The Third Motivation for Spherical Geometry

The Third Motivtion for Sphericl Geometry Yoichi Med med@keykiccu-tokicjp Deprtment of Mthemtics Toki University Jpn Astrct: Historiclly, sphericl geometry hs developed minly on the terrestril gloe nd the celestil gloe In this pper, we will introduce the third gloe visul gloe centered t our eyes When we see the externl world, we re under n illusion tht our view screen is plne Loclly it is true, however, glolly it is more nturl to think tht our view screen is sphere centered t our eyes In this concept, we will study visul ngles of rectngle in the spce An ngle in the spce chnges its visul ngle ccording to our viewpoint This visul ngle is esily mesured on the visul gloe nd there is very simple reltion mong the visul ngles of rectngle We will lso relize the visul ngle on two dimensionl Eucliden plne long with the dynmic geometry softwre Cri II Plus Furthermore, we will find out tht the simple reltion ove is lso true in hyperolic geometry Introduction An ngle in the three dimensionl Eucliden spce chnges its ppernce ccording to our viewpoint This ppernce is lso one of informtion out certin reltion etween the ngle nd the viewpoint [5-7] To nlyze this informtion, we will introduce two ides: visul ngle nd visul gloe Intuitively, the visul gloe is our view screen centered t the viewpoint, nd on this gloe we cn mesure visul ngles s the ngle of sphericl tringle In this pper, we will focus on the four visul ngles of rectngle in the spce Using sphericl trigonometry, we will find out simple reltion mong them The definition of visul ngle nd representtion of the visul ngle on the visul gloe re descried in Section We study the reltion mong the visul ngles of rectngle in Section 3 In Section 4, we will introduce how to relize visul ngles in plne long with dynmic geometry softwre The stereogrphic projection plys n importnt role This reliztion proposes n eductionl pproch to sphericl geometry without difficult clcultion such s sphericl trigonometry Furthermore, we will try to extend the ide of visul ngle to hyperolic geometry in Section 5 The reltion mong the visul ngles of rectngle is lso true Using the Poincre model, we cn esily confirm it y simple construction Visul Angle on the Visul Gloe In this section, we will introduce two importnt ides: visul ngle nd visul gloe Let us strt from the definition of visul ngle Definition Visul Angle) Let BAC e fixed ngle determined y three points A, B nd C in the three dimensionl Eucliden spce E 3 Figure ) The visul ngle from viewpoint O is defined s the dihedrl ngle of the two fces OAB nd OAC of the tetrhedron OABC The following proposition shows tht the visul gloe is useful to mesure visul ngles

Figure Visul ngle on the visul gloe Proposition Visul Gloe) Let A, B nd C e the projected points of A, B nd C from the viewpoint O onto the unit) sphere centered t O Figure ) Then the visul ngle of BAC is equl to the ngle A' of the sphericl tringle A' B' C' Proof Let V e vector tngent to the rc A B t A, nd W vector tngent to A C t A The ngle etween V nd W is equl to A' On the other hnd, since oth vectors V nd W re perpendiculr to the line OA, the ngle etween V nd W is lso equl to the dihedrl ngle of the two fces OAB nd OAC In this wy, it turns out tht it is etter to regrd our view screen s sphere centered t our viewpoint Of course, this screen is loclly Eucliden nd when we look outside in the smll re, it is enough to regrd our view screen s plne This visul gloe is the third motivtion for sphericl geometry When we look up the ceiling in the room, ech corner hs the visul ngle greter thn 90 Why the sum of four ngles of rectngle is greter thn 360? The reson is simple, ecuse our view screen is not plne ut sphere In the following rgument, we use only one sphericl trigonometry, tht is, the sphericl cosine lw for ngles [4] p59): cos A cos B cosc sin Bsin C cos where is the opposite side of the vertex A in the sphericl tringle ABC 3 Visul Angles of Rectngle In this section, let us consider the reltion mong four visul ngles of rectngle in the spce Figure 3 shows the centrl projection of rectngle in the spce onto the visul gloe Without loss of generlity, let us ssume tht rectngle is rrnged on the plne zconstnt nd two pirs of opposite sides re prllel to x-xis nd y-xis, respectively Then four projected sides re gret circles on the sphere pssing through X,0,0), X,0,0), Y0,,0) or Y0,,0) These points correspond to the vnishing points of four sides

