MP93 MITRE PRODUCT Fining Minimm Vales of Distances an Distance Ratios in ATS Spectm Management D. Leone C. Monticone D. Rica E. Snow Fank Bo Decembe 8
Te contents of tis mateial eflect te views of te ato an/o te Diecto of te Cente fo Avance Aviation System Development (CAASD, an o not necessaily eflect te views of te Feeal Aviation Aministation (FAA o te Depatment of Tanspotation (DOT. Neite te FAA no te DOT makes any waanty o gaantee, o pomise, epesse o implie, concening te content o accacy of te views epesse eein. Tis is te copyigt wok of Te MITRE Copoation an was poce fo te U.S. Govenment ne Contact Nmbe DTFA--C- an is sbject to Feeal Aviation Aministation Acqisition Management System Clase 3.5-3, Rigts in Data-Geneal, Alt. III an Alt. IV (Oct. 996. No ote se ote tan tat gante to te U.S. Govenment, o to tose acting on bealf of te U.S. Govenment, ne tat Clase is atoie witot te epess witten pemission of Te MITRE Copoation. Fo fte infomation, please contact Te MITRE Copoation, Contact Office, 755 Colsie Dive, McLean, VA (73 983-6. 8 Te MITRE Copoation. Te Govenment etains a noneclsive, oyalty-fee igt to pblis o epoce tis ocment, o to allow otes to o so, fo Govenment Pposes Only.
MP93 MITRE PRODUCT Fining Minimm Vales of Distances an Distance Ratios in ATS Spectm Management Sponso: Te Feeal Aviation Aministation Dept. No.: F43 Poject No.: Poject Task Appove fo Pblic Release; Distibtion Unlimite. D. Leone C. Monticone D. Rica E. Snow Fank Bo 8 Te MITRE Copoation. All Rigts Reseve. Decembe 8
Abstact Peventing intefeence among aio cicits se fo ai taffic sevices (ATS often eqies spectm manages to obseve cannel-assignment les base on minimm atios of te geat-cicle istances (GCDs tavese by esie an nesie (potentially intefeing signals. Eac cicit opeates witin a sevice volme of aispace aving a cicla o polygonal footpint on te sface of a speical Eat. Minimiing nesie-to-esie GCD atios wit espect to tese footpints can be significantly facilitate in many cases tog te se of steeogapic pojection. Instea of pefoming te minimiation sing te footpints on te sface of te spee, a steeogapic pojection of te footpints to te comple plane is pefome to tansfom te oiginal minimiation poblem into a simple poblem of minimiing a atio of istances in te comple plane. Tis atio can be epesse in tems of a single eal vaiable an ten minimie sing te Newton-Rapson meto. Tee ae ote assignment les tat eqie spectm manages to obseve cannel assignment les base on minimm istances between sevice volme footpints. Fining te minimm GCD istance between sevice volme footpints on te spee can be one in an efficient manne sing vecto analysis, an as a eslt a close fom soltion can be povie enabling te comptation to be pefome qickly. Pocees escibe in tis pape ave been incopoate into Spectm Pospecto, an atomate tool evelope by Te MITRE Copoation s Cente fo Avance Aviation System Development (CAASD to pefom spectm analysis sties fo te Feeal Aviation Aministation (FAA. iii
Acknowlegements Te atos wol like to tank Kisty Lany an Angela Signoe fo tei elp wit ocment pepaation. iv
Table of Contents Intoction. Backgon. Appoac Use of Steeogapic Pojection to Fin Minimm Geat-Cicle Distance Ratios. Cocannel Intefeence. Steeogapic Pojection.. Steeogapic Pojection of Cicles an Polygons on te Spee.. Compting Coal Lengts.3 Optimiation of te / Ratio.3. Sowing tat GCD an Coal / Ratios Have te Same Minimm Soltion - 7.3.. Minimiing Ove te Entie Bonay of a Cicle.3.. Minimiing Ove an Ac.3. Poving tat Minimm / Ratio of GCDs Occs on te Bonay of te Desie SV.3.3 Poof tat te Minimm / Ratio of GCDs Occs on te Bonay of te Unesie SV - - - - - -3-4 -6-7 -7 - -4-6.3.3. Poof tat te Minimm / (Maimm / Distance Ratio of Coal Lengts Occs on te Bonay of te Unesie s-cicle -6.3.3. Poof tat te Minimm / (Maimm / Distance Ratio of Coal Lengts Occs on te Bonay of te Unesie s-polygon -8 3 Minimiing / Ratios fo Ai-to-Ai Commnications 3. Two s-cicles 3. s-cicle an s-polygon 3.. Desie s-polygon an Unesie s-cicle 3.. Desie s-cicle an Unesie s-polygon 3... Unesie Ac Lies on te Desie Containment Ac 3... Unesie Ac Etenal to te Desie Containment Ac 3...3 Unesie Ac Lies Patly on te Desie Containment Ac 3..3 Desie s-polygon an Unesie s-polygon 3.3 Pojection of a Sie of Desie SV is a Staigt Line 3-3-3 3-5 3-5 3-6 3-7 3-3- 3-3-3 v
4 Minimiing / Ratios fo Gon-to-Ai, Gon-to-Gon, an Ai-to-Gon 4. Gon-to-Ai 4. Ai-to-Gon 4.3 Gon-to-Gon 5 Minimiing Geat-Cicle Distance between s-objects 5. Minimiing Geat-Cicle Distance between Two s-cicles 5. Minimiing Geat-Cicle Distance Between Two s-polygons an Between an s-cicle an an s-polygon 5-5.3 Point-to-Ac Algoitm 6 Conclsions 6- List of Refeences Appeni A Genealiation of / Ratio to Consie Vetical Distances Appeni B Spee Eqations of Cicles in te Comple Plane tat ae Pojections of Cicles on te B- Appeni C Fining Tangent Points of Sies of Cental Angle Foming Desie Containment Ac Appeni D Rotation of Points on te Spee to te Sot Pole Appeni E Fining Enpoints on Unesie Cicle of Desie Containment Ac Appeni F Ientifying Intesections of te Sies of te Unesie Ac s Cental Angle wit te Desie Cicle F- Appeni G Glossay 4-4- 4-4-5 5-5- 5-5 R- A- C- D- E- G- vi
List of Figes Fige -. Components of / GCD Ratio Fige -. Cocannel Intefeence between A/G Raio Cicits Fige -. Secto Footpints an GCDs Fige -3. Components of Wost-Case / GCD Ratio Fige -4. Steeogapic Pojection Fige -5. Steeogapic Pojection of a Geat Cicle Ac Fige -6. Repesentation of an s-polygon in te Comple Plane Fige -7. Minimiing Ratio of Coal Lengts Fige 3-. Desie Containment Ac Fige 3-. Configation fo Ai-to-Ai / Ratio Minimiation (Two s-cicles Fige 3-3. Configation fo Ai-to-Ai / Ratio Minimiation (Desie s-polygon, Unesie s-cicle 3-6 Fige 3-4. Configation fo Ai-to-Ai / Ratio Minimiation (Desie s-cicle, Unesie s- Polygon Wit Unesie Ac Intesections 3-8 Fige 3-5. Configation fo Ai-to-Ai / Ratio Minimiation (Desie s-cicle, Unesie s- Polygon Wit Collineaity Conition Ientification 3-9 Fige 3-6. Unesie Ac Does Not Lie on Desie Containment Ac Fige 3-7. Unesie Ac Patially Lies on Desie Containment Ac an Patially Otsie It 3- Fige 3-8. Desie Cental Angle an Unesie Cental Angle Ovelap Fige 3-9. Sie of Desie s-polygon Pojects as Staigt Line Segment on Line Intesecting Unesie Ac 3-4 Fige 3-. Sie of Desie s-polygon Pojects as Staigt Line Segment on Line Not Intesecting Unesie Ac Fige 4-. Gon-to-Ai Case Fige 4-. Footpint an Steeogapic Pojection fo Gon-to-Ai Case Fige 4-3. Ai-to-Gon Case Fige 4-4. Footpint an Steeogapic Pojection fo Ai-to-Gon Case Fige 4-5. Gon-to-Gon Case Fige 4-6. Footpint an Steeogapic Pojection fo Gon-to-Gon Case Fige 5-. GCD Between Two s-cicles - - - - -4-5 -6-9 3-3-4 3-3-3 3-6 4-4- 4-4 4-4 4-5 4-6 5- vii
Fige 5-. Intesecting Geat Cicles Fige 5-3. Sotest GCD between Acs of Geat Cicles Fige 5-4. Minimm GCD Occs at Enpoints Fige 5-5. Constct fo Fining Minimm GCD fom Fie Point to Sie of s-polygon Fige A-. Slant Ranges Fige B-. s Cicle Define by Spee an Intesecting Plane Fige B-. Ientifying Tee Points on Small s-cicle Fige C-. Desie Containment Ac an Points of Tangency on Desie Cicle Fige D-. Fining te Speical Cente of a Geat Cicle Fige E-. Enpoints of Desie Containment Ac Fige F-. Intesections on Desie Cicle 5-3 5-4 5-4 5-5 A- B- B-3 C- D- E- F- viii
