and g 0 , g1 :Y  Z are (, ) -continuous and (, ) -ho- H  xw,=. x (2) motopic. Then g 0  f 0 and g1  f1 are (, )-homo- u 2 topic in Z . Suppose n > 4 . Equation (2) implies

H (x(u1) mod n , w) {x1 , x2 , x3}, but this is impossible, for 3. Homotopy Properties of Digital Simple the following reasons.

Closed Curves • H (x(u1) mod n , w)  x1 , by choice of x1 .

• H (x(u1) mod n , w)  {x2 , x3} , from Equation (1), because

A classical theorem of Euclidean topology, due to L.E.J. n > 4 implies neither x2 nor x3 is  -adjacent to x0 . Brouwer, states that a d -dimensional sphere S d is not The contradiction arose from the assumption that contractible [19]. Theorem 3.3, below, is a digital analog, n > 4 . Therefore, we must have n  4 . Since a digital for d = 1 . We also present some related results in this simple closed curve is assumed to have at least 4 points, we must have n = 4 . section. In [8], an example is given of a simple closed c - Proposition 3.1 Let S be a digital simple closed 2 a curve S  Z 2 such that Z 2 \ S has 2 c -connected  -curve, a {0,1} . Let f : S  S be a ( , ) - 1 a 0 1 0 1 components, with | S |= 4 , such that S is c -contrac- continuous function. If | S |=| S | , then the following 2 0 1 tible (see Figure 2). By contrast, we have the following. are equivalent. Theorem 3.3 Let (S,) be a simple closed  -curve a) f is one-to-one. such that | S |> 4 . Then S is not  -contractible. b) f is onto. Proof: It follows from Theorem 3.2 that if | S |> 4 , c) f is a ( , ) -isomorphism. 0 1 then there cannot be a (,) -homotopy between 1 Proof: Since | S |=| S | , the equivalence of a) and b) S 0 1 and a constant map in S . follows from the fact that S0 is a finite set. That c) It is natural to ask whether we can obtain an analog of implies both a) and b) follows from the definition of Theorem 3.3 for higher dimensions. In order to do so, we isomorphism. Therefore, we can complete the proof by must decide what is an appropriate digital model for the showing that b) implies c). k -dimensional Euclidean sphere S k . The literature n1 Let S a = {xa,i }i=0 , where the points of Sa are circu- contains the following. • Let X be the set of all points p  Z k such that larly ordered, a {0,1} . Let x1,u  S1 and let 2k 1 x = f (x ) . Then the  -neighbors of x in S p is a c1 -neighbor of the origin. 0,v 1,u 1 1,u 1 Then ([10], Proposition 4.1) X is c -contractible. are x and x , and the  -neighbors of 2k k 1,(u1) mod n 1,(u1) mod n 0 Notice that this example generalizes the contractibility of x in S are x and x . Since f is a 0,v 0 0,(v1) mod n 0,(v1) mod n a 4-point digital simple closed curve [8] (see Figure 2 continuous bijection, our choice of x implies 0,v for the planar version); the contractibility seems due to fx{,}0,(vnvn 1) mod x 0,( 1) mod = {,}. x 1,( unun  1) mod x 1,( 1)mod the smallness of the image, rather than its form. • Let Bd I denote the boundary of a digital k -cube, Thus, k 1 i.e., for some integer n > 2 , fx{,}1,(unun 1) mod x 1,( 1) mod = {, x 0,( vnvn  1) mod x 0,( 1) mod }. Bd I = {( x , , x ) [0, n 1]k | for some i  {1, , k }, Since u was taken as an arbitrary index, f 1 is kkZ1 ( , )-continuous, so f is a ( , ) -isomorphism. xni {0, 1}}. 1 0 0 1

Figure 2. c2 -contraction of X 4 , a digital simple closed c2 -curve, via H:X44×[0,2]Z  X . (a) shows X 4 . Points are labeled by indices. We have Hx(,0)= x for all x  X 4 . Arrows show the “motion” of x23, x at the next step. (b) shows the results of the first step of the contraction: Hx(030 ,1) = H(x ,1) = x; Hx(121 ,1) =Hx ( ,1) =x. The arrow shows the motion at the next step of the contraction. (c) shows the results of the final step of the contraction: Hx,(2)i =x0 for all i {0,1,2,3} .

