Illumination problems on translation surfaces with planar infinities

Illumination problems on translation surfaces with planar infinities Nik ola y Dimitrov ∗ Abstract In the curre nt article we discus s an illumination pro b- lem prop osed by Urrutia and Zaks. The fo cus is on configuratio ns of finitely many t wo-sided mirr ors in the plane together with a source of light placed at an arbitrar y p oint. In this setting, w e study the r egions unilluminated b y the source. In the c ase of rational- π angles betw een the mir rors, a planar configuration gives rise to a s urface with a translation struc tur e and a num b er of planar infinities. W e show that on a sur face of t his t yp e with at lea st tw o infinit ies, one can find plent y of unilluminated re gions isometric to un b ounded plana r sector s . In addition, we establish that the non-bijectivit y of a certain circle map implies the existence o f un b ounded dark sector s for rational planar mirror configurations illuminated b y a ligh t- source. 1 In tro duc tion Consider a planar do main with a ligh t reflecting bo undary . Place a source of light a t a po int inside the domain. Assume that the source emits rays in all dir ections. Each ray follows a stra ight line a nd whenever it reaches the b oundar y it is reflected ac- cording to the rule that the angle of incidence equals the angle of reflection. A point from the domain is considered illu minate d by the sourc e whenever there is a ray that rea ches the p oint e ither directly or a fter a series o f reflections. In this s etting, o ne can a sk the following ques tions, also known as il lumination pr oblems . Question 1. If we plac e the sourc e of light at any ∗ Departmen t of Mathemat ics and Statistics, McGill Uni- v ersity , dimitrov@math.m cgill.ca p oint in the domain, wil l al l of the domai n b e il lu- minate d? If not, what c ould b e s aid ab out the non- il luminate d r e gions? Question 2. Is ther e a p oint fr om which the light sour c e c an il lum inate the ent ir e domain? These problems a re often attributed to E. Straus who p ose d them so metime in the ea r ly fifties and first published b y V. Klee in 1969 [5]. Some famo us ex- amples and interesting results are Penrose’s ro om [1], T ok a rsky’s e xample [5 ] as well as the ar ticle [3] b y Huber t, Schm oll and T roub etzkoy on illumination on V eec h surfaces. In 1991, J. Urrita and J. Zaks pro po sed the follow- ing problem [6]. Assume we ar e given a finite num b er of disjoin t compa ct line segmen ts in the plane eac h representing a mirror that reflects light on bo th sides (a tw o- sided mirro r). Let p 0 be an y po int on the S S a) b) P Figure 1: plane not incide nt to an y o f the segmen ts. Then, the complement of the set of mir rors is an unbounded do - main with light-reflecting b o undary and if w e place a so urce of light S a t p 0 we can p ose questions 1 and 2. Figure 1a depicts an exa mple of a tw o-sided mir- ror configuration with a light emitting source S . The 1 conv ex h ull of the mirrors is a polygon. If S is in the conv ex hull, one can construct a triangle P unillu- minated by S , lik e the shaded o ne on fig ure 1 b. T o do that, it is sufficient f or a m irr or seg ment to be an edge of t he conv ex h ull. In this pap er we are interested in finite t wo-sided mirror configura tions with the following prop er ty: any pair of lines determined by the mirror seg ments are either parallel o r intersect at a n angle which is a rational m ultiple of π . W e will call suc h a configura- tion a r ational mirr or c onfigur ation and the doma in obtained as a complement o f the mirro rs will b e called r ational mirr or domain . F or those, we will find condi- tions that will g uarantee the existence of unbounded unilluminated sectors in the plane (see definition 2). A ra tional mirr