Intoction Te Feeal Aviation Aministation (FAA as appoimately 53 vey ig feqency (VHF aio cannels available fo assignment to an ai taffic sevices (ATS aio system compising moe tan 7, ai/gon (A/G aio cicits. Eac cicit opeates witin a sevice volme (SV of aispace aving a cicla o polygonal footpint on te sface of a speical Eat. Etensive cannel ese is essential to satisfying sc a lage eman wit te esoces available. Howeve, peventing intefeence among te cicits often eqies spectm manages to obseve cannel-assignment les base on minimm atios of te geat-cicle istances (GCDs tavese by esie an nesie (potentially intefeing signals. Sc atios ae calle nesie-to-esie (/ GCD atios. In Fige -, G an G ae te GCDs (mease on te sface of te Eat se as meases of te istances tat te esie an nesie signals, espectively, mst tavese. G Desie Signal G Gon Raio Unesie Signal Desie Cicit s SV Footpint Unesie Cicit s SV Footpint Fige -. Components of / GCD Ratio Minimiing / GCD atios wit espect to tese footpints can be significantly facilitate tog te se of steeogapic pojection. Instea of pefoming te minimiation sing te footpints on te sface of te spee, a steeogapic pojection of te footpints to te comple plane is pefome to tansfom te oiginal minimiation poblem into a simple poblem of minimiing a atio of istances in te comple plane. Tis atio can be epesse in tems of a single eal vaiable an ten minimie sing te Newton-Rapson meto. Tee ae ote assignment les tat eqie spectm manages to consie assignments base on minimm GCD istances between sevice volme footpints. Fining te minimm GCD istance between sevice volme footpints on te spee can be one in an efficient manne sing vecto analysis, an as a eslt a close fom soltion can be povie enabling te comptation to be pefome qickly. -
Pocees escibe in tis epot ave been incopoate into Spectm Pospecto, an atomate tool evelope by Te MITRE Copoation s Cente fo Avance Aviation System Development (CAASD to pefom spectm analysis sties fo te Feeal Aviation Aministation (FAA [,,3,4]. Te optimiation pocees escibe in tis epot ae esigne to enable Spectm Pospecto to follow istance-base cannel assignment les as accately as possible witot ecessive nning time.. Backgon In te ATS envionment, pilots se aibone aios (ARs to commnicate wit tei esignate gon-base aios (GRs, wic ae se by ai taffic contolles. Eac cicit opeates witin a SV, a tee-imensional volme of aispace wose oiontal coss section at any altite as te same cicla o polygonal footpint on te sface of a speical Eat. Aicaft witin te same SV commnicate wit te same GR on a given cannel. Te GR oes not necessaily lie in te footpint of its SV. Becase spectm is a scace esoce, cannels ae ese, constaine by cetain stingent les tat mst be followe to pevent intefeence between te aios in one cicit an tose in ote cicits assigne te same cannel. Te poblem of etemining wete two cicits can be cocannel, i.e., assigne te same cannel, can be pose as an optimiation poblem involving cicles an polygons on te sface of a spee.. Appoac Becase te / atio as it stans is a complicate fnction (i.e., an invese tigonometic fnction of fo vaiables (te latites an longites of te two aicaft positions, attempting to fin a global minimm in a staigtfowa manne leas to a vey inefficient an cmbesome matematical pocee. Mc of te complication is e to te fact tat te atio is compise of GCDs on te sface of te spee. We ealie tat one possible simplifying step was to poject te components involve in te atio to te comple plane tog steeogapic pojection. Steeogapic pojection is a viable appoac becase cicles on te spee ae pojecte to cicles in te comple plane, allowing s to etain te simple stcte of te cicle, wic is ccial in simplifying te minimiation pocess. Fte, steeogapic pojection enables epessing te coal lengt between points on te spee in tems of istances between points in te comple plane. Ts, a atio of coal lengts can be epesse in tis way, an it tns ot, minimie in an efficient manne. Bt in tis epot we sow tat te soltion to te poblem of minimiing te atio of coal lengts is also te soltion to te actal poblem of minimiing te atio of GCDs. In tems of fining te minimm GCD between two SVs on te spee, we wee able to fin an efficient pocee sing vecto analysis to obtain a close-fom soltion to te poblem of fining te minimm GCD fom a point on te spee to an ac of a geat cicle. Using tis eslt an a feate of intesecting geat cicles, we wee able to fin an efficient pocee fo fining te minimm GCD between any two SVs on te spee. -
Use of Steeogapic Pojection to Fin Minimm Geat- Cicle Distance Ratios Tis section fist escibes te poblem of cocannel intefeence (CCI, wic gives ise to te nee fo te istance-atio calclations in tis epot. It ten escibes an efficient pocee fo fining minimm / GCD atios tog te se of steeogapic pojection.. Cocannel Intefeence Fige - sows an eample of a sitation wee CCI col aise between two A/G aio cicits opeating in neigboing SVs. In te eample, te esie cicit s AR eceives not only te esie signal fom its own GR, bt also a potentially intefeing nesie signal tansmitte by te AR of te nesie cicit. Tis constittes a case of potential ai-to-ai intefeence. Te esie cicit, of cose, can also intefee wit te nesie cicit, altog intefeence in tat iection is not sown in te fige. Ai-to-gon, gon-to-ai, o gon-to-gon intefeence migt also occ between te two cicits bt is not sown in te fige eite. Te altites of te aicaft above te eat ae sally small enog, in elation to te oiontal etent of te SV, tat te actal lengts of te signal pats ae vitally ientical to te associate GCDs. Weneve ealing wit SVs, only a oiontal slice at a given altite is consiee, an te bonay of te oiontal slice is pojecte to lie as a footpint on te spee as sown in Fige -. Any fte efeence to SVs will be to tei footpints. Also, all istances between any two points in te oiontal slices, egaless of altite, ae mease along te sface of te eat sing tei footpints as sown in Fige -. In Fige -, G an G ae te GCDs se as meases of te istances tat te esie an nesie signals mst tavese, espectively. Te cicits of Fige - ae close enog fo ARs opeating at tei eges at te maimm allowable altite to ave an nobstcte mtal aio line of sigt (RLOS, wic in ATS spectm management is geneally egae as an essential conition fo intefeence. Desie Cicit INTERFERENCE VICTIM UNDESIRED SIGNAL INTERFERENCE SOURCE Unesie Cicit DESIRED SIGNAL GROUND RADIO CONTROLLER GROUND RADIO CONTROLLER Fige -. Cocannel Intefeence between A/G Raio Cicits -
Fige -. Secto Footpints an GCDs Fo te poblem we ae consieing, it is often assme fo simplicity tat te esie an nesie aio signals ave ientical otpt powes, an tat fee-space popagation laws pevail ove bot te esie an te nesie signal pats. Une tese conitions, te les fo avoiing CCI can be epesse as istance-base cannel-assignment constaints. Pio to te evelopment of te pocee in tis epot, a wost-case / GCD atio was se, becase attempting to fin te etemm soltion was too comple an time-consming. Fige -3 sows te vales se to compte tis wost-case / atio. Te enominato is te GCD between te GR an te fatest point in te esie SV, an te nmeato te minimm GCD between te esie an nesie SVs. G, ma G, min is G, etm G,ma Gon Raio Desie Signal Desie Cicit s SV Footpint G, etm Unesie Signal G,min Unesie Cicit s SV Footpint Fige -3. Components of Wost-Case / GCD Ratio -
G Let be te minimm possible vale of te / atio, an let G, etm an G, etm be te G min vales of G an G tat poce tat etemm / atio. Ten we wol ave G G min G G, etm, etm G G,min,ma Ts, sing te wost-case atio wol be a lowe bon on te etemm atio an col eslt in a less efficient se of spectm. In te special case wen te GR is locate at te cente of a cicla esie SV an te nesie SV is also cicla, te minimm / atio is eqal to te wost-case atio, an it is easy to calclate. Fo cases ote tan tis special case, tis epot escibes an efficient pocee fo obtaining te te minimm / atio. In pesent FAA cannel-planning pactice, no pai of A/G cicits wit an nobstcte mtal G RLOS may opeate on te same cannel nless 5 fo tat pai i.e., nless te G min victim aio is at least 5. times as fa fom te nesie signal soce as fom te esie G tansmitte. Ts, mst be calclate to etemine its vale. G min Calclating tat atio on te spee fo eac of te millions of cicit pais tat eist in te U.S. ATS system can be a time-consming pocess. Tis epot escibes a moe efficient pocee tat tansfoms te optimiation poblem on te spee to an optimiation poblem in te comple plane tog te se of steeogapic pojection. Altog te eslts we povie in tis epot ae fo te atio of GCDs, Appeni A etens te eslts to te case wee SV altites ae incopoate wit te GCDs to poce a atio of slant anges.. Steeogapic Pojection Fige -4 illstates a steeogapic pojection, wic is a pojection fom points on te X, Y, Z on te spee wit ais R to a point iy in te comple plane ae [6, 7]: spee to points in te comple plane. Te pojection eqations tat tansfom a point ( R X R Z R y Y R Z ( -3
