A Spatial Version of the Theorem of the Angle of Circumference

Transcription

1 This is a pr-print o a contribution publishd in In L. Cocchiarlla (d): ICGG Procdings o th 18th Intrnational Conrnc on Gomtr and Graphics, 40th Annivrsar, Milan/Ital 2018, pp , publishd b Springr Intrnational Publishing AG 2019 (ISBN ). Th inal authnticatd vrsion is availabl onlin at A Spatial Vrsion o th Thorm o th Angl o Circumrnc Gorg Glasr 1, Boris Odhnal 1, and Hllmuth Stachl 2 1 Univrsit o Applid Arts Vinna, Vinna, Austria gorg.glasr@uni-ak.ac.at, boris.odhnal@uni-ak.ac.at, 2 TU Win, Vinna, Austria stachl@dmg.tuwin.ac.at Abstract. W tr a gnraliation o th thorm o th angl o circumrnc to a vrsion in thr-dimnsional Euclidan spac and ask or pairs (ε, ϕ) o plans passing through two (dirnt skw) straight lins ε and ϕ such that th angl α nclosd b ε and ϕ is constant. It turns out that th st o all such intrsction lins is a quartic ruld surac with bing its doubl curv. W shall stud th surac and its proprtis togthr with crtain spcial apparancs showing up or spcial valus o som shap paramtrs such as th slop o and (with rspct to a id plan) or th angl α. Kwords: ruld surac, angl o circumrnc, quartic ruld surac, Thaloid, isoptic surac 1 Introduction Th thorm o th angl o circumrnc stats that a straight lin sgmnt (boundd b two points E and F) in th Euclidan plan is sn at a constant angl α rom an point o a pair o circular arcs passing through E and F. Espciall, i th visual angl α is a right angl, th pair o circls bcoms on circl with diamtr EF, usuall rrrd to as th Thals circl. It would b natural to gnrali th thorm o th angl o circumrnc in Euclidan thr-spac b asking or all points that s a straight lin sgmnt boundd b two points E and F undr a constant angl α. Th locus o all such points is an algbraic surac o dgr our. It is obvious that th lattr surac has a rotational smmtr - th ais o th rotation coincids with th straight lin [E,F] - and this isoptic surac can b obtaind b rotating th pair o circular arcs through E and F, and is thror, a torus, s Fig. 1. Isoptic curvs o sphrical conics ar also wll-known, s [1]. In this contribution, w tr a lin gomtric gnraliation. W ask or th st o all intrsction lins r o plans rom two pncils. Th lins r can b considrd as on-dimnsional sing a pair o straight lins undr a constant angl. In Sction 2, w shall driv th quation o th ruld surac carring all th lins that s a pair (,) o (skw) straight lins undr constant angl. From

2 2 Gorg Glasr t al. E α α F Fig. 1. A possibl gnraliation o th thorm o th angl o circumrnc in thrdimnsional Euclidan spac. th quation o w can driv som proprtis o th surac which shall b th contnts o Sction 3. Finall, in Sction 4 w look at spcial cass o th ruld surac that appar i ithr th as and rach a spcial rlativ position or i th angl α attains spcial valus or i vn both is th cas. 2 Equation o th ruld surac It is avorabl to rprsnt points in Euclidan thr-spac R 3 b Cartsian coordinats (,,). It mans no rstriction to assum that th as and o th two pncils o plans ar givn b d 0 d 0 = 0 +t 1, = 0 +u 1 (1) 0 k 0 k paramtridbralparamtrst,u R.Hrand inth ollowing, = 2d R is th distanc btwn th straight lins, and k R is thir slop with rspct to th plan = 0, s Fig. 2. Sinc g = (0,1,k) and h = (0,1, k) ar th dirctions o th lins and, th normals o th plans ε and ϕ ar linar combinations o g 1 = (0, k,1), g 2 = (1,0,0) or h 1 = (0,k,1), h 2 = g 2, rspctivl. With λ,µ R w lt n ε = g 1 +λg 2, n ϕ = h 1 +µh 2. (2) Now, w can writ down th condition <)(ε,ϕ) = <)(n ε,n ϕ ) = α b valuating n ε,n ϕ 2 = A 2 n ε,n ε n ϕ,n ϕ