x . This completes the induction proof for the See Figure 3 for the planar version. Then ([7], Coro- (1)modvj n Claim 2, which, in turn, completes the induction proof llary 5.9) for n >2, BdI is not c -contractible. k 1 for the Claim 1. Thus, the assertion is established. The next result may be interpreted as stating that for n > 4 , a map homotopic to 1 must be a “rotation” of S 4. Antipodal Maps the points of S . Theorem 3.4 Let S be a simple closed  -curve A classical theorem of Euclidean topology, due to K. such that | S |= n > 4 . Let f : S  S be a (,) - Borsuk, states that a continuous antipodal map f : S d continuous function such that f is (,) -homotopic S d from the -dimensional unit sphere to itself is to 1 . Then, for some integer j , we have  d S not homotopic to a constant map [19]. In this section, we f (x ) = x for all i [0,n 1] . i (i j ) mod n Z obtain a digital analog, Theorem 4.16, for d = 1 . We say a set X  Z d is symmetric with respect to Proof: Let H : S [0,m]Z  S be a (,) -homo- the origin if X satisfies the property that topy from 1S to f . For t [0,m]Z , let H t : S  S be the induced map. The assertion follows from the x  X if and only if  x  X. following. Claim 1: For each t [0,m] , there is an integer j d d Z Suppose we have X  Z 0 , YZ 1 , and X is sym- such that H (x ) = x for all i [0,n 1] . t i (i j ) mod n Z metric with respect to the origin. A function f : XY To prove Claim 1, we argue by mathematical induc- is called antipodal-preserving or an tion on t . For t = 0 , we can clearly take j = 0 . Now, suppose the claim is valid for t [0,u]Z such that 0  u < m . Then, in particular, there is an integer j such that H u (xi ) = x(i j ) mod n for all i [0,n 1]Z . The continuity properties of homotopy imply Hxu10() {,,}xxx(j 1)modnj ( j 1)mod n.

Without loss of generality, H u1 (x0 ) = x j . This is the initial case of the following:

Claim 2: H u1 (xk ) = x(k j ) mod n for all k [0,n 1]Z . Suppose the equation of Claim 2 is true for all k [0,v]Z , for some v [0,n  2]Z . In particular,

H u1 (xv ) = x(v j ) mod n . (3)

By the continuity properties of homotopy, H u1 (xv1 ) is adjacent to or coincides with H u1 (xv ) = x(v j ) mod n and with

Hxuv()=1(1)mod x v j n . Thus, Hxuv11(){ x ()mod vjn  , x(1)modvj n} . Since H u1 must also be an isomorphism by Theorem 3.2, from Equation (3), Hx()= uv11 Figure 3. BdI2 , the “boundary” of a digital square.

This completes an induction argument from which we conclude that  (n1)/2 n1 S = {xi }i=0 {xi }i=n(n1)/2

is (cl ,c1 ) -isomorphic to a digital arc, with the endpoints  of S being x(n1)/2 and xn(n1)/2 , such that

1 i  (n 1)/2 implies xi and xni are non-adjacent antipodes. • If n is odd, then n 1 is even, so S = S . This is 7 Figure 4. A digital simple closed c1 -curve Sx={ii }=0 and a a contradiction, since S is not a simple closed cl - curve. (,)cc11-continuous map f : SS such that f is (,)cc11  -homotopic to 1 . According to Theorem 3.4, such a map • If n is even, then S = S \{xn/2 } . Since S is S symmetric with respect to the origin, we must have that f must “rotate” the members of S . In (a), points of S x is the origin. But since S is a simple closed are labeled by indices. In (b), each yS is labeled by the n/2 c -curve in which x is the origin, this is a contra- index of the point x  S such that f ()=xy. Here, we l 0 i i diction. have fx()=ii x(2)mod8 for all i . Whether n is even or odd, the assumption that the origin belongs to S yields a contradiction. Hence, the antipodal map if f (x) =  f (x) for all x  X [19]. origin is not a member of S . n1 In this section, we study properties of continuous Lemma 4.4 Let S = {xi }i=0 be a digital simple closed d antipodal maps between digital simple closed curves. cl -curve in Z such that the points of S are d Lemma 4.1 Let p0 , p1 be cl -adjacent points in Z , circularly ordered. If S is symmetric with respect to

