Sectrix Curves on the Sphere

Transcription

1 riginal scintific papr Accptd LÁSZLÓ NÉMETH Sctri Curvs on th Sphr Sctri Curvs on th Sphr ABSTRACT In this papr w introduc a class of curvs drivd from a gomtrical construction. Ths planar curvs ar th gnralization of th lss-known sctri of Cva. W also prsnt thr variations of th sctri curvs on th sphr with using th gomtrical construction on th sphr, with th strographic projction and with a so-calld rolld transformation. K words: sctri, folium, Chbshv polnomial, curvs on sphr MSC200: 5N20 Sktris na sfri SAŽETAK U ovom članku uvodimo klasu krivulja izvdnih gomtrijskom konstrukcijom. Takv ravninsk krivulj su gnralizacija manj poznatih Cvinih sktrisa. Takod - r, prikazujmo tri varijacij sktrisa na sfri, koristći gomtrijsku konstrukciju na sfri, strografsku projkciju i takozvano valjano prslikavanj. Ključn rijči: sktrisa, folium, Chbshvjv polinom, krivulj na sfri Sctri on th plan Th Sctri of Cva is a lss-known planar curv ([7, p. 3-35]), that is dfind with th polar quation 2 3 ρ = a + 2a sinkϕcos(k + )ϕ, a > 0, k N, ϕ [0,2π]. sinϕ () Figur shows its shap whr a = and k = 2. It has two prpndicular as of smmtr. In this articl w us this curv in cas a =. Figur : Sctri of Cva (k = 2) If k = thn w gt th so-calld Cva Ccloid (Figur 2). It was dvisd b Cva, who trmd it th ccloidum anomalarum ([2, p. 29], [8]). Its polar quation is ρ = + 2cos2ϕ, ϕ [0,2π]. (2) Figur 2: Cva Ccloid (k = or n = 2k + = 3). In [3] a gomtrical construction was dfind from which a gnralization of sctri of Cva coms. Lt b a lin givn b th origin and angl α btwn ais + and, as th angl of polar coordinats of (Figur 3). Lt th point A 0 coincid with. Lt th point A b givn on such that th distanc btwn th points and A is. Lt th point A 2 b on ais such that th distanc of A and A 2 is qual also to and A 2 if it is possibl. Thn lt th nw point A 3 b on th lin again such that A 2 A 3 = and A 3 A if it is possibl. Rcursivl, w can dfin th point A i (i 2) on th lin or on ais if i is odd or vn, rspctivl, whr A i A i = and A i A i 2 if it is possibl. For all α th point A i ists. Figur 3 shows th first si points. If α is small nough thn A i is btwn points and A i+2. Lt angl A i+ A i b α i, thn α i = iα can b provd asil. If A is on th ais w obtain a similar gomtric construction (Figur ). Ths constructions gav a nw proof for som trigonomtric connctions [5]. 2

2 Th paramtric quation sstm of th orbits of th points in [3] is dtrmind not onl whn th point A is on lin, but also whn it is on ais. In cas of vrtics A n (n ) th paramtric quation sstm of th curvs is n (α) = n (α) = cosα U n (cosα) sinα U n (cosα), and th polar quation of th curvs whn α [0,2π] is ρ n (α) = U n (cosα), () whr U n () is th Chbshv polnomial of th scond kind. (Som orbits can b sn on Figur 3 and.) Th rcursiv dfinition of th Chbshv polnomials of th scond kind U l () is U 0 () =, U () = 2, U l+ () = 2U l () U l (), l. Whn th substitution = cosϕ givs th prssions sinlϕ = sinϕ U l (cosϕ) [6]. A 5 A 3 α α A 2α 2α α α 3α 3α 5α 5α 3 5 A 2 Figur 3: Gnralizd sctri of Cva in cas n = 5. A A 5α 5α 2 3α 3α α αa 2α 2α α α 6α 2 A 3 A 5 6 A A 6 A 6 (3) (5) Proof. Sinc sin(2k + )α = sin(k + k + )α if ϕ = α thn w hav = sinkαcos(k + )α + coskαsin(k + )α = sinkαcos(k + )α +coskα(sinkαcosα + coskαsinα) = sinkαcos(k + )α +sinkαcoskαcosα + cos 2 kαsinα = sinkαcos(k + )α +sinkα(cos(k + )α + sinαsinkα) +cos 2 kαsinα = 2sinkαcos(k + )α + sin 2 kαsinα +cos 2 kαsinα = 2sinkαcos(k + )α + sinα, U 2k (cosα) = sin(2k + )α 2sinkαcos(k + )α + sinα = sinα sinα = sinkαcos(k + )α + 2. sinα Th Cartsian quation of curvs (without th point in th origin) dfind with (3) or () is = Un 2 2, (6) whr W tnd th quation (6) to ngativ valus n. Th dfinition of th Chbshv-polnomials for ngativ inds with dfinition (5) is U l () = 2U l () U l+ (), l <. (7) Now, U () = 0 and U m () = U m 2 (), (m 2) and implis n 0. If n = 2k + thn quation (6) givs th sctri of Cva. thrwis, if n = 2k w gt th union of curvs in cas n and n (s Figur 5 and 6). Figur : Gnralizd sctri of Cva in cas n = 6. Figur 6: Union of curvs in cas n = and n =. Morovr, th polar quation of folium curv is Figur 5: Folium curvs in cas n = and n =. Lmma If n = 2k + th curvs dfind b quations () and () for a = ar th sam. (Compar th Figurs and 3.) ρ = cosϕ ( asin 2 ϕ b ), ϕ [0,2π], (8) and th curv dfind b quations () is th folium curv if n = and a = 2, b =, as U 5 (cosα) = U 3 (cosα) = 8cos 3 α + cosα = cosα(2cos 2 α ) = cosα(8sin 2 α ) (s Figur 5). 3