Te invese eqations fom te comple plane to te spee ae: X Y 4R 4R 4R 4R y ( Z R 4R (,y Fige -4. Steeogapic Pojection.. Steeogapic Pojection of Cicles an Polygons on te Spee It is known tat te steeogapic pojection maps cicles on te spee (s-cicles to cicles in te comple plane, an tat te inteio of an s-cicle is mappe to te inteio of its pojecte cicle in te comple plane. Appeni B sows te eivation of te eqation of a cicle in te comple plane tat is te steeogapic pojection of an s-cicle on te spee. Tese s-cicles can be eite geat s-cicles (i.e., tose wit cente at [,,R] o small s-cicles (i.e., tose wit centes elsewee. Pojections of geat cicles contain te oigin of te comple plane in tei inteios. We also nee to sow tat te inteios of polygons on te spee (s-polygons ae pojecte to te inteios of tei pojections in te comple plane. Since, in te cases we ae consieing -4
now, eac sie of an s-polygon is an ac segment of a geat cicle on te spee, te epesentation in te comple plane of a sie is an ac segment of a cicle. Tis is sown in Fige -5. An eception to te sitation wee te sie of an s-polygon is pojecte to an ac of a cicle in te comple plane is wen te sie lies along a meiian, i.e., wen te two vetices of te sie ave te same longite. In tis case te pojection of te s-polygon sie is a line segment. In tis epot we eal wit te moe pevalent case wee te sie of an s-polygon oes not lie along a meiian. Te concepts an pocees evelope ee also ave been aapte to te eceptional case. A geat cicle patitions a spee into two emispees. We efine te inteio of a geat cicle as te spee s soten emispee, an te eteio as its noten emispee. Fige -5 also sows tat any point in te inteio (eteio of a geat cicle is mappe ne steeogapic pojection to te inteio (eteio of te pojection of tat geat cicle in te comple plane. Teefoe, fo any point on te spee tat is insie an s-polygon, its elationsips (i.e., inteio o eteio wit espect to te geat cicles on wic te polygon sies lie ae maintaine wit espect to te pojecte cicles of tose geat cicles in te comple plane. Tis means tat te pojection of a point on te spee witin an s-polygon will be witin te cicla polygon (a polygon wose sies ae cicla ac segments tat is te pojection of te s-polygon (see Fige -6. Tat s-cicles an s-polygons ae mappe to cicles an cicla polygons an tat inteios ae mappe to inteios ae eslts enabling te application of te Maimm Mols Pinciple, wic we will enconte late. N(,,R Geat Cicle Ac Segment Pojecte Cicla Ac Segment Fige -5. Steeogapic Pojection of a Geat Cicle Ac -5
Comple Plane Fige -6. Repesentation of an s-polygon in te Comple Plane.. Compting Coal Lengts Usefl to o optimiation pocee is te comptation of coal lengt (D, wic fo any, Y Z X, Y Z on te spee is efine as two points ( X an (,, ( X X ( Y Y ( Z D (3 Z It can be sown tat D can be compte in te comple plane, sing te steeogapic, Y Z X, Y Z, as [6, 7] pojections of te points ( X an (,, D 4R (4 4R 4R Tis eqation will be sefl in minimiing te / atio in an efficient manne. -6
.3 Optimiation of te / Ratio Woking wit atios of coal lengts ate tan atios of GCDs enables s to evelop a moe efficient minimiation pocee. Te fist step of o stategy is to sow te eqivalence of minimiing a / atio of GCDs an its coesponing atio of coal lengts. Te secon step is to sow tat te minimm soltion lies on te bonaies of te esie an nesie SVs..3. Sowing tat GCD an Coal / Ratios Have te Same Minimm Soltion In te applications we iscss late, te / atio is eqie to be minimie ove te entie bonay of a cicle, o jst ove an ac of a cicle. Tese two cases ae ealt wit sepaately to sow tat te / GCD atio an te coal lengt / atio ave te same minimm soltion in eac case..3.. Minimiing Ove te Entie Bonay of a Cicle Let G ( P, P y be te GCD between two points P an solve is: Py on te spee. Te poblem we wis to G Min ( P, P ( P P P, P G, (5 wee ( X, Y Z, P ( X, Y, Z, P ( X, Y, Z P, P is on te bonay of te nesie SV, P is on te bonay of te esie SV, an P on te P is te fie location of te esie gon aio. Tat te soltion of (5 occs fo bonay of te esie SV an fo Let D ( P, P an ( P on te bonay of te nesie SV will be sown late. D P, P be te nesie an esie coal lengts, espectively. We will sow tat te soltion to te minimiation poblem: D Min ( P, P ( P P P, P D, (6 is also te soltion to te minimiation poblem (5. Tat (6 implies (5 means tat if te atio P P, ten te atio of coesponing GCDs is of coal lengts is minimie at some point (, also minimie at ( P P., -7
We se te following elationsip between coal lengt an GCD. Let S be te lengt of te co connecting te enpoints of an ac on te spee of ais R wit cental angleα, so α S Rsin. Te lengt of te coesponing ac, i.e., te GCD between te enpoints of te ac, is S Rα R sin. Te atio of GCDs can be epesse as: R G G ( P, P ( P, P R sin R sin D D ( P, P R ( P, P R sin sin D D ( P, P R ( P, P R (7 Witot loss of geneality, we can assme tat te spee as nit ais an tat te poblem can be simplifie by otating te nesie s-cicle so tat its cente on te spee is at te Sot Pole, sing te eqations in Appeni D. Ten its steeogapic pojection as its cente at te oigin of te comple plane. Using te same otation eqations, te esie SV an te esie GR location ae otate so tat te nesie SV an te esie SV an its GR maintain te same geogapical elationsips as befoe te otations an ten ae steeogapically pojecte. Te steeogapic pojections of te nesie an esie s-cicles o s-polygons ae efee to as te nesie an esie cicles o polygons, espectively. Te stategy to sow tat te same pai of points minimies bot atios is to sow tat fo eac fie point in o on te esie pojecte s-cicle o geat cicle, te point on te bonay of te nesie pojecte s-cicle o geat cicle tat minimies te atio of coal lengts also minimies te atio of GCDs. Using (4 te atio of coal lengts can be witten as D D ( P 4 4 4, P ( P, P 4 4 4 4 4 (8 Now we fi at any point in o on te esie cicle, bt otsie an abitaily small isc of ais ε centee at. Te eason fo ecling tis isc is tat te atio appoaces infinity as appoaces. Teefoe, te minimm fo te atio wol occ otsie tis isc. Assme tat is on an abitay cicle c of ais containe witin te nesie cicle, -8
as sown in Fige -7. Tis abitay cicle wol be te steeogapic pojection of an abitay s-cicle sc containe witin te nesie SV on te spee. ( cos( θ isin( θ Ten, R, wee R is te ais of te nesie cicle. Since an ae fie, an since is constant, te atio simplifies a geat eal to (, P K (, P D P D P (9 4 wee K is a positive constant eqal to 4 Desie Comple Plane Abitay Cicle ( sc Unesie θ R (, P ( y ( y P, P, P ε ( y ( y,, Fo eac fie P, P minimies te nesie-to-esie atio of coallengts Fige -7. Minimiing Ratio of Coal Lengts Denotes steeogapic pojection -9
Te atio of coal lengts can ten be epesse in tems of te single vaiable θ as ( ( θ, P D P (, P D P ( θ y sin( θ K K y cos ( An becase is a constant, te atio of coal lengts is simply a constant times a fnction of θ fo fie P, i.e., ( ( θ, P D P (, P D P ( θ K K f ( Te atio of GCDs can be epesse sing (8 an applying (4 to te coal lengt in te nmeato as ( ( θ, P D P sin D P K sin 3 D( P, P sin sin K K 3 y cos( θ sin( θ y ( ( θ, P K sin ( K ( ( wee K an K 3 ae positive constants eqal to (, D P sin P K, K 3 4 4 (3 Tis sows tat if te atio of coal lengts is ρ ( θ K f ( θ ρ ( θ K sin ( K f ( θ GCD 3 cl, ten te atio of GCDs is -