or doma in can b e ” unfolded” in to a surface that car ries a flat structur e with conical sin- gularities a nd trivial ho lonomy g roup [2], [4]. This means that the surface has a sp ecia l atlas with the prop erty that aw ay from the cone p oints, the transi- tion maps betw een tw o charts from the atlas are Eu- clidean translations. In the litera ture, such an o b ject is calle d a tr anslation surfac e . As a r esult, the piece- wise linear tra jectory of a light r ay in the o r iginal domain becomes a smoo th geo desic on the flat sur- face. Thus, one can think of a ligh t source placed at a no nsingular po int on the surfac e , emitting geo desic rays in all directio ns. Any other p oint on the surface is considered il luminate d if there is a smo oth g eo desic connecting the s ource to the p oint. In this way , one can ask questions 1 and 2 f or the translation sur face. Notice that there ar e regions on it isometric to com- plement s of compa ct sets in the plane. W e will call a surface with such geometry a tr anslation surfac e with planar infinities . A tra nslation surface with pla nar infinities gives rise to a pair ( X , ω ) where X is a closed surface with a co mplex structure and ω is a meromo rphic differ- ent ial on X with only double p oles and z e ro residues. The zeroes of ω are t he cone points o f the flat struc- ture [2],[4], and a round each pole the surface lo o ks like the complemen t of a compact set in the plane. The converse is also true . A pair ( X, ω ) of a closed Riemann surface and a meromorphic differe ntial with only double p o les and zer o residues induces a trans- lation structure o n X with pla nar infin ities. Definition 1. The p air ( X , ω ) is c al le d a tr anslation surfac e with pla nar infinities whenever t he fol lowing c onditions hold: (1) X is a close d surfac e with a c omplex stru ctur e; (2) ω is a mer omorphic differ ential on X ; (3) Every p ole of ω is of or der exactly 2 and the r esidue a t that p ole is zer o. We will r efer t o t he p oles of ω as planar infinities. In this study we will b e interested in a sp ecial t yp e of domains b oth on a tr anslation surface wit h planar infinities and in the plane. Definition 2. a) L et l 1 and l 2 b e two half -lines in the plane b oth st arting form a p oint p 0 and going t o infinity. L et θ b e the angle b etwe en l 1 and l 2 at the vertex p 0 , m e asur e d c ounter clo ckwise fr om l 1 to l 2 . Then, the op en r e gion C b ounde d by l 1 and l 2 , whose internal angle at p 0 is θ , is c al le d an infinite se ctor of angle θ (se e figur e 2a). b) An op en sub domain C of a t r anslation surfac e with planar infinities ( X , ω ) is c al le d an infinite se ctor of angle θ whenever t her e ex ists a chart fr om the tr ans- lation atlas of ( X , ω ) that maps C isometric al ly to a planar infinite se ctor of angle θ like the one define d in p oint a. p 0 l 2 l 1 θ C a) b) p 0 q l l q p 0 α α (α) (α) = l q (0) l p 0 (0) Figure 2: On any translation s ur face ( X , ω ) one can alw ays find an orien table foliation F ω with singularities, whose leaves are geo desics. Indeed, let us foliate the E uclidean plane into horizo nt al straig ht lines, or i- ent ed as usual from left to right. Since ea ch transi- tion map b etw een tw o charts is a Euclidean transla - tion, it sends horizontal lines to horizontal lines (line 2 orientation pr e served). Th us, pulling bac k on to the surface the plana r horizo nt al foliation from a ll trans- lation charts defines globa lly the desired folia tion F ω . Moreov er, the singularities of F ω are the cone p oints of the sur face ( X , ω ), i.e. the zero es of the differential ω . W e ca ll F ω the horizontal foliation of the surface and its le av es - the horizontal ge o desics of the sur - face. A t each non-singula r p oint p 0 of ( X , ω ) the oriented