1  l  d . Then p0 and p1 are not antipodal. the origin, then n is even. Proof: The hypothesis implies that there is an index i Proof: By Lemma 4.3, the origin is not a member of th such that p0 and p1 differ by 1 in the i coordinate: S , so every member of S is distinct from its antipode.

| p0,i  p1,i |= 1. If p0 and p1 are antipodal, this would Therefore, n must be even. n1 imply {p0,i , p1,i } = {1/2,1/2}, which is impossible. Lemma 4.5 Let S = {xi }i=0 be a digital simple closed d d Lemma 4.2 Let p0 , p1 be cl -adjacent points in Z , cl -curve in Z such that the points of S are cir-

1  l  d . Then  p0 and  p1 are cl -adjacent. cularly ordered. If S is symmetric with respect to the

Proof: Elementary, and left to the reader. origin, then for all i we have xi = x(in/2) mod n . n1 Lemma 4.3 Let S = {xi }i=0 be a digital simple closed Proof: Suppose there is a simple closed cl -curve d n1 cl -curve in Z such that the points of S are cir- S = {xi }i=0 that is symmetric with respect to the origin cularly ordered. If S is symmetric with respect to the such that the points of S are circularly ordered, such origin, then the origin is not a member of S . that there exist indices u,v such that xu and xv are Proof: Suppose the origin is a member of S . Without antipodes and v  (u  n/2) mod n . Without loss of loss of generality, x0 is the origin, and therefore is its generality, we can assume u = 0 , v  n/2 . own antipode. Then, from Lemma 4.2, x1 is antipodal to either xv1

By Lemma 4.2, x1 and xn1 are antipodes; by or x(v1) mod n . Without loss of generality, x1 and xv1

Lemma 4.1, these points are not cl -adjacent. This are antipodal. If v 1 = 1 , this is a contradiction of establishes the base case of an induction argument: Lemma 4.3; or, if v 1 = 2 , this is a contradiction of 1 1 S1 = {xi }i=0 {xn j } j=1 is a connected subset of S , such Lemma 4.1; otherwise, we inductively repeat the that x1 and xn1 are non-adjacent antipodes; hence, argument above with the antipodal (by Lemma 4.2) pair S is (c ,c ) -isomorphic to a digital interval. Now, 1 l 1 (x2 , xv2 ) , etc., until similarly we obtain a contradiction suppose for some integer k , 1 k < (n 1)/2 , of Lemma 4.3 or of Lemma 4.1. The assertion follows. k k S k = {xi }i=0 {xn j } j=1 is (cl ,c1 ) -isomorphic to a digital We have the following. interval, with endpoints x and x , such that x di k nk m Theorem 4.6 Let Si  Z be simple closed and x are non-adjacent antipodes for all nm  i -curves, i {0,1}, each symmetric with respect to the m {1,,k} . Then, by Lemma 4.2, x and x k1 nk1 origin. Let f : S0  S1 be a ( 0 ,1 ) -continuous are antipodes, and by Lemma 4.1, these points are not antipodal map. Then f is onto. c -adjacent. Thus, S = {x }k1 {x }k1 is (c ,c ) - n1 l k1 i i=0 n j j=1 l 1 Proof: Let S1 = {xi }i=0 , where the points of S1 are isomorphic to a digital interval, with endpoints x1 and circularly ordered. Without loss of generality, there xn1 . exists p  S0 such that f ( p) = x0 . Since f is anti-