θ θ θ Ts, ρ ( θ K f ( θ, wen f ( θ θ Also, ρ ( θ cl. GCD K K 3 f θ K f 3 ( θ ( θ θ f wen ( θ ρ GCD is also a minimm. Tis is a eslt of applying te fist eivative test, an te fact tat K, K, an K 3 If te eteme point of ρ cl ( θ is a minimm, ten te eteme point of ( θ ae positive constants. Teefoe te vale of θ tat minimies te / atio of coal lengts on an abitay cicle of ais witin te nesie cicle also minimies te / atio of GCDs on tat abitay cicle. Tis vale of θ epens on an on an abitay cicle of ais. We can wite it as θ ( P, sppessing te notation fo its epenence on te abitay s-cicle. Since te same vale of θ ( P minimies ρ cl ( θ an ρ GCD ( θ an since P is an abitay point, we can say tat D( P, P Min an G( P, P Min ave te same soltion fo P, P D( P, P P, P G( P, P P in o on te esie s-cicle an P on te nesie abitay s-cicle sc. Becase te cicle witin te nesie cicle is abitay, we can say tat D( P, P Min an G( P, P Min P, P D( P, P P, P G( P, P ave te same soltion fo any points in o on te esie an nesie s-cicles.. P.3.. Minimiing Ove an Ac Now we fin te minimm / atio ove an ac of a cicle, instea of te wole cicle as peviosly. Tis aises wen te nesie SV is an s-polygon. Ts, let s consie tat te esie SV is an s-cicle as peviosly an tat te nesie SV is now an s-polygon. Recall tat eac sie of an s-polygon is an ac of a geat cicle on te spee, an its pojection is te ac of a cicle in te comple plane. We pefom te otations escibe peviosly fo an entie cicle so te geat cicle is otate to te eqato, an its pojection is a cicle centee at te oigin of te comple plane. In tis case te soltion of ( θ θ f, as iscsse peviosly fo te entie cicle, may fall otsie of te nesie ac in te comple plane. Une tis conition, we want to sow tat D( P, P G( P, P Min an Min ave te same soltion, since tis wol allow s to P ε Ac, P D( P, P P ε Ac, P G( P, P solve te simple poblem of minimiing te atio of coal lengts. -
To pove tis we se te following eslt, epicte in Fige -8, tat fom any point etenal to a cicle in te comple plane, te closest point on te bonay of tat cicle to te etenal point is te intesection wit te cicle of a line connecting te etenal point an te cente of te cicle. Ts, te eslt is tat Min ( θ, ( θ θ η wee ( θ an. Tis latte eqality epesses te conition tat, an te cente, of te cicle ae collinea. Te soltion tat minimies te istance will also minimie te ρ θ K θ fo te same ρ cl θ is a positive atio of coal lengts cl ( ( constant mltiple K of te istance. In aition, ( ρ GCD ( θ K ( θ e to te fact tat ( θ K sin 3, fo te same ρ GCD inceases as K3 ( θ since ( will also minimie te atio of GCDs, wee K an K3 ae positive constants, inceases. Comple Plane Etenal Point Closest Point to Etenal Point Fige -8. Closest Point on Cicle to Etenal Point Anote vey sefl eslt is tat te istance fom te cicle to te etenal point monotonically inceases as a point ( θ on te cicle moves away in eite iection fom ( θ, te closest point to te etenal point. Tis also implies tat bot / atios, ρ cl ( θ an ρ GCD ( θ, monotonically incease as θ moves away fom θ. -
Tis eslt sffices to pove tat if te ac on te nesie cicle oes not contain te location of te global minimm soltion ove all θ, θ π fo a fie P, ten it mst occ at one of te vetices, an it mst be te vete tat is neaest te location of te global minimm soltion. In Fige -9, te minimm occs at P vete,. Comple Plane Desie Unesie Fin minimm / atio on encicle ac minimies te nesie-to-esie atio of coal lengts ove minimies te nesie-to-esie atio of coal lengts ove ac Fige -9. Minimm Distance Ratio Ove an Ac Denotes steeogapic pojection Note tat te above poof sows tat te same point optimies te two atios consiee above in o on te bonay of te esie cicle. Tis is impotant since, wen we sow tat te optimal soltion fo te atio of coal lengts mst occ on te bonay of te esie cicle, it mst also be te of te atio of GCDs. -3
.3. Poving tat Minimm / Ratio of GCDs Occs on te Bonay of te Desie SV Now we pove tat te optimal soltion mst occ on te bonay of te esie SV. In all of te poofs above we ave assme tat P can occ anywee in o on te bonay of te esie SV. We now pove tat it mst occ on te bonay of te esie SV fo a fie location on te bonay of te nesie SV. (It sffices to consie a point on te bonay of te nesie SV e to Section.3.3. Becase we sowe in te pevios section te eqivalence of minimiing te / atio of GCDs an coal lengts, we will pove te eslt fo a atio of coal lengts, wic is a mc easie task. To begin, we steeogapically poject te SVs ne consieation to te comple plane. Using te eslt of Section.3., we nee only consie te following atio of coal lengts sing (4: D D ( P, P ( P, P 4R 4R 4R 4R 4R 4R (4 4R 4R Since te tem 4R is a constant e to being te fie location of te esie aio, tis implies tat if we wis to minimie te / atio ove a egion in te comple plane, we sol solve te poblem Min (5, R 4 Fo a fie point of te pojecte nesie SV, we ave to etemine if Min 4R (6-4
occs on te bonay of te pojecte esie SV in te comple plane, wic wol geatly simplify te poblem. Te vaiable mst be consiee ove te bonay an inteio of te pojecte esie s-polygon o s-cicle. It wol be convenient to se te Minimm Mols Pinciple [8] to sow tat te minimm occs on te bonay an not in te inteio, bt te Minimm Mols Pinciple applies only to analytic fnctions (i.e., iffeentiable fnctions of a comple vaiable. Te fnction to be minimie is not analytic becase te aicaft can be anywee in te esie s-polygon o s-cicle, an ts it can be iectly ove te esie aio site, wic means tat, te enominato, can be eo if te aio site is witin te SV. To esolve tis poblem, we can invet te epession an fin Ma 4R (7 Since becomes 4R is a positive constant, we enote it by K so tat te maimiation poblem Ma K K Ma (8 an ts K can be effectively ignoe in te maimiation. Te fnction g( (9 is an analytic fnction in an on te pojection of te esie s-polygon o s-cicle. Teefoe, te Maimm Mols Pinciple [8] can be se, wic says tat Ma g( Ma ( occs on te bonay of te pojection of te esie s-polygon o s-cicle. -5
Tis poves tat fo a fie in te pojection of an nesie s-polygon o s-cicle, te maimm of g( in an on te esie s-polygon o s-cicle occs on te bonay of te pojection of te esie s-polygon o s-cicle. Teefoe, fo fie te minimm of te / atio of coal lengts occs on te bonay of te pojection of te esie SV an ence on te bonay of te esie s-polygon o s-cicle. In aition, fo fie te / atio of GCDs also takes its minimm on te bonay of te esie s-polygon o s-cicle, since bot atios ave te same minimm soltion as sown in Section.3.. Since was abitay, we can state tat te / atio of GCDs takes its minimm on te bonay of te esie SV. Appeni A sows tat te eslts of tis section also apply to atios of slant anges, mentione ealie..3.3 Poof tat te Minimm / Ratio of GCDs Occs on te Bonay of te Unesie SV We ave sown tat te minimm / atio occs on te bonay of te esie SV. Now we pove tat it also occs on te bonay of te nesie SV. As in te pevios section, we will pove tat te eslt fo te / atio of coal lengts, an we will also wok wit te invete / atio o / atio. We will pove te eslt fist fo an s-cicle an ten fo an s-polygon..3.3. Poof tat te Minimm / (Maimm / Distance Ratio of Coal Lengts Occs on te Bonay of te Unesie s-cicle We begin by letting P be any fie point of te esie SV. We se te eqations in Appeni D to otate te nesie s-cicle so tat its cente is at te Sot Pole. To maintain te same geogapical elationsips wit te nesie s-cicle, P an P (te location of te esie GR nego te same otation. Ten we steeogapically poject te nesie s-cicle an esie SV to te comple plane. Te / atio of coal lengts can be epesse sing (4 an simplifie as: D D ( P 4, P R ( P, P ( 4R Let be te pojection of P an be te pojection of te pojection of te otate nesie s-cicle, as sown in Fige -. P. We note tat lies otsie -6