3 A spatial vrsion o th thorm o th angl o circumrnc 3 k d 1 k d 1 Fig.2. Choic o a Cartsian coordinat sstm and th maning o d and k. whr u,v dnots th canonical scalar product o two vctors u,v R 3 and A := cosα. This givs (1 k 2 +λµ) 2 = A 2 (1+k 2 +λ 2 )(1+k 2 +µ 2 ). (3) Th plans rom ithr pncil hav normal vctors givn in (2), and thus, th hav th quations ε : λ k + = dλ, (4) ϕ : µ+k + = dµ. Soar,both λandµcanvarrlinr,andthus,th plansεandϕintrsctin th lins o a hprbolic linar lin congrunc with as and, paramtrid b r(t,λ,µ) = d k k µ µk +t 2k µ λ k(λ+µ) (5) whr t R is th paramtr on th lins in th congrunc. Th ruld surac w ar aiming at is prcisl that subst o th congrunc (5) whr λ and µ ar subjct to (3). Th quation o th ruld surac in trms o Cartsian coordinats is obtaind rom th paramtriation (5) b liminating all paramtrs t, λ, µ: Assum r = (r,r,r ). Thn, w liminat t rom r, r, and r b computing th rsultants r 1 := rs( r, r,t), r 2 := rs( r, r,t). In th nt stp, w liminat λ rom both, r 1 and r 2 using (3) which rsults in two urthr polnomials r 1 R[,,µ] and r 2 R[,,µ]. It would mak no dirnc i w liminat µ irst. Finall, th rsultant o r 1 and r 2 with rspct

4 4 Gorg Glasr t al. to µ contains a non-trivial actor which is th quation o. (Th trivial actors o th lattr rsultant ar dtctd b substituting (5) and vriing that th do not vanish.) So, w obtain th ollowing quation o : σ 1 σ 2 ( 2 d 2 ) 2 B 2 ( 2 k 2 2 ) σ 3 (d 2 2 +k )+2σ 4 (d 2 k )+ 8A 2 dk(1+k 2 ) = 0 (6) with th abbrviations σ 1,2 := A k 2 ±k 2 +A 1, σ 3,4 := A 2 k 2 k 2 +A 2 ±1, and B 2 = 1 A 2 (or B = sinα). Summariing, w can stat: Thorm 1. Th isoptic ruld surac as th st o intrsction lins r o plans ε, ϕ rom two pncils with as, and <)(ε,ϕ) = α = const. is th algbraic ruld surac with th quation (6) and is, in gnral, o dgr our. Fig. 3 shows an ampl o a ruld surac togthr with th as and o th pncils o plans. In th cas A = 0 which is quivalnt to α = π 2, thr ists a gnration o b mans o a projctiv mapping κ rom th pncil o plans about to th pncil o plans about. Th projctivit κ assigns to ach plan ε (through ) prcisl on plan ϕ (through ) such that ε ϕ. Thus, th lins ε κ(ε) orm a rgulus, i.., on amil o straight lins on a (rgular) ruld quadric. Insrting A = 0 into (6) rturns th quation o th rgular ruld quadric which is in an cas a hprboloid (with multiplicit two): ((1 k 2 ) 2 k d 2 (k 2 1)) 2 = 0 (7) an ampl o which is shown in Fig Proprtis o From th construction o it is clar that th lins and ar part o th surac. Morovr, th union o ths lins is th doubl curv o. Hnc, is o Sturm tp 1, c. [2]. Sinc is o Sturm tp 1, it is lliptic. Each plan ε in th pncil about intrscts along with multiplicit 2. Sinc ach such plan ε contains at last on gnrator, th rmaining part o ε has to b a straight lin s too. Th lin s is a urthr gnrator o. A similar statmnt can b mad about th plans ϕ through. Th tangnt plans o mt along (planar) cubic curvs which ar ithr rational or lliptic, s Fig. 5. Th quartic ruld surac carris no rgular conic: An plan ε through a pair o intrscting gnrators g 1, g 2 shars on