Copyright © 2010 SciRes. AM 382 L. BOXER podal, it follows from Lemma 4.5 that f ( p) = x . GHxHx(1)= ( )= ( ), n/2 xi 10ii Since f is continuous and S is  -connected, it Gm(1)=()=(), H x Hx 0 0 xi mi1 mi follows that one of the 1 -paths in S1 from x0 to a(1) = a(0) = 0 = a(m) = a(m 1) . Note that the con- x is contained in f (S ) . Without loss of generality, tinuity of H implies that n/2 0 x0 {x }n/2  f (S ) . For n/2 < j < n , there exists q  S j j=0 0 0 {at ( 1), at ( 1)} { at ( )  1, at ( ), at ( )  1}. such that f (q) = x jn/2 , and from Lemma 4.5, Suppose H (,x ) = y so Gt()= y . xx =  = fq ( ) t i j xiY[()]modjat n jjn /2 If a(t 1) = a(t) 1, then = (since f is antipodal ) f ( q ) f ( S0 ). is  -adjacent Gtx (1)= y[(1)]modjat n = y [()1]mod jat   n Y Thus, f (S ) = S . iYY 0 1 to . di Gtx () Corollary 4.7 Let Si  Z be simple closed  i - i If a(t 1) = a(t) , then G (t 1) = G (t) . curves, i {0,1}, each symmetric with respect to the xi xi origin. Let f : S0  S1 be a ( 0 ,1 ) -continuous anti- If a(t 1) = a(t) 1, then podal map. If | S |=| S | , then f is a (,) -iso- is  -adjacent 0 1 Gtx (1)= y[(1)]modjat n = y [()1]mod jat   n Y morphism. iYY to G (t) . Proof: By Theorem 4.6, f is onto. The assertion xi Similarly, G (t 1) is  -adjacent to, or equal to, follows from Proposition 3.1. xi Y nX 1 Gt().Therefore, the induced function G is (c , ) - Proposition 4.8 Let X = {x } be a simple closed xi xi 1 Y i i=0 continuous.  X -curve with circularly ordered points. Let Y = nY 1 Therefore, G is a homotopy between H 0 and H m {}yii=0 be a simple closed  Y -curve with circularly such that G(x0 ,t) = H 0 (x0 ) for all t [0,m]Z . ordered points. Suppose H : X [0,m]Z  Y is a Let (X , X ) and (Y, Y ) be digital images and let ( X , Y ) -homotopy between the ( X , Y ) -continuous y Y . We denote by y : XY (or, for short, y ) the maps H 0 , H m such that H 0 (x0 ) = H m (x0 ) . Then there constant map yx()= y for all x  X . is a ( , ) -homotopy GX:[0,] m  Y between X Y Z Lemma 4.9 Let (X , X ) and (Y, Y ) be digital

H 0 and H m such that Gx(,)= t H ()x for all 0 00 simple closed curves and let f : X  Y be ( X , Y ) - t [0,m]Z homotopic to a constant map. Then for any x0  X , Proof: Without loss of generality, H 0 (x0 ) = y0 . Let f : (X , x0 )  (Y, f (x0 )) is ( X , Y ) -pointed homotopic a :[0,m]Z  [0,m]Z be defined by a(t) = i if to the constant map f (x0 ) . H (x0 ,t) = yi . Proof: Let F : X [0,m0 ]Z  Y be a ( X , Y ) - Let G : X [0,m]Z  Y be defined by homotopy between f and the constant map y0 for G(x ,t) = y , if H (x ,t) = y . i [ ja(t )]mod nY i j some y0 Y . Since Y is  Y -connected, there is a

path p :[0,m1 ]Z  Y from y0 to f (x0 ) . Then the Roughly, we may think of G(xi ,t) as rotating map H : X [0,m0  m1 ]Z  Y defined by H (xi ,t) counter to the rotation of H (x0 ,t) , so that the image under G of x at time t is constant with F(x,t) if 0  t  m0 ; 0 H (x,t) = respect to t . We show below that G is a homotopy. In  p(t  m0 ) if m0  t  m0  m1 , particular, a(0) = a(m) = 0 , so G = H and 0 0 is a homotopy between f and f (x ) . The assertion G = H ; and, for all t [0,m] , 0 m m Z follows from Proposition 4.8. Gx( , t ) = y = y . 0[()()]mod0at at nY Given a digital image (X ,) and a positive integer n , for x  X we define [20] For each xi  X and t [0,m]Z , if H t (xi ) = y j then we have the following. Nxn (,)= {} x • By the continuity of H , we have t {|y X there is a path in X Hx({ , x }) { y , yy , }. ti( 1)mod nXX ( i 1)mod n ( j  1)mod njj Y (  1)mod n Y from y to x of length at most n}. Therefore, Gx()= y and ti[()]mod jatn Y The covering space and the lifting of maps are notions Gx, x  borrowed from algebraic topology that have been t inin1modXX  1mod  important in digital algebraic topology. We have the yyy,, . following. jatnjatnjatn1 modYY   mod  1  mod Y Definition 4.10 [20] Let (E, E ) and (B, B ) be Therefore, G is ( , ) -continuous. t X Y digital images. Let p : E  B be a ( E , B ) -conti- • Let G :[0,m]  Y be the induced function de- nuous function. Suppose for each b  B there exists xi Z fined by G (t) = G(x ,t) . To simplify the following, let   N such that xi i