horizontal geo desic l p 0 (0) f ro m F ω defines a p ositive horizontal dir e ction a t p 0 . The c o unterclock- wise angle α b etw een l p 0 (0) and a n ar bitrary o riented geo desic l p 0 ( α ) through p 0 is called the di r e ction o f l p 0 ( α ) at p 0 (see figure 2b). F rom now on, l p 0 ( α ) de- notes the ge o desic ray on ( X , ω ) star ting from p 0 ∈ X and going in the direction of ang le α . It is important to emphasize that, sinc e we a re working with a trans- lation surface, the in tersection of t he geo desic l p 0 ( α ) with a ny other horizo ntal geo desic l q (0) will alwa ys form the same angle α, a s shown lo cally on figur e 2b. In other words, just like in the plane, a ge o desic o n ( X, ω ) do es not changes its angle with resp ect to the horizontal direction. Since a dir e c tion at any non- singular p oint p ∈ X is defined as an angle α ∈ R mo d 2 π , we can iden tify the set of all direc tio ns at p with the unit cir cle S 1 = { z ∈ C : | z | = 1 } . The po int 1 ∈ S 1 gives the horizontal direction α = 0. 2 Results It is natural to ask questio ns abo ut the behavior of the geo des ics on a s urface. The first question we will address is the following. On a tr anslation surfac e with planar infinities, wher e do most ge o desics emanating fr om a n ons ingu lar p oint go? As it tur ns out, almos t all of them fall on to the p oles of the surface. Same is true for an y rational mirror config uration in the plane. Theorem 1. The fol lowing two statements ar e true: (1) L et ( X , ω ) b e a tr anslation surfac e with planar infinities and let p 0 ∈ X b e non-singular. Then the set of al l dir e ctions α ∈ S 1 for which t he ge o desic p assing thr ou gh p 0 in dir e ction of α go es to one of the p oles of ω is op en and dense in the cir cle S 1 ; (2) Assu me we ar e given a r ational mirr or c onfigu- r ation in the plane and let p 0 b e a p oint not lying on any of t he mirr ors. Then the s et of al l dir e ctions α ∈ S 1 for which the pie c e-wise line ar r efle cte d tr a- je ct ory starting fr om p 0 in t he dir e ction of α go es to infinity is op en and dense in the cir cle S 1 . The next r esult e s tablishes the existence of infinite unilluminated sec tors and large unbounded regions on translation surfa c es with mor e than one pla nar infinit y . Theorem 2. L et ( X , ω ) b e a tr anslation surfac e with at le ast two planar infin ities. Then, for any p oint p 0 on X \ ( z er oes ( ω ) ∪ p oles ( ω )) ther e exists an infinite se ctor C on ( X , ω ) unil luminate d by p 0 , i.e. for any p oint p ∈ C ther e is no smo oth ge o desic on ( X , ω ) that c onne ct s p 0 to p . Mor e over, ther e ex ists a re gion on ( X, ω ) c onsisting of unil luminate d, non-overlapping infinite se ctors of total angle 2 π ( k − 1 ) , wher e k is the numb er of p oles of ω . The main idea s used in the proo f of theorem 2 c a n be adjusted to the s tudy of illuminatio n pro blems for rational mirror configura tions in the plane. F or instance, an interesting question put in an every day language, is the following. How big of an obje ct c an b e hidden fr om a stationary observer in a r ational mirr or domain? Can we hi de a c ar? A whole p arking lot of c ars? Precis e ly s p ea king, we would like to find a bas ic co nditio n that will ensure the existence of an infinite unilluminated s e ctor for a light sour ce placed at a p o int inside a rational mir ror domain. Let D be a rationa l mirror domain and let p 0 ∈ D . Draw a large e no ugh cir cle K , so that its interior con- tains the mirror s from the configur ation and the light source at the p oint p 0 . Denote by U p 0 the op en dense set of all directions whic h go to infinit y , provided by theorem 1. F or an angle α ∈ U p 0 ⊂ S 1 follow the straight line l p 0 ( α ) starting form p 0 in direction of α . Whenev er t he line r eaches a mirror it is reflected, changing its direction. In this w ay , a piecewise lin- ear tra jector y is formed, which at some po int lea ves the disc b ounded by K never to come back to it. Denote b y f p 0 ( α ) the angle betw een the horizo nt al direction of C and the po rtion of the tra jector y that is outside the circle K . As a result, w e obtain a map f p 0 : U p 0 − → S 1 . F or a picture of the construction o f f p 0 see figure 3. 