5 A spatial vrsion o th thorm o th angl o circumrnc 5 ϕ r ε Fig.3. Th quartic isoptic ruld surac o a pair o skw straight lins and. o th as, sa, with. Thror, th rmaining part o ε \ {,g 1,g 2 } has to b a straight lin l and l is a singular conic, c. [2]. I w prorm th projctiv closur o th Euclidan thr-spac, thn w can look s intrsction with th idal plan, s Fig. 6. Th idal points E and F o th straight lins and ar th onl doubl points o th lliptic quartic. An quation o can obtaind rom (6) b rmoving all trms o dgr thr and lss: : (σ 1 σ k 2 σ σ 4 2 ) 2 B 2 (k ) 2 = 0. (8) Thn, w intrprt : : as homognous coordinats o points in th plan at ininit and not that in s quation d dos not show up. 4 Spcial cass W can pct cptional apparancs o th quartic ruld surac i w choos spcial valus or A, d, or k. Although all ths valus ar originall assumd to b ral and spciall A 1, w nd not rstrictd ourslvs to ral valus. In th ollowing, w shall discuss som o ths spcial choics that lad to somtims unpctd suracs which, vntuall, ar thn no longr ruld suracs with ral rulings.

6 6 Gorg Glasr t al. Fig.4. A on-shtd hprboloid appars i A = 0, d,k R, c. (7). 4.1 On-shtd hprboloids. Intrscting lins and. In th vr bginning, w mad th natural assumption 2d = 0, i.., th lins and ar skw. I w allow d = 0, thn (6) simpliis to (8) which coms as no surpris, sinc (8) is indpndnt o d. Thror, (8) can also b viwd as th quation o a quartic con Γ manating rom (0,0,0). Obviousl, and ar gnrators with multiplicit two. Th con Γ is th asmptotic con o an ampl o which is displad in Fig. 7. Furthr, i k = 0 (togthr with d = 0 this actuall mans = ), thn Γ dgnratsandbcomsthpairoisotropicplans = 0with multiplicit two. I w allow A = 0, th con Γ bcoms th quadratic con (k 2 1) 2 +k = 0 (9) with multiplicit two. Th quadratic con (9) is a normal con (c. [1, p ]) and it is th asmptotic con o th doubl hprboloid (7) bing th spcial orm o i A = 0. An intrsting cas occurs i k = ±i (bsids d = 0), i.., th as and o th pncils o plans ar isotropic lins. In this cas, splits into two singular quadrics: 2 2 +(1 A) 2 +(1 A) 2 = 0, 2 2 +(1+A) 2 +(1+A) 2 (10) = 0.

7 A spatial vrsion o th thorm o th angl o circumrnc 7 τ c r Fig.5. Th quartic ruld surac carris a two-paramtr amil o cubic curvs which com as th intrsction o with its tangnt plans. Hr: τ = r c whr r τ is a ruling, τ is a tangnt plan through r, and c is th cubic curv. Y E Z F X Fig.6. Th intrsction o with th plan at ininit is an lliptic quartic curv with th two doubl points E and F which ar th idal points o s doubl curv.

8 8 Gorg Glasr t al. Fig.7. Intrscting lins and produc a quartic con which is also th asmptotic con o th gnric (non-dgnrat) quartic ruld surac. On o ths bcoms a plan with multiplicit two i ithr A = +1 or A = 1 whil th othr on bcoms an isotropic con = 0. Th cas A < 1 turns both o th quadrics into cons without an ral points bsids th common vrt (0,0,0). A > 1 corrsponds to purl imaginar angls α i ln(a+ A 2 1) (mod 2π). Nvrthlss, insrting A > 1 into (10) maks ithr th irst or th scond quadric a con with ral points whil th othr still has onl on ral point, naml th vrt (0,0,0). Paralll lins and. Th cas o paralllas and is clarlan trusion o th planar igur o th thorm o th angl o circumrnc. Thus, th ruld surac (6) will split into two clindrs 1 and 2 o rvolution rctd on thos circular arcs in th [,]-plan which ar th locus o all points sing th lin sgmnt EF (with E,F = (±d,0,0)) undr constant angl α (c. Fig. 8). From (6) w ind th quation o th dgnrat quartic b rplacing k with 1/K and subsquntl stting K = 0. (Othrwis, w would hav to st k =.) This rsults in th pctd pair o clindrs o rvolution 1,2 : ± 2dA 1 A 2 d2 = 0.