Figure 3 Centrl projection of rectngle on the visul gloe Theorem 3Visul Angles of Rectngle) Four visul ngles A, B, C nd D of rectngle ABCD in the spce stisfy the following simple reltion: cos A 'cosc' cos B'cos D' Proof Proposition enles us to consider the visul ngles on the visul gloe s in Figure 3 Note tht two gret circles A B nd C D pss through X nd X In the sme wy, two gret circles B C nd D A pss through Y nd Y see, Figure 3) Let α A' XY B' X ) Y, β B' Y X ) C' Y ) X ), γ C' X ) Y ) D' X Y ), δ D' Y ) X A' YX Figure 3 Orthogonl projection of sphericl qudrngle from the North Pole

Appling the sphericl cosine lw for ngles to the sphericl tringle A XY, cos A' cosα cosδ sinα sinδ cos XY Since XY /, cos A ' cosα cosδ In the sme wy, the following four equtions re derived in totl: cos A' cosα cosδ, cos B' cos β cosα, cosc' cosγ cos β, cos D' cosδ cosγ Now the eqution cos A 'cosc' cos B'cos D' is trivil This completes the proof Remrk 3 The sum of four visul ngles of rectngle represents the rtio of occupied field of vision to the visul gloe, tht is [3] pp78-79, [4] p5): the field of vision of rectngle ABCD) A ' B' C' D' For exmple, in the cse of huge rectngle, ll visul ngles re nerly equl to, so the field of vision is lmost, tht is, the re of the hemisphere On the other hnd, in the cse tht rectngle is very smll for the oserver, the sum of four visul ngles is nerly equl to, so the field of vision is nerly equl to 0 4 Reliztion of Visul Angle in the Plne In this section, we will introduce simple method how to relize visul ngle in the plne We cnnot mesure the exct visul ngle y the orthogonl projected imge s in Figure 3 To relize the exct visul ngle, we use the stereogrphic projection from the South Pole to the xyplne This projection hs very importnt properties: conformlity nd circle-to-circle correspondence As in Figure 4, four projected gret circles re circles rcs) pssing through vnishing points X, X, Y nd Y on the equtor These four vnishing points re fixed under the stereogrphic projection Drwing ritrry four rcs pssing through four points X, X, Y nd Y, we cn esily check the reltion cos A 'cosc' cos B'cos D' y mesuring the four ngles A, B, C nd D in the plne Figure 4 Stereogrphic projection of sphericl qudrngle

5 Geodesic Rectngle in Hyperolic Geometry In this section, we will try to pply the result ove to nother geometry, hyperolic geometry To do this, let us introduce n ide of geodesic rectngle Definition 5Geodesic Rectngle) A qudrngle ABCD is clled Geodesic Rectngle if nd only if there re two orthogonl geodesics g nd g such tht AB g, CD g, BC g nd DA g Remrk 5 In Eucliden geometry, rectngle is geodesic rectngle, however, two orthogonl geodesics re not unique The following theorem chrcterizes geodesic rectngle in sphericl geometry Theorem 5Geodesic Rectngle in S ) Let A, B, C nd D e four non-colliner points in S A qudrngle ABCD in S is geodesic rectngle if nd only if cos AcosC cos B cos D Proof If qudrngle ABCD in S is geodesic rectngle, we cn set up system of coordintes with the intersection of orthogonl geodesics t the North Pole s in Figure 3 Using the sme technique in the proof of Theorem 3, it is esy to show tht geodesic rectngle stisfies cos AcosC cos B cos D To prove the converse, let us show tht there exists two orthogonl geodesics s in Definition 5 Let P e the nerer point to D of AB CD nd Q the nerer point to D of BC DA First, let us show PQ / In fct, y the lw of cosines for ngles, cos A cosα cosγ sinα sin γ cosl, cos B cosα cosδ sinα sinδ cosl, cosc cos β cosδ sin β sinδ cosl, cos D cos β cosγ sin β sin γ cosl, where α APQ, β DPQ, γ DQP, δ CQP nd l PQ The eqution cos AcosC cos B cos D nd direct computtion imply tht cos l sin α β )sin δ γ ) 0, tht is PQ / Tke point R s one of poles of the geodesic PQ, then the sphericl tringle PQR is right regulr sphericl tringle Regrding Q s the North pole nd PR s the equtor, the longitude DA pssing through Q is perpendiculr to the equtor PR In this wy, it is found tht PR nd QR re two orthogonl geodesics of the qudrngle ABCD This completes the proof Now, let us consider geodesic rectngle in hyperolic geometry H Here just recll the following proposition y Lmert without proof see, [] pp 56-57) Proposition 5Lmert qudrilterl) Let OABC e qudrngle in H If O A C / Figure 5), then cos B sinh OA sinh OC Remrk 5 In sphericl geometry, there is similr eqution Let OABC e qudrngle in the unit sphere S If O A C /, then cos B sin OA sin OC This fct is directly derived from the proof of Theorem 3