3 α horizontal direction p 0 S 1 D K α f p 0 α f p 0 α S 1 f p 0 Figure 3: The map f p 0 is defined almost everywhere on the unit circle. In fa c t, its domain U p 0 is op en and dense in S 1 . Moreover, f p 0 is a rotation when r estricted to any co nnected comp onent of U p 0 . Our hope is that finding ways to study the combinatorial prop erties o f f p 0 may facilitate the sear ch for un b ounded unillumi- nated sectors in ra tional mir ror domains. Theorem 3. Assume we ar e given a r ational mirr or c onfigur ation. F or an arbitr ary p oint p 0 , not lo c ate d on any of the mirr ors, c onstruct t he cir cle map f p 0 as explain in the pr evious two p ar agr aphs (se e also figur e 3). If f p 0 is n ot inje ctive, then ther e exist s an infinite se ctor in t he plane u nil lu minate d by p 0 . 3 T ranslation surfaces. In the curren t section we discuss tr anslation sur faces and show how to construct one from a rationa l mirror configuratio n. T o illustr ate the idea better, we apply the proce dure to an example. V ar ious des criptions. A translation surfac e is a closed s urface X with a finite set of po ints Σ ⊂ X , called sing ula rities, and a cover of X \ Σ by op en charts { ( W a , ϕ a ) | W a ⊆ X \ Σ , ϕ a : W a → C } having the prop erty that whenever W a ∩ W b 6 = ∅ the transitio n map b etw een the tw o charts ( W a , ϕ a ) and ( W b , ϕ b ) is a Euclidean tr a nslation, i.e. z b = ϕ − 1 b ◦ ϕ a ( z a ) = z a + c . In our study , Σ par titions into t wo subsets Σ 0 and Σ ∞ . Each point from Σ 0 has a co ne angle of 2 π N , where N is a p o s itive integer. Each p oint p ∞ form Σ ∞ has a n op en neighbo rho o d W ′ ⊂ X with a map ϕ ∞ : W ′ \ { p ∞ } → C such that ( W ′ \ { p ∞ } , ϕ ∞ ) is a trans lation chart from the atlas. Also, the set C \ ϕ ∞ ( W ′ \ { p ∞ } ) is compact. Thus, the colle c tion Σ ∞ contains all planar infinities o n the surface. Since all trans la tions are holomorphic maps, the translation atlas induces a co mplex structure on X (for de ta ils s ee [2] and [4]). Moreov er, the differential dz a in each ϕ ( W a ) ⊂ C ca n be pulled back as a holo- morphic differential ω a = ϕ ∗ a dz a in the corr esp onding W a . But if z b = ϕ − 1 b ◦ ϕ a ( z a ) = z a + c then dz b = dz a . Hence, ω a = ω b in a ny intersection W a ∩ W b 6 = ∅ which gives rise to a g lobal holomorphic differential ω o n X \ Σ. Moreov er, ω extends to the singular set Σ so that Σ 0 bec omes the set o f zero es of ω and Σ ∞ bec omes the s et of all p oles of ω . The la tter are all double and with residue 0. So we se e that a tra nslation surface with planar infinities induces a pair ( X, ω ) of a co mpa ct Riemann surface without bo undary together with an appropr iate mer o morphic differential. T o r e cov er the transla tion atlas fro m a pair ( X , ω ), one ca n cover X \ (zero es( ω )) with top ologica l discs W a . On ea ch of them define the chart ϕ a ( p ) = R p p a ω , where p a ∈ W a is fixed and p v a ries in W a . As ω is either holomo rphic or meromorphic with a double po le a nd residue 0 inside the top ologic al disc W a , the path of integration in W a \ p oles ( ω ) is ar bitrary . If W a ∩ W b 6 = ∅ then z b = Z p p b ω = Z p p a ω + Z p a p b ω = z a + c for p ∈ W a ∩ W b . Thus, we hav e obtained the desir ed translation atlas. As w e can see, the description of a tra nslation surfa c e with planar infinities which w