• for some   N and some index set M , and e0  E . Let (B, B ) be a digital image and 1 1 b  B . Let p : E  B be a ( , ) -covering map p (N (b, )) = N (e , ) with e  p (b); 0 E B B  E i i iM such that p(e0 ) = b0 . Then a  B -path f :[0,m]Z  B beginning at b has a unique lifting with respect to p • if i, j  M , i  j , then ~ 0 N (e , )  N (e , ) =  ; and to a path f in E starting at e0 . E i E j • the restriction map p | : N (e , )  N (b, ) For a positive integer n , a ( E , B ) -covering map N (ei , ) E i B E p : E  B is called a radius n local isomorphism is a ( E , B ) -isomorphism for all i  M . Then the map p is a ( , ) - covering map, and [21] if for each b  B and epb 1 (), p | E B N (,)en the pair (E, p) is a ( , ) - covering (or covering  Nbn(, ) is an isomorphism. It was observed inE [14] E B B space). that every covering is a radius 1 local isomorphism, but The following is a somewhat simpler characterization there are coverings that are not radius 2 local isomor- of the digital covering than given in Definition 4.10. phisms. Theorem 4.11 [14] Let (E, ) and (B, ) be E B Theorem 4.15 [21] Let (E, E ) be a digital image digital images. Let p : E  B be a ( , ) - E B and e0  E . Let (B, B ) be a digital image and continuous function. Then the map p is a ( , ) - E B b0  B . Let p : E  B be a ( E , B ) -covering map covering map if and only if for each b  B , there is an such that p(e0 ) = b0 . Suppose p is a radius 2 local index set M such that isomorphism. For  E -paths g 0 , g1 :[0,m]Z  E that • p 1 (N (b,1)) =  N (e ,1) , with e  p 1 (b) ; B iM E i i start at e0 , if there is a  B -homotopy in B from • if i, j  M , i  j , then N (e ,1)  N (e ,1) =  ; E i E j p  g 0 to p  g1 that holds the endpoints fixed, then and g 0 (m) = g1 (m) , and there is a  E -homotopy in E • the restriction map p |N (e ,1) : N (ei ,1)  N (b,1) E i E B from g 0 to g1 that holds the endpoints fixed. nX 1 is a ( E , B ) -isomorphism for all i  M . Theorem 4.16 Let X = {xi }i=0 be a simple closed The following is a minor generalization of an example  X -curve with points circularly ordered, that is of a digital covering map given in [20]. symmetric with respect to the origin. Let Y = {y }nY 1 be m1 d i i=0 Example 4.12 Let C = {ci }i=0  Z be a circularly a simple closed  Y -curve with points circularly ordered, ordered simple closed  -curve. Let p : Z  C be that is symmetric with respect to the origin. Let defined by p(z) = z mod m for all z  Z . Then p is f : X  Y be a ( , ) -continuous antipodal map. If a (c ,) -covering map (see Figure 5). X Y 1 | Y |> 5 , then f is not ( , ) -nullhomotopic. Definition 4.13 [20] For digital images (E, ) , 0 1 E Proof: Suppose there is such a function f that (B, ) , and (X , ) , let p : E  B be a ( , ) B X E B is ( , ) -nullhomotopic. From Lemma 4.9, f is covering map, and let f : X  B be ( , ) - 0 1 X B pointed homotopic to f (x ) . continuous. A lifting of f with respect to p is a 0 Let b :[0,n ]  X be defined by b(t) = x . ( , ) -continuous function F : X  E such that X Z t mod nX X E Let p : Z  Y be the (c , ) -covering map defined p  F = f . 1 Y by p(z) = y (see Example 4.12). By Proposition See Figure 6 for an illustration of Definition 4.13. z mod nY 2.2, f b and f (x ) b are (c , ) -homotopic paths Theorem 4.14 [20] Let (E, ) be a digital image 0 1 Y E in Y . From Theorem 4.14, these functions have unique

