{ } λ are real THE COSINE RULE II FOR A SPHERICAL TRIANGLE ON THE DUAL UNIT SPHERE S % 2 1. INTRODUCTION 2. DUAL NUMBERS AND DUAL VECTORS

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1 Mthemticl nd Computtionl Applictions, Vol 10, No, pp 1-20, 2005 Assocition for Scientific Reserch THE COSINE RULE II FOR A SPHERICAL TRIANGLE ON THE DUAL UNIT SPHERE S % 2 M Kzz 1, H H Uğurlu 2 nd A Özdemir 1 1 Cell Byr Üniversitesi Fen Edebiyt Fkültesi Mtemtik Bölümü, Mnis, Türkiye 2 Gzi Üniversitesi, Gzi Eğitim Fkültesi, Ort Öğretim Fen ve Mtemtik Alnlr Eğitimi Bölümü, Ankr, Türkiye mustfkzz@byredutr, hugurlu@gziedutr, ozdemir@hotmilcom Abstrct- In this work, we proved the Cosine Rule II for sphericl tringle on the dul unit sphere S 2 % Keywords- Cosine Rule II, dul unit sphere, dul sphericl tringle 1 INTRODUCTION Dul numbers hd been introduced by W K Clifford ( s tool for his geometricl investigtions After him, E Study used dul numbers nd dul vectors in his reserch on line geometry nd kinemtics (see [] He devoted specil ttention to the representtion of directed line by dul vectors nd defined the mpping tht is sid with his nme There exists one to one correspondence between the vectors of dul unit sphere S % 2 nd the directed lines of the spce R of lines (EStudy s mpping In plne geometry it is studied points, lines, tringles, etc On the sphere, there re points, but there re no stright lines, t lest not in the usul sense However, stright lines in the plne re chrcterized by the fct tht they re the shortest pths between points The curves on the sphere with sme property re the gret circles Thus it is nturl to use gret circles s replcements for lines The Sine Rule I nd Cosine Rule I for the dul nd rel sphericl trigonometries hve been well known for long time (see, [1], [4], [5] In this study, we prove the Cosine Rule II for sphericl tringle on the dul unit sphere S % 2 2 DUAL NUMBERS AND DUAL VECTORS Definition 21 A dul number hs the form ˆ λ : λ+ ελ, where λ nd λ re rel numbers nd ε stnds for the dul unit which is subject to the rules: 2 ε 0, ε 0, 0ε ε0 0, 1ε ε1 ε We denote the set of dul numbers by D : D ˆ λ λ+ ελ λ, λ R, ε 2 0 { } Equlity, ddition nd multipliction re defined in D by

2 14 M Kzz, H H Uğurlu nd A Özdemir λ+ ελ β + εβ if nd only if λ β nd λ β, nd ( λ ελ ( β εβ ( λ β ε( λ β , ( λ+ ελ ( β + εβ λβ + ε( λβ + λ β, respectively Then it is esy to show tht ( D, +, is commuttive ring with unity The numbers ελ ( λ R re divisors of 0 We note tht if λ nd β re two nonzero elements of ring R such tht λβ 0, then λ nd β re divisors of 0 (or 0 divisors Moreover, if given by ˆ λ λ+ ελ, ˆ β β+ εβ D with β 0 then the division is ˆ λ λ+ ελ λ λ λβ + ε( ˆ 2 β β + εβ β β β Now let f be differentible function Then the Mclurin series generted by f is f ( x ˆ f ( x+ ε x f ( x + ε x f ( x, where f ( x is the derivtive of f Then we hve Cos ( x+ ε x Cos x ε x Sin x, (1 Sin( x+ ε x Sin x+ ε x Cos x, (2 x + +, ( > 0 ( 2 x x ε x x ε x The norm of ˆx of dul number ˆx x+ ε x defined by ˆx x x xˆ x 2 xx + ε + ε Then the formul ( llows us to write ˆx x+ ε x x + ε x ( x 0 x x Thus we hve ˆx, if ˆx > 0 ˆx 0, if xˆ 0 ˆx, if xˆ < 0