e gav e in the b eginning of the curren t se ction is equiv- alent to definition 1. The hor iz ontal foliation F ω on X , mentioned in the int ro duction, is defined as follows. L e t F C be the fo- liation of horizontal lines { z ∈ C | I m( z ) = s } , s ∈ R 4 in C o riented from left to right (see figur e 2b ). De- fine the pulled-back loca l foliation F a = ϕ ∗ a F C in each W a . Observe that F C is inv ariant with resp ec t to any translation, i.e. the tra nslations map a ny horizon tal line to a horizontal line. Hence, F a = F b on ea ch W a ∩ W b 6 = ∅ . Thus, a ll lo cal foliations fit together in a global foliation F ω on X with geo desic leav es and sing ularities Σ. The or iented leav es o f F ω deter- mine g lobally a horizontal direction on ( X , ω ). Since translations are Euclidea n isometr ies, the Euclide a n metric on C induces a E uclidean metric on X \ Σ. In this metric geo desics that do not go thr ough singula r - ities are isometric to straight lines in C . The notio n of a direction at a non-sing ular po int p ∈ X is as defined in the intro duction. It is the counterclo ckwise angle betw een the horizontal leaf and an o riented geo des ic bo th passing thro ugh p . Finally , an oriented geo des ic alwa ys for ms the same ang le with an y horizontal leaf it intersects, s o it never self- intersects, except p oss i- bly to clo se up. z 1 z 2 I 2 D 1 * z 1 z 2 z 1 = + c I 2 + - I 1 - σ 1 reflect with in a horizontal line glue σ 1 + σ 1 D 2 * σ 1 ( ) I 1 - σ 1 ( ) I 2 + σ 1 ( ) Figure 4: Construction. Assume we hav e a configuration o f disjoint co mpact line segmen ts I 1 , ..., I m in the plane C , which we regard as tw o-sided mirror s. The a ngle betw een a ny tw o of them is a rational-multiple of π . Observe that if one of the mir rors fo rms a rational- π angle with the rest of the mirrors, then immediately follows that any pa ir of mir rors forms a rational- π angle. This is a consequence of the fact that in an Euclidean triangle the a ng les at the vertices sum up to π. T o understand be tter the construction that follo ws, one could hav e a simple toy-example in mind. Let us hav e tw o p erp endicular mirrors I 1 and I 2 in the plane C lik e the ones depicted on figure 4 . Begin by slicing C a long the segments I 1 , ..., I n to obtain a closed s litted domain D ∗ in which ev- ery mirro r s e gment I k is doubled in order to obtain t wo parallel co pies I + k and I − k that form the b ound- ary comp o nent of the surface D ∗ around the slit I k . F or an intuitiv e geometric picture of D ∗ in the case of the toy-example, lo ok a t figure 4. Then D ∗ is homeomorphic to a once-punctured s pher e with n disjoint op en discs remov ed, as sho wn on figure 5 for the case o f tw o ortho g onal mirro rs. In par ticula r, ∂ D ∗ = ⊔ n k =1 ( I + k ∪ I − k ). α 1 2 α α α 3 4 ∞ 1 ∞ ∞ ∞ 2 3 4 Figure 5: F or eac h segment I k , fix the line l k ⊂ C thr o ugh 0 ∈ C parallel to I k . Denote by σ k the refle c tion o f C in l k . The gro up G generated by all σ k , k = 1 , .., n is a finite group. If α 1 is a g eneric direction in C , then G ( α 1 ) = { g ( α 1 ) | g ∈ G } = { α 1 , ..., α m } is a n or bit of maximal length m ≤ n . In our example G ∼ = Z 4 and a generic o rbit has 4 element s. P ick m copies D ∗ j of D ∗ each with a c hoice o f a direction α j in it. I f you prefer more formally , let D ∗ j = ( D ∗ , α j ). On figure 5, in the c a se of the toy-example, we can see a topo logical mo del of these four slitted planes with a choice of direction on each of them. W e glue D ∗ i to D ∗ j if and only if there is a segmen t I k ⊂ C whose corres po nding reflection σ k satisfies σ k ( α i ) = α j . The gluing is done in the follo wing wa y . T ake D ∗ i and σ k ( D ∗ j ). Glue th e edge I + k ⊂ D ∗ i to the edge σ k ( I + k ) ⊂ σ k ( D ∗ j ) and the edge I − k ⊂ D ∗ i to the