Since is fie, 4 R is a constant. We will sow tat te maimm of te following epession occs on te bonay of te pojection of te nesie s-cicle, wee is any point in o on te pojection of te nesie s-cicle: Ma 4R ( As Fige - sows, fo any point, int in te inteio of te pojection of te nesie s-cicle, tee is always a point, B on its bonay tat is close to te etenal point. Ts, we see tat, B >, int an, B <, int. It follows ten tat fo any, int tee eists a, sc tat B 4R, B, B > 4R, int,int Becase an ae fie, tis establises tat te soltion of ( occs on te bonay of te pojection of te nesie s-cicle. Since P was picke as any point of te esie SV, ts making an abitay point of te pojection of te esie SV, we concle tat te maimm / atio o te minimm / atio occs on te bonay of te nesie s-cicle. -7
Comple Plane (, Fige -. Illstation of Poof tat Minimm Occs on Unesie Cicle, an Not in its Inteio.3.3. Poof tat te Minimm / (Maimm / Distance Ratio of Coal Lengts Occs on te Bonay of te Unesie s-polygon Now assme tat te nesie SV is an s-polygon an tat P is te point of te esie SV an P is te point of te nesie SV tat minimie te / atio of coal lengts on te spee. Fte assme tat P is an inteio point of te nesie s-polygon. Ten te geat cicle tat connects Let te vetices of tis sie be lies to te eqato so tat P v an P an P mst intesect one of te sies of te s-polygon. P v. Rotate te geat cicle on wic te s-polygon sie P is in te noten emispee. Apply te same otation to P, an P. Ten steeogapically poject te geat cicle an te points P, P an P to te comple plane. Te pojection of te geat cicle is a cicle centee at te oigin in te comple plane an lies otsie te cicle. Ts, we obtain te configation of Fige -. P, -8
Comple Plane (, Denotes steeogapic pojection Fige -. Illstation of Poof tat Minimm Occs on Unesie Polygon, an Not in its Inteio We wok wit te / atio as in te pevios section, stating wit (. Since 4 R is a constant. Ts, te soltion maimie te / atio:, is fie, tat minimies te / atio wol 4R (3 Now let, be te intesection of te line connecting B te easoning of te pevios section, an < an, B wit te ac >, so, B v, v. Using, B 4R 4, B > R -9
Tis contaicts te assmption tat maimies (3. Teefoe, cannot lie in te inteio of te pojection of te nesie s-polygon bt mst lie on its bonay. Tis implies tat P cannot be in te inteio of te nesie s-polygon bt mst be on its bonay. Ts, we concle tat te minimm / atio of coal lengts occs on te bonay of te nesie s-polygon. -
3 Minimiing / Ratios fo Ai-to-Ai Commnications Tee ae fo cases to consie: ai-to-ai, gon-to-ai, ai-to-gon, an gon-to-gon. In tis section we pesent only te ai-to-ai case, wic is te most ifficlt to analye becase neite te intefeence soce no te intefeence victim as a fie location. Te concepts an pocees evelope fo te ai-to-ai case ave been aapte to te ote tee cases involving te gon aio an ae pesente in Section 4.. Becase tee ae two vaiable points, te location (latite an longite of te aicaft in te esie sevice volme an te location of te one in te nesie sevice volme, te optimiation poblem can potentially become one of tying to minimie a fnction of fo vaiables. Howeve, we will sow tat te poblem can always be ece to minimiing a fnction of a single vaiable. In oe to convet te fo-imensional minimiation poblem into a one-imensional minimiation poblem, some sefl eslts mst fist be evelope. Te constction sown in Fige 3- in te comple plane is sefl fo o application. Wen te esie an nesie SVs ae s-cicles, te minimiation is pefome ove te entie bonaies of bot cicles. In te comple plane, we efine te Desie Containment Ac as te ac on te nesie cicle sbtene by te cental angle of te nesie cicle wose sies ae tangent to te esie cicle. Te Desie Containment Ac is so name becase te sies of its efining cental angle contain te esie cicle. Appeni C sows ow to fin te lines foming te cental angle of te nesie cicle tat sbtens te Desie Containment Ac. A feate of te Desie Containment Ac is tat a line awn fom any selecte point on te bonay of te esie cicle to te cente of te nesie cicle will intesect it. Te point of intesection is, in fact, te closest point on te nesie cicle to te selecte point on te esie cicle. Any line awn fom anywee on te bonay of te esie cicle to any point on te nesie cicle etenal to tis ac will not pass tog te cente of te nesie cicle. 3-
Desie Unesie Desie Containment Ac Fige 3-. Desie Containment Ac Fo te case wee te esie an nesie sevice volmes ae bot s-cicles, it was sown peviosly (see Fige -8 tat fo eac fie point on te esie cicle, te point on te nesie cicle tat minimies te atio of coal lengts, lies on te line connecting wit te cente of te nesie cicle. It is te intesection of tis line wit te nesie cicle. Ts, becase te cente of te nesie cicle is at te oigin of te comple plane as sown in Fige 3-, is a constant eqal to te ais of te cicle. Also, ecall tat becase of collineaity. Une tese conitions an sing (6 an (8, (6 can be simplifie to te following minimiation poblem, on te igt an sie, of one comple vaiable : Min, Min (4 3-
(Note tat sing te eslt of Section.3., te oiginal poblem involving a atio of GCDs as been ece to one involving a atio of istances in te comple plane.. Since lies on a cicle, it can be paametie in tems of te angle fome by te ais of te cicle of wic it is an enpoint an te oiontal ais to te igt of te cente (see Fige 3-. Ts, minimiation poblem (4 can be fte simplifie so tat te minimiation can be one wit espect to a single eal vaiable. Teefoe, solving poblem (4 is jst a matte of minimiing ove te angle coesponing to points on te esie cicle. Note tat te angle is now associate wit te esie cicle ate tan wit te nesie cicle as in o pevios iscssion. Tis is fte iscsse in te following section. 3. Two s-cicles Tee ae two cases to consie in tis sitation: te esie aio site is at te cente of te esie s-cicle, an te esie aio site is not at te cente of te esie s-cicle. Tis section iscsses te latte case. Te fome case can be pefome in a mc simple manne base on fining te minimm GCD between te two s-cicles, wic is escibe in Section 5.. Fo te latte case, te nesie s-cicle is otate to te Sot Pole (see Appeni D an ten pojecte to te comple plane to obtain te configation of Fige 3-. Te esie s-cicle an te esie aio site location nego te same otation, an ten te pocee in Appeni B is se to fin te cente an ais of te esie cicle s pojection in te comple plane. Te tansfome esie aio site location is also pojecte to te comple plane. As jst iscsse, we nee only fin te soltion to (4 to solve (6, an (5 an (6 ave te same soltion as sown in Section.3.. Ts, solving (4, involving a atio of istances, solves (5, involving a atio of GCDs, wic is te poblem we set ot to solve. Fige 3- sows te configation se to minimie te / atio. 3-3
Comple Plane Desie Unesie (, Fige 3-. Configation fo Ai-to-Ai / Ratio Minimiation (Two s-cicles Te / atio can ten be epesse sing (4 as follows: ( s cos( θ ( k s sin( θ, θ π (5 ( s cos( θ ( k y s sin( θ Te single comple vaiable as been paametie in tems of te angle θ coesponing to points on te esie cicle as sown in eqation (5 an in Fige 3-. As iscsse in te, is jst te poblem of pevios section, fining te minimm / atio ove all ( minimiing (5 wit espect to θ. Tis can be one sing te Newton-Rapson meto [9]. Ts, te fact tat te minimiation can be one ove a single eal vaiable is a geat simplification. 3-4
3. s-cicle an s-polygon In minimiing te / atio wen ealing wit two iffeent types of sevice volmes, namely an s-cicle an an s-polygon, tee ae two scenaios to consie: (i te s-polygon is te esie sevice volme; an (ii te s-cicle is te esie sevice volme. Te pocee fo fining te minimm / atio fo a sie of te s-polygon is escibe below. Te minimm atio is fon fo eac sie an te minimm of tose minima is te soltion fo te entie poblem. 3.. Desie s-polygon an Unesie s-cicle Minimiing te / atio in tis case is vey simila to te two s-cicle case covee in Section 3.. Te iffeence is tat instea of aving an entie cicle fo te esie SV, we ave te ac of a cicle, wic is te pojection of an s-polygon sie, as sown in Fige 3-3. (Te esie cicle of Fige 3-3 is in actality te pojection of a geat cicle an ts wol incle te oigin of te comple plane in its inteio. Fige 3-3 oes not sow te esie cicle containing te oigin of te comple plane in oe to moe clealy sow te etails of te esie ac an its associate angles. In oe to etemine te θ inteval fo te ac, te, of te enpoints of te pojecte sie of te s- Catesian cooinates ( a y a, an ( b y b polygon as sown in Fige 3-3 ae se to fin te anglesθ, i a b by means of te following eqation: wee s is te ais of te esie cicle. i, i θ i Cos, i a, b s (6 As in te two s-cicle case, te poblem is to minimie te epession sown in (7, wic is te same as (5 since collineaity ols fo all points on te ac, bt now wee te angle θ is esticte to lie between te angles θ an θ. a b ( s cos( θ ( k s sin( θ ( s cos( θ ( k y s sin( θ, θ θ θ a b (7 3-5
Desie Comple Plane Unesie (, Fige 3-3. Configation fo Ai-to-Ai / Ratio Minimiation (Desie s-polygon, Unesie s-cicle If te optimal soltion to (7 is fon to lie otsie of te inteval θ a θ θb, ten as sown peviosly sing Fige -9, te optimal soltion mst be at one of te enpoints, coesponing to eite θ a o θ b. Eac one mst be cecke to etemine wic one povies te smalle vale fo te igt an sie of (7. 3.. Desie s-cicle an Unesie s-polygon Tee ae two cases to consie in tis sitation: te esie aio site is at te cente of te esie s-cicle, an te esie aio site is not at te cente of te esie s-cicle. Tis section iscsses te latte case. Te fome case can be pefome in a mc simple manne base on fining te minimm GCD istance between te esie s-cicle an te nesie s-polygon an is escibe in Section 5.. Te case escibe now is moe complicate tan te cases iscsse in Sections 3. an 3.. becase te collineaity conition ( enabling a simplification of te minimiation pocess is no longe gaantee fo evey point on te esie cicle. Figes 3-4 an 3-5 sow te pojections fom te spee to te comple plane of te esie s-cicle an one sie of te nesie s-polygon. Te pojection of a sie of te nesie s-polygon will be 3-6
efee to as an nesie ac. Tis nesie ac wol be on a cicle (a pojection of te geat cicle pon wic te s-polygonal sie lies in te comple plane. To cente tis cicle at te oigin of te comple plane, te speical cente (see Appeni D of te geat cicle on te spee is otate to te Sot Pole, sing te otation eqations in Appeni D. Te cente of te esie cicle an te esie aio site location nego te same otation befoe being pojecte. Tee ae fo cases to consie: (i te nesie ac completely ovelays te Desie Containment Ac so tat it is eqal to it o oveetens it on one o bot sies; (ii te entie nesie ac lies on te Desie Containment Ac as a sbac, i.e., neite enpoint of te nesie ac coincies wit an enpoint of te Desie Containment Ac; (iii no potion of te nesie ac lies on te Desie Containment Ac; (iv te nesie ac lies patly on te Desie Containment Ac. Te fist case is easily solve, since a line connecting any point on te esie cicle wit te oigin of te nesie cicle wol intesect te potion of te nesie ac ovelaying te Desie Containment Ac. Ts, tee wol always be a point on te nesie ac tat is collinea wit, wic togete wit wol minimie te / atio. Teefoe, te soltion is fon as in Section 3. by minimiing (5. 3... Unesie Ac Lies on te Desie Containment Ac In oe to etemine te location of te nesie ac wit espect to te Desie Containment Ac (see Fige 3-4, te intesections ( i, yi, i, wit te nesie cicle of te (ase lines tog te oigin of te comple plane (te nesie cicle s cente an te enpoints of te nesie ac mst be fon. Te meto se to fin tese intesection points is povie in Appeni E. Fige 3-5 sows tee ae two acs, igligte by ases, of te esie cicle fo wic te collineaity conition can be se. Te enpoints of tese acs ae fon as te intesection points wit te esie cicle of te sies of te cental angle of te nesie cicle tat sbtens te nesie ac. To eac enpoint ( i, yi, i, of te nesie ac, tee coespon two intesection points on te esie cicle ( i, j, yi, j, j,. Te meto to fin tese intesection points is povie in Appeni F. Te coesponing θ, i, ; j, ae fon sing eqation (6 wit θi eplace by θ i, j an i eplace by i, j. i, j 3-7
Desie Comple Plane Unesie Unesie Ac Fige 3-4. Configation fo Ai-to-Ai / Ratio Minimiation (Desie s-cicle, Unesie s-polygon Wit Unesie Ac Intesections θ an θ,θ,, sown in Fige 3-4 mst also be consiee as viable caniates fo minimiing te / atio. As sown in Fige 3-5, wen te point falls otsie te acs of te esie cicle wee te collineaity conition applies, te point on te nesie cicle tat is collinea wit an minimies te / atio fo is (, wic is not on te nesie ac. As sown befoe, te / atio fo a given monotonically inceases as a point moves away fom, so its minimm vale on te nesie ac wol be attaine at,vete,, te closest enpoint of te nesie ac to. Since te entie esie cicle mst be consiee, points otsie te intevals (,θ,, ( 3-8
Desie Vete Vete Comple Plane Unesie Wee collineaity can be applie Fige 3-5. Configation fo Ai-to-Ai / Ratio Minimiation (Desie s-cicle, Unesie s-polygon Wit Collineaity Conition Ientification Tis leas to te following pocee fo fining te soltion tat minimies te / atio fo points of te esie cicle wose closest points on te nesie cicle o not lie on te nesie ac (i.e., collineaity can t be applie. Refeing to Fige 3-5, fo points on te esie cicle between (, y,, an (,, y, se enpt,vete, as te enpoint, an fo points between (, y,, an (, y,, se enpt,vete,. Te optimiation poblem fo tese two instances is ten fomlate as Min, ac Min enpt Min θ ( s cos( θ enpt k s sin( θ y ( s cos( θ ( k s sin( θ y ( enpt (8 3-9
Ts, again, to fin te soltion eqies minimiing a fnction of a single vaiable, wic can be one sing te Newton-Rapson meto. To smmaie te pocee fo te case wee te entie nesie ac lies on te Desie Containment Ac as a sbac, tee ae fo minimiation sb-poblems tat mst be solve: poblem (8 ove te ac conteclockwise fom (, y,, to (,, y, sing enpt,vete an again ove te ac conteclockwise fom (, y,, to (, y,, sing enpt,vete ; poblem (7 ove te igligte ac fom (,, y, to (, y,, an again ove te igligte ac fom (,, y, to (,, y,. Te minimm of te soltions fom tese fo sbpoblems is ten selecte as te soltion fo te polygon sie ne consieation. 3... Unesie Ac Etenal to te Desie Containment Ac Now consie te case wee te nesie ac lies entiely otsie te Desie Containment Ac, as sown in Fige 3-6. Desie Comple Plane Unesie Fige 3-6. Unesie Ac Does Not Lie on Desie Containment Ac 3-
Fige 3-6 illstates tat te same agment as above can be se to sow tat te optimal,. Te enpoint closest to te Desie Containment Ac is soltion mst be of te fom ( enpt selecte an te optimiation poblem (8 above is solve wit θ π to povie te soltion fo te polygon sie ne consieation. 3...3 Unesie Ac Lies Patly on te Desie Containment Ac If te nesie ac patially ovelays te Desie Containment Ac as sown in Fige 3-7, ten te potion of te esie cicle coesponing to tat ovelaying section of te nesie ac is wee te collineaity conition ols. Teefoe, fo te igligte potion of te esie cicle between ( θ an ( θ, te epession in (7 mst be minimie ove te inteval θ θ θ. Desie Comple Plane Unesie Fige 3-7. Unesie Ac Patially Lies on Desie Containment Ac an Patially Otsie It 3-