Figure 5 Lmert qudrilterl Theorem 5Geodesic Rectngle in H ) If qudrngle ABCD in H is geodesic rectngle, then four ngles stisfy the following simple reltion: cos AcosC cos B cos D Proof We cn see tht geodesic rectngle is composed of four Lmert qudrilterls s in Figure 5 Let P AB g, Q BC g, R CD g, S DA g nd O g g Applying Proposition 5 to ech Lmert qudrilterl, one hs cos A sinh OS sinh OP, cos B sinh OPsinh OQ, cosc sinh OQsinh OR, cos D sinh ORsinh OS Now the eqution cos AcosC cos B cos D is trivil This completes the proof Figure 5 Geodesic rectngle in the Poincre model In this wy, the simple ide of visul ngle leds us to not only sphericl geometry ut lso hyperolic geometry In ddition, it turns out tht sphericl nd hyperolic geometries re connected y the ide of geodesic rectngles At the end of this pper, let us introduce few interesting properties etween sphericl nd hyperolic geometries from the spect of complementry ngle

Exmple 5Complementry Angles in the Circle) Let A e n ritrry point in the unit circle s in Figure 53 First, regrding this point A s stereogrphic projected point on the unit sphere S, drw two rcs XA X) nd YA Y) where X,0), X,0), Y0,) nd Y0, ) Let α e the ngle etween the rcs XA nd YA In the next, regrding the point A s point in the Poincre model of H, drw two geodesics AP nd AQ perpendiculr to X X) nd Y Y), respectively Let β e the ngle etween the rcs AP nd AQ Then two ngles α nd β lwys stisfy tht α β One cn esily check this property y construction Figure 53 Complementry ngles in sphericl nd hyperolic geometries Exmple 5Complementry Geodesic Rectngles) For n ritrry geodesic rectngle ABCD in S, there exists geodesic rectngle ABCD in H such tht AABBCCDD s in Figure 54 The res of these two rectngles re equl In fct, Figure 54 Complementry geodesic rectngles

) ABCD re D C B A D C B A D C B A D C B A re The construction of complementry geodesic rectngle is little it complicted Figure 55left) shows prt of Figure 54 The rc AD is geodesic of S perpendiculr to x-xis one of orthogonl geodesics) t S On the other hnd, the rc AD is corresponding geodesic of H perpendiculr to x-xis t H The ide is to mke the point H in H from S in S such tht Note tht OS nd OH re mesured y sphericl nd hyperolic metrics, respectively If we identify R OS sinh OH sin with C, these metrics re given s [] p 9 nd p 60) z dz ds nd z dz ds Let S,0) nd H,0) in Eucliden coordintes Then, log, tn 0 0 z dz OH z dz OS therefore, sin OS nd sinh OH The eqution OH OS sinh sin implies tht Figure 55right) shows the construction of H from S :,0),,,,,0) 3 H P P P S Using this technique, we cn esily construct pir of complementry geodesics Figure 55 Construction of H in H from S in S

6 Conclusion In this pper, we hve proposed one of nturl pproches to not only sphericl geometry ut lso hyperolic geometry Visul ngle is the strt point Visul gloe is our view screen nd the visul ngle is mesured on this sphere Four visul ngles of rectngle in the spce hve simple reltion, which is extended to hyperolic geometry On the other hnd, stereogrphic projection enles us to construct the visul ngle on plne Here is simple question: wht is the men of visul ngle in hyperolic geometry? This nturl question is yet to e investigted We hve lredy found out few interesting things, complementry ngle nd complementry geodesic rectngle These things my give us clue for deep understnding nd further study of sphericl nd hyperolic geometries References [] Berdon, A 983) The Geometry of Discrete Groups Springer-Verlg New York [] Berdon, A 99) Itertion of Rtionl Functions Springer-Verlg New York [3] Berger, M 977) Geometry II Springer-Verlg Berlin Heidelerg [4] Jennings, G 994) Modern Geometry with Applictions Springer-Verlg New York [5] Med, Y 00) Viewing Cue nd Its Visul Angles Discrete nd Computtionl Geometry, JCDCG 00, LNCS 866) pp9-99 Springer) [6] Med, Y nd Mehr, H 00) Oserving n Angle from Vrious Viewpoints Discrete nd Computtionl Geometry, JCDCG 00, LNCS 866) pp00-03 Springer) [7] Mori, M nd Med, Y 005) Three Visul Angles of Three Dimensionl Orthogonl Axes nd Their Visuliztion Proceeding of the Tenth Asin Technology Conference in Mthemtics, ATCM005, pp 35-3