edge of σ k ( I − k ) ⊂ σ k ( D ∗ j ). O n figure 4 of the toy-example, we hav e chosen i = 1 a nd j = 2. The upp er edge I + 1 ⊂ D 1 of the cut I 1 is glued to the lo wer edge σ 1 ( I + 1 ) ⊂ σ 1 ( D ∗ 2 ) of the cut σ 1 ( I 1 ). Analogously , the lower edge I − 1 from D ∗ 1 is glued to 5 ∞ j ∞ j ω (X, ) ω (X, ) ~ ~ φ φ 1 - W Figure 6: upper edg e σ ( I − 1 ) from σ 1 ( D ∗ 2 ). Both D ∗ i and σ k ( D ∗ j ) ar e naturally tr a nslation sur- faces with piecewise ge o desic b o undaries, global co or- dinates z i and z j , a nd differentials dz i and dz j resp ec- tively . Segments I k and σ k ( I k ) are e qual and parallel, hence the gluing map is a trans lation z j = z i + c (see the gluing o f the shaded piec e s on figure 4). Therefor e the res ulting s ur face made out of D ∗ i and σ k ( D ∗ j ) has a transla tion structure. Moreover, dz j = dz i along the gluing locus, so there is a well-defined holo mo r- phic differential on the new surface which extends meromorphica lly to bo th of its infinity p oints. Now, follo w the describ ed g luing pro cedure for all cuts on the piece s D ∗ j , whe r e j = 1 , .., m . The final result is a closed Riemann surface X and a mero mor- phic differen tial ω with only double p oles and ze ro residues, as w ell as simple zero es with cone ang le 4 π . F or the example of the tw o orthogo nal mirror s, figure 5 illustr ates ho w the four pieces D ∗ 1 , ..., D ∗ 4 fit to gether to form a co mpact torus X with a complex s tr ucture and a meromor phic differen tial ω on X . Ther e are eight simple zero es of ω and four double p ole s . The zero es ar e obtained fro m identifying pairs of black vertices on the s e gments I k form figur e 4. The cone angle at ea ch zer o is 4 π and the r esidue at each p ole is 0 a s desired. 4 Pro ofs Pro of of theorem 1. F rom now on ( X , ω ) is an a r- bitrary transla tio n surface with planar infinities and p 0 ∈ X \ (zero es( ω ) ∪ p oles ( ω ) a ny fixed po int . The idea is to cut out a rectangle around ea ch p ole ∞ j ∈ po les( ω ) a nd replace it by a one-handle. Indeed, choose a small to p o logical disc W aro und ∞ j and map it to C by ϕ ( p ) = R p q 0 ω where p v aries in W and q 0 ∈ W is fixed. Notice, ϕ is well defined as the residue at ∞ j is 0, so the path of integration is irrelev ant. T he image ϕ ( W ) ⊂ C is the complement of a compact set (the total sha ded reg ion o n figur e 6 stretching to infinity). Draw a rectang le Q ⊂ ϕ ( W ) as shown on figure 6 a nd remov e its exterior (the darker r egion). On the surface, w e r emov e the darker rectangular domain containing ∞ j . Then glue to- gether the low er horizontal edge of Q to the upp er and the left to the r ight, like gluing a torus. The gluing maps a r e clea rly a vertical and a horiz o ntal tra nsla- tion resp ectively . Therefore w e obtain a handle with a transla tion structure co mpatible with the structure on the rest of the surface (see figur e 6). By doing this for each ∞ j , we obta in a compac t translation surface ( ˜ X , ˜ ω ) of genus( ˜ X ) = genus( X ) + ♯ (po les( ω )), where ˜ ω is no w holomo rphic (has no po les). A lot is known ab out the behavior of the geo desics on suc h surfaces [2], [4], [7], so we us e this knowledge in our adv ant ag e. Let ˜ Λ p 0 be the set of all directions θ ∈ S 1 for which the g eo desic ˜ l p 0 ( θ ) on ˜ X is closed or hits a zero of ˜ ω . Also, let ˜ Ξ b e the set of all dire ctions θ ∈ S 1 for which the geo des ic flo w of ( ˜ X , ˜ ω ) in direc tion of θ is minimal [4] (e.g. an er go dic flow is minim al [2],[4]). Then ˜ Λ p 0 is countable but dense in S 1 (see [7]) a nd ˜ Ξ is dense and of full measure in S 1 (see [4], [2]). As a res ult, the set ˜ Θ p 0 = ˜ Ξ \ ˜ Λ p 0 consists of all θ ∈ S 1 for which the geode s ic ra y ˜ l p 0 ( θ ) is dense in ˜ X . Moreover, ˜ Θ p 0 is dense and of full