Containment Ac, is te closest point to ( Becase te entie esie cicle mst be consiee, points otsie te igligte potion of te esie cicle also mst be consiee. On te nesie ac, vete being on te Desie coesponing to any on te nonigligte potion of te esie cicle. Ts te minimiation poblem (8 is solve sing vete as te enpoint ove te inteval θ θ θ. Te minimm of te soltions to (7 an (8 is selecte as te soltion fo te polygon sie ne consieation. 3..3 Desie s-polygon an Unesie s-polygon Fo te case wee bot SVs ae s-polygons, te optimiation pocee mst be applie to eac pai of sies, one sie fom te esie s-polygon an te ote sie fom te nesie s-polygon. Ts, we ae always ealing wit geat cicles. In all cases te cente of te geat cicle containing te sie ne consieation of te nesie s-polygon is otate to te Sot Pole. Wen tese same otation eqations ae applie to te geat cicle containing te sie ne consieation of te esie s-polygon, te two new vetices of tat sie may ave te same longites. Ts, wen tat sie is pojecte to te comple plane, we obtain a staigt line instea of a cicle. Tis case is covee in Section 3.3. Te case of two s-polygons is teate in a manne simila to te case of a esie s-cicle an an nesie s-polygon iscsse in Section 3... Te iffeence is tat te esie s-cicle is eplace by te ac of a geat cicle as sown in Fige 3-8 afte pojection to te comple plane. Te same fo elationsips escibe in Section 3.. tat can occ between an nesie ac an a esie cicle can occ also between an nesie an a esie ac. Collineaity can be se wee te cental angle of te nesie cicle tat sbtens te nesie ac (te nesie cental angle an te cental angle of te nesie cicle tat sbtens te esie ac (te esie cental angle ovelap. Wee collineaity cannot be se, te vete of te nesie ac closest to te esie ac is se in te / atio to be minimie. Fige 3-8 sows an eample of te case tat is simila to case (iv, ientifie in Section 3.., an sown in Fige 3-7. To fin te soltion, te / atio (7 is minimie ove te inteval θa θ θ,, wee collineaity ols, an te minimiation poblem (8 is solve ove te inteval θ, θ sing,vete,, wee collineaity oes not ol. Te angles θ a, θ,, an θb ae efine wit, (see Fige 3-3. Fo eac minimiation poblem, te enpoints of te espect to ( k inteval ove wic te minimiation is pefome mst be cecke weneve te eivative as no eo vale in te inteval. Fom te soltions obtaine fom minimiation poblems (7 an (8, te smalle one is selecte as te soltion fo te sie ne consieation. Te minimm / atio is fon fo eac pai of sies, an te minimm of tese minima is te soltion to te entie poblem. θ b 3-
Desie Comple Plane Unesie Fige 3-8. Desie Cental Angle an Unesie Cental Angle Ovelap 3.3 Pojection of a Sie of Desie SV is a Staigt Line In tis case, we assme tat afte te geat cicle on wic te nesie ac lies as been otate to te eqato, te sie ne consieation of te esie s-polygon, afte negoing te same otation, lies along a meiian. Tis means tat te two vetices of te esie s-polygon sie ave te same longites, an te geat cicle containing tis sie passes tog bot Poles. A steeogapic pojection of tis sie wol be a staigt line segment in te comple plane passing tog te oigin of te comple plane as sown in Fige 3-9. Fige 3-9 sows te case wee te esie line segment falls witin te cental angle of te nesie ac. 3-3
Raio Site Comple Plane Desie Unesie (, Fige 3-9. Sie of Desie s-polygon Pojects as Staigt Line Segment on Line Intesecting Unesie Ac Te enpoints of te esie line segment ae easily obtaine as te steeogapic pojection of te enpoints of te geat cicle ac segment sown in Fige 3-9, wose latites an longites ae known. Since te line segment is on a line passing tog te oigin of te iθ comple plane, any point on te line segment can be epesse as se wee s is vaiable an bone by te magnites of te enpoints of te line segment, an θ is a constant since a vecto fom te oigin to any point along te line segment makes te same angle wit te eal ais of te comple plane. Fo any point on te line segment, te point on te nesie ac tat minimies te / atio is te point wic is te intesection of te line tog te oigin containing te esie line segment wit te nesie ac. Since te collineaity conition applies, te nesie istance can be epesse as s R, 3-4
wee R is te ais of te nesie cicle. Te esie istance can be epesse as s e iθ s e iθ s s s s cos ( θ θ. Ts, we mst solve te poblem Min s s s s R s s cos ( θ θ,, enpt s, enpt (9 Te minimiation wol be pefome ove te single vaiable s, since all te ote vaiables ae constant, an can be implemente sing te Newton-Rapson meto. All of te above also applies in te case wen tee is an nesie s-cicle instea of an nesie s-polygon, te cente of te nesie s-cicle is otate to te Sot Pole, an te sie ne consieation of te esie s-polygon negoes te same otation an lies along a meiian. Fige 3- sows te case wee te esie line segment falls otsie te cental angle of te nesie ac. In tis case te enpoint on te nesie ac closest to is selecte to solve te minimiation poblem Min s s s s enpt s s s s s enpt cos cos ( θ θ ( θ θ enpt,, enpt s, enpt (3 As befoe te minimiation wol be pefome ove te single vaiable s, since all te ote vaiables ae constant, an Newton-Rapson wol be applie. Note tat in bot poblems (9 an (3, te minimiation is applie to te atio of coal lengts; oweve, te soltion obtaine is also te soltion to te poblem wee te atio of coal lengts is eplace wit te atio of GCDs, wic is wat we eally esie to minimie. Te poof povie in Section 3. can be sown to apply to tis case wee te pojecte esie sie is a staigt line segment. 3-5
Raio Site Comple Plane Desie (, Unesie Fige 3-. Sie of Desie s-polygon Pojects as Staigt Line Segment on Line Not Intesecting Unesie Ac 3-6
4 Minimiing / Ratios fo Gon-to-Ai, Gon-to- Gon, an Ai-to-Gon Tis section coves te / atio fo te cases wee te gon aio eite tansmits te intefeing signal o it is te victim of te intefeing signal. 4. Gon-to-Ai Te gon-to-ai case sown in Fige 4- coves te sitation wee a aio at a gon site sppoting a given SV A cases intefeence wit an aicaft flying in a iffeent SV B becase SV A an SV B ae cocannel. In tis instance, te nesie signal is fom te gon aio. Fige 4-(a sows te footpint associate wit Fige 4- incling te esie an nesie GCDs. Tis case falls ne te case of an nesie s-cicle an a esie s-cicle, wee te nesie s-cicle as ecease to a cicle of ais eo, i.e., it is jst a point. Fige 4-(b sows te steeogapic pojection to te comple plane fo tis case, wee it is assme tat te location of te nesie GR as been otate to te Sot Pole. Fo te case of a esie s-polygon, te pojection of a sie wol be te ac of a cicle. In te case of a esie s-cicle, te minimm / atio can be fon as a soltion to (5 wit. Fo a esie s-polygon, te minimm / atio fo a given sie is te soltion to (7 wit. Te minimm fo all sies is ten te soltion. Unesie Desie CONTROLLER GROUND RADIO CONTROLLER GROUND RADIO Fige 4-. Gon-to-Ai Case 4-
Unesie G Desie G GCD (a Footpint on Spee Desie Comple Plane Unesie (, (b Pojection to Comple Plane Fige 4-. Footpint an Steeogapic Pojection fo Gon-to-Ai Case 4. Ai-to-Gon Te ai-to-gon case sown in Fige 4-3 coves te sitation wee an aicaft in SV B cases intefeence to a aio site sppoting SV A becase SV A an SV B ae cocannel. Fige 4-4(a sows te footpint associate wit Fige 4-3 incling te esie an nesie GCDs. Fige 4-4(b sows te steeogapic pojection to te comple plane. In tis case te gon aio fo SV A is tying to eceive a signal fom an aicaft in SV A, bt is being intefee wit by a signal fom an aicaft in SV B. In tis case te / atio always as te same nmeato, wic is te nesie coal lengt between te point on te nesie cicle an. Tis is becase is collinea wit an te cente of te nesie cicle, an teefoe is te point wee te nesie coal lengt is te smallest (see Fige -9. It will neve cange becase te aio site is fie. Notice tat te cente of te esie cicle is at te oigin of te comple plane (becase we otate te cente of te esie s-cicle to te Sot Pole, instea of te cente of te nesie cicle as in all te pevios cases. Tis is one to simplify te minimiation of te / atio of coal lengts, wic can be epesse as 4-
4-3 ( ( 4 4 4 4 4 4 4 4,, R R R R R R R R P P D P D P (3 Te tems, 4 R, an 4 R ae all constants, an R is te ais of te spee fom wic te pojection is one. Te latte tem is a constant becase is a point on a cicle wit cente at te oigin of te comple plane. Teefoe, te poblem becomes one of minimiing te epession ( ( K P P D P D P,, Tis is eqivalent to solving te poblem ( ( ( ( sin cos y s s Ma Ma θ θ θ (3 wic as te soltion π θ y Tan. Tis coespons to te point on te esie cicle wic is fatest fom.