3 The Cosine Rule II for Sphericl Tringle on the Dul Unit Sphere S % 2 15 Let D be the set of ll triples of dul numbers: D { % (,, i D, i 1, 2, } 1 2 The elements of D re clled dul vectors A dul vector ~ my be expressed in the form % +ε, where nd re the vectors of R Now let %, +ε b % b+ εb D nd ˆλ λ + ελ D then we define % + b % + b + ε( + b, ˆ λ% λ + ε ( λ + λ 1 Then D becomes unitry D -module with these opertions It is clled D -module or dul spce by The inner product of two dul vectors % +ε, b % b+ εb D is defined < % % >< >+ ε < >+< >, b, b (, b, b, where <, b> is the known inner product of the vectors nd b in the -dimentionl vector spce R The cross product of two dul vectors % +ε nd b % b+ εb D is defined by % % + ε + b b ( b b, where b is the known cross product in R Lemm 22 Let, % b, % c, % d % D Then we hve < % b, % c % > det(, % b %, c %, % b % b %, % ( % b % c % < %, c % > b % +< b, % c % >, % < % b, % c % d% > <, % c % >< b, % d% >+<, % d% >< b, % c % >, < % b, % % > 0, nd < % b, % b % > 0

4 16 M Kzz, H H Uğurlu nd A Özdemir % Definition 2 Let +ε D Then ~ is sid to be dul unit vector if the vectors nd stisfy the following equtions <,> 1, <, > 0 The set of ll dul unit vectors is clled the dul unit sphere, nd is denoted by % (for more detils, see [], [5] S 2 Theorem 24 (E Study s Mpping The dul unit vectors of the dul unit sphere S % 2 re in one to one correspondence with the directed lines of the -spce R lines [] THE COSINE RULE II FOR A DUAL SPHERICAL TRIANGLE In this section we prove the Cosine Rule II for dul sphericl tringles Let A, % nd C ~ be three points on dul unit sphere S % 2 given by the linerly independent dul unit vectors % +ε, b % b+εb nd c % c+εc, respectively These points together with the gret-circle-rcs AB, % % BC, % % CA %% form dul sphericl tringle ABC % % % see Figure 1 We will suppose tht det(, b, c > 0 We denote the dul unit vectors with the sme sense s b % c, % c % % nd % b % by n %, n % b nd n ~ c, respectively The side ~ of ABC % % % is defined s the dul ngle for which < b, % c % > Cos, % b % c % n % Sin % It cn be given similr definitions for the other sides b ~ nd c ~ of <, % b % > Cos c, % % b % n % c Sin c, % < c, % % > Cos b, % c % % n % Sin b % % c % b ABC % % % Thus we hve Since we cn write n % n +ε n, it is obvious tht n is the rel unit vector hving the sme sense s b c If % +ε, we hve Sin > 0 This mens tht Sin % Sin %, nd similrly Sinb % Sinb %, Sinc % Sin c % It is obvious tht, % b % nd c ~ re the dul unit vectors hving the sme sense s n % b n % c, n % c n % nd n % n % b, respectively

5 The Cosine Rule II for Sphericl Tringle on the Dul Unit Sphere S % 2 17 Definition 1 The ngle α% of The ngles β ~ nd γ ~ of ABC % % % is defined s the dul ngle given by < n %, n % > Cos α%, n % n % % Sinα% b c b c ABC % % % cn be defined similrly (for detils, see [5] S % 2 Figure 1: Dul Sphericl Tringle Lemm 2 Let ABC % % % be sphericl tringle on the dul unit sphere S % 2 Then the Sine Rule nd Cosine Rule I re given by respectively [4] Sin % α Sin % β Sin % γ, Sin % Sinb% Sinc% Cos % Cosb% Cos c% Cos % α Sinb% Sin c% (4, (5 In the sme wy, we obtin Cosine Rule I for β ~ nd γ ~ s follows: Cosb% Cos Cosc Cos % % % β, Sin % Sinc% Cosc% Cos% Cosb% Cos % γ Sin % Sinb% (6 (7 Using the equtions (1-(2 we obtin the following corollries:

6 18 M Kzz, H H Uğurlu nd A Özdemir Corollry The rel nd dul prt of the formul (4 re given by Sinα Sinβ Sinγ, (8 Sin Sinb Sin c Cos Sin Cos Sin Cos Sin α α Cot α β β b Cot b β γ γ c Cot c γ, Sin Sin Sinb Sinb Sin c Sinc respectively In corollry, the rel prt is known s the Sine Rule for sphericl tringle Corollry 4 The rel nd dul prts of the formuls (5, (6 nd (7 re given by Cos Cos bcos c Sin Cosα, Sinα ( b Cosγ + c Cosβ ; Sinb Sinc α Sinb Sinc Cos b Cos Cos c Sinb Cosβ, Sinβ ( Cosγ + c Cosα b ; Sin Sinc β Sin Sinc Cos c Cos Cosb Sinc Cosγ, Sinγ ( Cosγ + b Cosβ c, Sin Sinb γ Sin Sinb respectively In corollry 4, the rel prts give the Cosine Rule I for sphericl tringle Now we stte nd prove the correspondence of Cosine Rule II for hyperbolic sphericl trigonometry given in [1] Lemm 5 (The Dul Cosine Rule II Let ABC % % % be sphericl tringle on the dul unit sphere S % 2 Then the Cosine Rule II is given by Cos % α Cos % β+ Cos % γ Cos c% Sin % α Sin % β (9 Proof: For brevity, let Cosine Rule I yields ~ A, B ~ nd C ~ be Cos %, Cosb % nd Cos c%, respectively Then the Cos % α A% BC % % 1 ( ( 1 1 1, (10

7 The Cosine Rule II for Sphericl Tringle on the Dul Unit Sphere S % 2 19 Cos % β Cos % γ AC % % ( 1 A% ( 1 AB % % ( 1 A% ( 1, (11 (12 Since Cos % α+ Sin % α 1, it follows tht 2 Sin % α ( 1 ( 1, 2 where 1+ 2ABC % % % A% We note tht D ~ is positive nd symmetric in ~ A, B ~ nd C ~ Then we obtin Sin % α Sin % β Sin % γ 1 ( ( ( 1 A% ( 1 ( 1 A% ( 1, (1, (14 (15 If we write the formuls (10-(15 in the right side of the formul (9, then the equlity is stisfied: ( A% BC % %( AC % % + ( AB % %( 1 ( 1 A% ( 1 ( D ( 1 ( 1 A% ( 1 Cos % αcos % β+ Cos % γ Sin % αsin % β % AB % % A% + ABC % % % + AB % % + ABC % % % 2 ( 1+ 2ABC % % % A% Cos c% 2 2

8 20 M Kzz, H H Uğurlu nd A Özdemir In the sme wy, we cn give the similr formuls for Cos b % nd Cos % s follows: Cos b% Cos % Cos % γ Cos % α+ Cos % β ; Sin % αsin % γ Cos % βcos % γ + Cos % α Sin % β Sin % γ (16 (17 Using the equtions (1 nd (2, we obtin the following corollry: Corollry 6 The rel nd dul prts of the formuls (9, (16 nd (17 re given by CosαCosβ+ Cosγ Sinγ Cos c, Sinc ( β Cos + α Cos b+ γ ; α β SinαSinβ c Sin Sin Cosγ Cosα+ Cosβ Sinβ Cos b, Sinb ( α Cos c+ γ Cos + β ; γ α α γ Sin Sin b Sin Sin CosβCosγ + Cosα Sinα Cos, Sin ( γ Cos b+ β Cos c+ α, SinβSinγ SinβSinγ respectively In Corollry 6, the rel prts re known s the Cosine Rule II for sphericl tringle REFERENCES 1 A F Berdon, The Geometry of Discrete Groups, Springer-Verlg, New York, Berlin, I N Bronstein; K A Semendjjew; G Musiol nd H Mühlig, Tshenbuch der Mthemtik, Verlg Hrri Deutsch, Frnkfurt, 1995 E Study, Geometrie der Dynmen, Leipzig, HH Uğurlu nd H Gündoğn, The Cosine Hyperbolic nd Sine Hyperbolic Rules for Dul Hyperbolic Sphericl Trigonometry, Mthemticl nd Computtionl Applictions 5(, , G R Veldkmp, On the Use of Dul Numbers, Vectors nd Mtrices in Instntneous, Sptil Kinemtics, Mechnism nd Mchine Theory 11, , 1975