meas ure in S 1 . There fo re, for any θ ∈ ˜ Θ p 0 the corresp onding geo desic ray l p 0 ( θ ) on the original surface ( X, ω ) hits a p ole of ω . Let U p 0 ⊂ S 1 be the s et of all directions θ ∈ S 1 with the prop erty that the geo desic ray l p 0 ( θ ) o n ( X, ω ) in the direction of θ reac hes a p o le of ω . Since the geo desic flo w on ( X, ω ) dep ends contin uously on the initial p oint and direction, the condition that a geo desic ray reaches a planar infinit y is o p en. There- fore, for each θ ∈ U p 0 there exists an ope n circular int erv al ( α, β ) ⊂ U p 0 that contains θ and for a ny θ ′ ∈ ( α, β ) the ray l p 0 ( θ ′ ) also reaches the same infin- it y . Hence, U p 0 is op en in S 1 . Moreov er, the dense set of full measure ˜ Θ p 0 is contained in U p 0 . Therefore, U p 0 is open and dense s et of full measure in S 1 . The se c o nd part of theor em 1 fo llows from the first one. If we are given a rationa l mirror config uration, 6 unfold it into a trans lation surface with pla nar infini- ties ( X, ω ) as describ ed earlier . Then, the infinit y of the mirror do main lifts to the set of poles of ω on X and w e apply the first part of the theorem. Pro of of theorem 2. As an o pen dense s ubset of S 1 , the co nstructed U p 0 is a countable disjoint union of op en circular interv als ( α j , β j ) ⊂ S 1 , i.e. U p 0 = ⊔ ∞ j =1 ( α j , β j ). By construction, the geodesic rays l p 0 ( θ ) emitted from p 0 in all directions θ ∈ ( α j , β j ) go to the same pole of ω . Fix some j and take a subinterv a l ( α ∗ , β ∗ ) ⊆ ( α j , β j ) (it may even be conv enient to choo se ( α ∗ , β ∗ ) = ( α j , β j )). Ch o o se ( α ∗ , β ∗ ) so that its meas ure is les s than π . No- tice, that fo r every θ ∈ ( α ∗ , β ∗ ), each ray l p 0 ( θ ) on X g o es to the same ∞ ∗ ∈ p oles( ω ). I n partic- ular, ∞ ∗ = ∞ 3 on figure 7. As ♯ (p oles( ω )) ≥ 2, take another ∞ ∈ p ole s( ω ) \ {∞ ∗ } and call it ∞ 1 just like o n our picture b elow. Choose a ”small” top ologica l disc W around ∞ 1 with the pro p e rty W ∩ (zero es( ω ) ∪ po les( ω )) = {∞ 1 } . Define the trans- lation c hart ϕ ( p ) = R p q 0 ω , wher e p v aries in W and q 0 ∈ W is fixed. The zero residue at ∞ 1 guaranties independenc e of the in tegral on the path betw een q 0 and p in W . On figure 7 we ha ve a lso provided an analogo us c har t ψ around p 0 . F rom now on, we use the same notations in W as the o nes in ϕ ( W ). Thus, we iden tify W with ϕ ( W ). In C the domain W look s like the complement of a c ompact set (the shade d re- gion on figure 7). Let K ⊂ W be a E uclidean cir cle in C centered at O and con taining C \ W in its interior. Abusing nota tio n, let α ∗ and β ∗ be the t w o points on the circle K suc h that the coun ter-clo c kwise a ngles betw een the p ositive hor izontal line through O in C and the ra dii Oα ∗ and O β ∗ are resp ectively α ∗ and β ∗ . Let p oints T 1 and T 2 on K be such that co unter- clo ckwise ∡ α ∗ OT 1 = ∡ T 2 Oβ ∗ = π 2 . Draw the lines t 1 and t 2 tangent to circle K at T 1 and T 2 resp ectively . Then they b o und an infinite sector C , depicted on figure 7 a s a darker shaded reg io n. W e claim that that C ⊂ X is not illuminated by p 0 . Assume that for some po int p ∈ C there exists θ ∈ S 1 such that the geo desic l p 0 ( θ ) ⊂ X star ing from p 0 in the direction of θ passes thro ugh p . Then, clearly l p 0 ( θ ) g o es to ∞ 1 . As already commented in ∞ 1 ∞ 2 ∞ 3 ∞ 4 p 0 W φ ψ β * θ l p 0 ( ) β * l p 0 ( ) α * Figure 7: the introduction, an y smo oth geo des ic on a transla- tion sur face forms the s ame angle with the horizontal direction at every p oint it passes through. In par - ticular, the angle betw een l p 0 ( θ ) and the horizontal direction in the c hart W as well as near the point p 0 is alwa ys θ . By lo ok ing at the picture of the chart W on figure 7, w e see that θ ∈ ( α ∗ , β ∗ ) in W . Hence θ ∈ ( α ∗ , β ∗ ) ⊂ S 1 at the point p 0 as well. By the choice o f the c ir cular in terv al ( α ∗ , β ∗ ), the geo desic ray l p 0 ( θ ) sho uld go to ∞ ∗ 6 = ∞ 1 . But a geo desic r ay can o nly rea ch one po le of ω , s o we get to a contra- diction. Therefor e , the infinite sector C on ( X , ω ) is not illuminated b y p 0 ∈ X . T o conclude the pro of, notice that for each circula r int erv al ( α ∗ , β ∗ ) ⊂ U p 0 the unilluminated se ctor C near ∞ 1 can b e also constructed around an y o ther po le ∞ 6 = ∞ ∗ of ω , i.e. there are k − 1 unillumi- nated copies of C . Partition U p 0 int o disjo int s ubin- terv als for which we can apply the constructio n of unilluminated infinite sectors from the preceding tw o paragr aphs. Th us, the the total sum of the angle s of all unilluminated sectors constructed on ( X , ω ) is k − 1 times the tota l measur e of U p 0 ⊂ S 1 which is 2 π . Hence, the to tal angle is 2 π ( k − 1 ). Pro of of theorem 3. Le t D ⊂ C b e a rational mir- ror domain a nd p 0 ∈ D (see figure 1 o r 3). Recall the finite gr oup G gener ated by all reflections in the lines through 0 ∈ C parallel to the mirrors. It a cts o n S 1 by rotations. Let f p 0 : U p 0 → S 1 be the ma p describ ed at the end of subsection ” Main results” (see also fig - ure 3) and assume it is not injective. Then, there are 7 θ 1 6 = θ 2 from U p 0 such that f p 0 ( θ 1 ) = f p 0 ( θ 2 ). T ake the finite orbit G ( θ 1 ) = { g ( θ ) ∈ S 1 | g ∈ G } . Then θ ∈ G ( θ 1 ) if and only if f p 0 ( θ ) ∈ G ( θ 1 ) so θ 2 ∈ G ( θ 1 ). Hence, the r estriction f | G ( θ 1 ) : G ( θ 1 ) → G ( θ 1 ) is not bijectiv e a nd there is θ ∗ ∈ G ( θ 1 ) suc h that θ ∗ ∈ U p 0 \ f p 0 ( U p 0 ). S ince f p 0 is a restriction of a rotation o n e ach connected comp onent of U p 0 , there is ( α ∗ , β ∗ ) ∋ θ ∗ such that ( α ∗ , β ∗ ) ⊂ U p 0 \ f p 0 ( U p 0 ). Remem b er the circle K from figure 3 that encom- passes the mirrors a nd p 0 . Using the circ ula r interv al ( α ∗ , β ∗ ), w e ca n carry out a bsolutely the same c o n- struction as the one in the chart W describ ed in the pro of of theorem 2. F or a picture of this construction lo ok at the rightmost larg e shaded area W on figure 7. Observe that the nota tions of the current pro of match the picture’s notations so that we can use it directly , thinking that the set of mirrors is in the lit- tle white elliptic region co ntaining the center O . W e claim that the infinite sector C (the darker shaded area) is not illuminated by the source p 0 ∈ D . In- deed, assume there is a light ray emitt ed by p 0 that reaches s ome p ∈ C . Then, fro m the picture, the direction of this ray is θ ∈ ( α ∗ , β ∗ ). But the light ray started from p 0 in so me direction θ 0 ∈ S 1 , so θ = f p 0 ( θ 0 ) whic h is a contradiction. References [1] Cro ft, H. T.; F alconer, K. J.; and Guy , R. K., “Unsolved Pro ble ms in Geometry”, New Y o rk, Springer-V erla g, 199 1 [2] Hub ert, P .; Schmidt, T. A., A n intr o duction to V e e ch surfac es , Handbo ok of dynamical systems. V ol. 1B, Els e vier B. V., Amsterdam, 2006, pp. 501-5 26 [3] Hub ert, P .; Schmoll, M.; T roubetzkoy , S., Mo d- ular fib ers and il lumination pr oblems , Int. Math. Res. Not. IMRN 20 0 8, no. 8, Art. ID r nn011, 42 pp. [4] Masur , H., Er go dic the ory of tr anslation surfac es , Handbo ok o f dy na mical sy stems. V ol. 1B, E lse- vier B. V., Amsterdam, 2006, pp. 5 27-5 4 7 [5] T ok arsk y , G. W., Polygonal r o oms not il luminable fr om every p oint , Amer. Math. Monthly 102 , 1995, pp . 867– 879. [6] Urrutia, J., Op en pr oblems on mirr ors , p ersona l webpage, URL: http://w ww.ma tem.unam.mx / ∼ urr utia/o penprob/Mirrors/ [7] V orob ets, Y., Planar structur es and bil liar ds in r ational p olygons: the V e e ch alternative , Russ. Math. Surv. 5 1 , 1996, no. 5, pp. 7 79-81 7 , 8