9 A spatial vrsion o th thorm o th angl o circumrnc 9 Fig.8. A pair o clindrs o rvolution as th isoptic ruld surac o paralll lins and. In ordr to ind ral suracs i, th valus or A ar rstrictd to A 0. Th Thaloid = d 2 (clindr orvolution)throughand cannot b obtaind dirctl rom th clindrs quations, sinc thn A = 1. Othr quadrics. Th as and o th pncils o plans can b chosn as isotropic lins. Thror, w lt k = i. (Th choic k = i producs th sam rsult.) Again, w ind that(6) dgnrats and splits into quadratic polnomials: Q 1 : 2 2 +(1 A) 2 +(1 A) 2 = 2d 2, Q 2 : 2 2 +(1+A) 2 +(1+A) 2 = 2d 2. (11) Th cas d = 0 was discussd arlir, so w hav d 0 in th ollowing. Indpndnt o th choic o A and rgardlss o th rgularit, both quadrics Q 1 and Q 2 hav th -as or thir common ais o rvolution. In th vr spcial cas A = ±1, th pair o quadrics (11) contains prcisl th singular quadric 2 d 2 = ( d)(+d) = 0 (a pair o (ral) paralll plans) and th Euclidan sphr = d 2 with radius d cntrd at (0,0,0) touching th plans at (±d,0,0). I A > 1, th pair (Q 1,Q 2 ) o quadrics consists o a two-shtd hprboloid o rvolution and an llipsoid o rvolution. Finall, w obtain two llipsoids o rvolution i A < 1. Th ollowing tabl summaris th spcial and dgnrat cass o dpnding on spcial choics o A, k, d.

10 10 Gorg Glasr t al. A = 0 k = 0 k = i k = (d ) 2 = 0 (2d ) 2 = 0 (d ) 2 = 0 right clindr, µ = 2 llipsoid, µ = 2 right clindr, µ = 2 d = 0 d = 0 d = 0 ( ) 2 = 0 ( ) 2 = 0 ( ) 2 = 0 compl. conj. plans, con, no ral compl. conj. plans, µ = 2 point (0,0,0), µ = 2 µ = 2 d = i d = i d = i ( ) 2 = 0 ( ) 2 = 0 ( ) 2 = 0 right clindr, llipsoid, right clindr, no ral points, µ = 2 no ral point, µ = 2 no ral point, µ = 2 A = 1 k = 0 k = i k = 2 = 0 ( 2 d 2 )(d )=0 2 = 0 plan, µ = 2 sphr tangnt plans plan, µ = 2 d = 0 d = 0 d = 0 2 ( )=0 mpt st ral doubl plan mpt st isotropic con d = i d = i d = i 2 = 0 ( 2 +1)( ) 2 =0 2 = 0 plan, µ = 2 sphr, no ral point plan, µ = 2 compl. tang. plans Tabl 1. Spcial shaps o causd b A = 0,1, α = 0, π, d = 0, k = 0,,i; 2 th intgr µ dnots th multiplicitis o th componnts. Rrncs 1. G. Glasr, H. Stachl, B. Odhnal: Th Univrs o Conics. From th ancint Grks to 21 st cntur dvlopmnts. Springr-Spktrum, Springr-Vrlag, Hidlbrg, E. Müllr: Vorlsungn übr Darstllnd Gomtri. III. Band: Konstruktiv Bhandlung dr Rgllächn. Dutick, Lipig und Win, H. Wilitnr: Spill Ebn Kurv. Sammlung Schubrt LVI, G.J. Göschn sch Vrlagshandlung, Lipig, 1908.