Figure 5. A simple closed c1 -curve C and a covering by the digital line Z . Members of C are labeled by their respective indices. A point zZ is labeled by the index of point of C to which the covering map sends z .

5 Figure 6. Example of lifting. Bb={ii }=0 is a simple closed c2 -curve whose members are labeled by their indices. EZ= has its points z labeled above by their coordinates and labeled below by the index i such that p()=zbi (note p is given by 11 the formula pz()= bz mod 6 ). Xx={mm } =0 is a simple closed c2 -curve that has points labeled by a pair mn, such that m is the index of the point, f ()=xbmn (thus, f is defined by fx()=mm bmod 6 ), and F()=xmm . Since p is a covering map (by Example 4.12) and pF = f, F is a lifting of f with respect to p .

d liftings with respect to p to paths F0 and F1 , x,x  S such that f (x) = f (x) [17]. For d = 1 , respectively, in Z , each starting at 0 p 1 ( f (b(0))) . the following is a digital analog. Since | Y |> 5 , N ( f (x ),2)  Y , so p is a radius 2 Y 0 Theorem 5.1 Let S be a digital simple closed local isomorphism. From Theorem 4.15, F and F n1 0 1 cl -curve with S = {xi }i=0 , where the points of S are must end at the same point. Indeed, this point must be 0 , circularly ordered. Suppose S is symmetric with for the uniqueness of F1 implies F1 must be the cons- respect to the origin. Suppose f : S  Z is a (cl ,c1 ) - tant map 0 . continuous function. Then there is a pair of antipodes Since f is an antipodal map, we must have x,x  S such that | f (x)  f (x) | 1.

| F0 (t  nX /2)  F0 (t) | = nY /2 for all t [0,nX /2 1]Z . (4) Proof: By Lemma 4.4, n is even. Consider the

function g :[0,n 1]Z  Z defined by gi()= f ( xi ) Since F0 (0) = 0 , F0 (nX /2) {nY /2, nY /2} . Without loss of generality, fx()(/2)modin n . By Lemma 4.5, we are done if, for some i , g(i) = 0 . Therefore, we assume for all i that F0 (nX /2) = nY /2. (5) g(i)  0. (6) Since F is continuous and nY > 5 , a simple induction argument based on Equations (4) and (5) shows that Clearly, for all i , F (t  n /2) > F (t) for all t [0,n /2 1] . But since 0 X 0 X Z g[(i  n/2) mod n] = g(i). (7) F0 (nX ) = 0 , this implies with Equation (4) that The continuity of f implies that F0 (nX /2) = nY /2 , which contradicts Equation (5). The assertion follows from the contradiction. | g(i)  g(i 1) | 2. (8)

5. Antipodes Mapped Together It follows from Equation (7) that g takes both positive and negative values, so from inequality (6), there is A classical result of topology is that if f is a conti- an index j such that g( j) and g( j 1) have oppo- nuous map from the d -dimensional unit sphere S d to site sign; without loss of generality, g( j) > 0 and Euclidean d -space R d , then there is a pair of antipodes g( j 1) < 0 . From inequality (8), it follows that

that digital simple closed curves of more than 4 points are not contractible; that a continuous antipodal map from a digital simple closed curve to itself is not nullhomotopic; and that a continuous map from a digital simple closed curve to the digital line must map a pair of antipodes within 1 of each other. Except as indicated concerning whether or not a digital model of a sphere is contractible, it is not known at the current writing whether these results extend to higher dimensional digital models of Euclidean spheres. We thank the anonymous referees for their helpful suggestions.

7. References

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