Desie Unesie CONTROLLER Fige 4-3. Ai-to-Gon Case Desie G G Unesie GCD (a Footpint on Spee Comple Plane Unesie Desie (,k (, (b Pojection to Comple Plane Fige 4-4. Footpint an Steeogapic Pojection fo Ai-to-Gon Case 4-4
In te above eamples te SVs ae s-cicles. Te same pocee can be applie to an s-cicle an an s-polygon o to two s-polygons since te pojection of an s-polygon sie lies on a cicle in te comple plane. Te pocee mst be applie to eac of te sies of an s-polygon, an fo eac pojecte sie te optimiation wol be esticte to an ac of te cicle ate tan to te entie cicle. If te nesie pojection is an ac, ten wol be eplace by te closest point on te nesie ac to to compte te minimm nesie coal lengt. 4.3 Gon-to-Gon Te gon-to-gon case is sown in Fige 4-5. Fige 4-6(a sows te footpint associate wit Fige 4-5 incling te esie an nesie GCDs. Fige 4-6(b sows te steeogapic pojection to te comple plane. Tis case epesents anote instance wee te cente of te esie cicle is at te oigin of te comple plane becase te cente of te esie s-cicle is otate to te Sot Pole. Te minimm / atio in tis case can be fon by applying te fomlation fo te ai-to-gon case an sing (3 wee is se in place of an 4R B is se in place of 4 R. Note tat A B can be easily compte becase te cooinates of te two aio sites ae povie. If te esie SV is an s-polygon, ten eac sie mst be pojecte to obtain an ac in oe to geneate te soltion, an te optimiation of (3 is esticte to te ac. Te maimm ove all te soltions of (3 fo eac sie is te soltion. A B Desie Unesie GROUND RADIO CONTROLLER GROUND RADIO CONTROLLER Fige 4-5. Gon-to-Gon Case 4-5
Desie G G Unesie GCD (a Footpint on Spee Comple Plane Desie A (, B Unesie (b Pojection to Comple Plane Fige 4-6. Footpint an Steeogapic Pojection fo Gon-to-Gon Case 4-6
5 Minimiing Geat-Cicle Distance between s-objects Minimiing te GCD between te footpints on te spee of two s-objects is a elatively easy task in compaison to minimiing te / atio. We will iscss minimiing te GCD in te following sections between two s-cicles, an s-cicle an an s-polygon, an two s-polygons. Tee ae two basic applications tat eqie fining te minimm GCD between s-objects. Te fist is in minimiing te / atio in te case of a esie s-cicle wit te aio site at its cente an an nesie s-polygon as mentione in Section 3.., an te secon is to etemine wete ajacent cannels can be se in a pai of SVs base on te minimm GCD between tem. In te cent system, if te minimm GCD between two sevice volmes is less tan.6 nmi, ten tei assignments mst be sepaate by at least two cannels. Tis le, to pevent ajacent-cannel intefeence (ACI, is efee to as te ajacent-cannel assignment le. Te eason wy minimiing te GCD between sevice volme footpints on te spee applies to te fist application is te following. In te case wee te aio site is at te cente of te esie s-cicle, te enominato of te / atio is a constant eqal to te ais of te s-cicle, an ts minimiing te / atio eqies only minimiing te nesie GCD. 5. Minimiing Geat-Cicle Distance between Two s-cicles Fige 5- sows fining te minimm GCD between two s-cicles of a given ais. In tis case, it is not necessay o elpfl to pefom a steeogapic pojection. Te minimm GCD is easily obtaine by compting te GCD between te centes of te two s-cicles an sbtacting te two aii. Ts te minimm GCD between te cicles C an C wol be compte as Min ByC, ByC (, C G( ( Lat, Lon, ( Lat, Lon G C (33 5-
Fige 5-. GCD Between Two s-cicles 5. Minimiing Geat-Cicle Distance Between Two s-polygons an Between an s-cicle an an s-polygon Te appoac fo minimiing te GCD between two s-polygons is base on te fact tat any two geat cicles intesect as sown in Fige 5-. It can be seen tat te GCD between te two geat cicles is monotonic as one moves away fom te point of intesection in eite iection. Fo geat-cicle acs, Fige 5-3 illstates tat te minimm occs at an enpoint of one of te acs, an wol be te sotest istance between an enpoint of one of te acs an te ote ac. We ave evelope a vey efficient point-to-ac algoitm, eplaine in te net section, to etemine te minimm istance between a point etenal to an ac of a geat cicle an tat ac. Given tat pocee an given any two acs of geat cicles, an appoac to fining te minimm istance between te acs wol be te following. Using te point-to-ac algoitm, fin te minimm istance between eac enpoint of eac ac an te ote ac. Te smallest of tese fo minimm istances wol be te minimm GCD between te two acs. If it is fon tat no sotest GCD ac fom any enpoint of eite ac intesects te ote ac, ten te sotest GCD wol occ between a pai of enpoints fom te two acs as sown in Fige 5-4. 5-
To fin te minimm GCD between two s-polygons, te minimm GCD fom eac sie (ac of one of te s-polygons is fon to te ote s-polygon, an ten te GCD fom eac sie of te ote s-polygon is fon to te fist s-polygon. Te minimm of te GCDs ove all pais of sies of te two s-polygons is te soltion. Te appoac to fin te minimm GCD between an s-cicle an an s-polygon also elies on te point-to-ac algoitm. In tis case te sotest GCD fom te cente of te cicle to eac sie of te s-polygon is fon, an te ais of te s-cicle is sbtacte. Te smallest of tese minimms is te soltion. Fige 5-. Intesecting Geat Cicles 5-3
Fige 5-3. Sotest GCD between Acs of Geat Cicles A B Fige 5-4. Minimm GCD Occs at Enpoints 5-4
5.3 Point-to-Ac Algoitm Fige 5-5 sows te constcts se to etemine te minimm GCD fom a fie point, y to te sie of an s-polygon, wee te sie is a geat cicle ac between te points (, (, y an (, y,, Te fist case is wen (, y. Tee ae two cases to consie tat ae escibe in tis section. lies on te geat cicle of te polygon sie. Ten eite it lies, on te polygon sie o otsie of it. In te fist instance, te minimm GCD is eo, an in te, y, y secon instance, te minimm GCD is te minimm of te GCDs fom (, to (, an (, y., Fige 5-5. Constct fo Fining Minimm GCD fom Fie Point to Sie of s-polygon, on,, fo wic planes A an B ae otogonal. Once fon, tis point is pojecte along a vecto fom te oigin of te spee to te point ( η v, ηyv, ηv on te sface of te spee wee η. Te pocee fo v yv v etemining if two planes A an B ae otogonal is to fin te nomal vectos VA an V B to te planes an ten eqie tat tese vectos be otogonal. Two non-paallel vectos lying in,,, y,,, y, ; an two In te secon case, te poblem can be pose as te poblem of fining te point ( v yv, v te co connecting (, y an (, y plane A ae te vectos fom ( to ( an fom ( to (, v v v 5-5