3 Hr ar som Cartsian quations from (6): n = : = (circl) n = 2: ( ) 2 = 2 n = 3: ( ) 3 = (3 2 2 ) 2 (Cva ccloid, Figur 2) (two circls) n = : ( ) = 6 ( 2 2 ) 2 (union of foliums, Figur 6) n = 5: ( ) 5 = ( ) 2 (sctri of Cva, Figur ). Th two biggst loops ar vr similar to th loops of th lmniscats. In Figur 7 w can s that th angls btwn an two lins m i (i =,...,2n) ar th multipls of π/(2n), whr m i is a tangnt lin of th curv in th origin or a lin gos through th origin and on of th trma of th curv ( = n or = ±). For som mor dtails, for mor figurs and for a gnralization s [3, ]. Ths curvs can also b considrd as th gnralizations of th wllknown ros curvs (s in []). π 2n Figur 7: Curv with som proprtis in cas n = 5.. Sctri on th sphr In this subsction w dtrmin th orbit of th point A n (n ) with similar conditions as in Sction on a sphr. W considr a sphr with radius with quation z 2 =. Lt th as ξ and ψ of th coordinat sstm on it, with origin K(,0,0), b main circls according to Figur 8. z Lt also b a main circl through point K and lt th rotation angl btwn th ais ξ and th lin b α, whr 0 α 2π. Morovr, lt th distanc btwn two conscutiv points b d, whr 0 < d < π/2. Figur 9 dmonstrats th construction in a plan with coordinat as ξ and ψ. (Compar Figurs 8 and 9.) Morovr, Figur 9 shows an odd cas whn A lis on and n = 5. Lt α i (i ) b th angls A i A i A i+ and A i A i+ A i and lt 2β i b th angl A i A i A i+. (If α i < π/2 thn th point A i (i 2) is furthr from th origin thn A i 2.) b2 A d 2 a b b A b A b A b 3 A a a5 a2 a b3 A b A b A b 5 5 A 2 A 5 5 A 6 Figur 9: Construction in th plan with coordinat as ξ and ψ. Lt th orthogonal projction of A i to ξ or in cas if i is odd or vn, rspctivl, b A i. W dnot b b i and a i th sphrical sgmnts A i A i = A i+a i and A ia i, rspctivl. Now w dtrmin th angls α i rcursivl. Lmma 2 If i 2 thn α i = π 2β i α i 2, whr α 0 = 0, α = α and β i = arccot(cosd tanα i ). Proof: From th triangl A 0 A A 2 w obtain at point A that α 2 = π 2β (s Figur 9). W suppos th lmma holds for an j from 2 up to i. From th triangl A i 2 A i A i (i 3) w obtain at point A i that α i 2 + α i = π 2β i and b th us of th sphrical trigonomtric idntit cotα i cotβ i = cosd in th right angld triangl A i 2 A i A i w gt th lmma. From th triangl A i A i A i using th sphrical trigonomtr th nt lmma holds. Lmma 3 sina i = sind sinα i, tanb i = tand cosα i. Figur 8: Construction on th sphr. Thorm follows from th summation of th lmmas.

4 Thorm Th quation sstm of th sctri on th sphr is whr n (α) = cosψ n (α)cosξ n (α) n (α) = cosψ n (α)sinξ n (α) z n (α) = sinψ n (α), ξ n (α) = ψ n (α) = a n. { 2(b + b b n 2 ) + b n if n = 2k +, + 2(b 2 + b + + b n 2 ) + b n if n = 2k, W mntion that from th triangl A na n th quation sinξ n (α) = cotα tanψ n (α) givs th implicit connction btwn th coordinats. B using th paramtric quations from Thorm, w obtain th visualizations of th curvs in th softwar Mapl 7 (from Maplsoft). In this articl, w improv th qualit of Mapl-graphics b r-rndring in th softwar PV-Ra. Figurs 0 2 show som curvs on th sphr in cas n = 3, and 5. Figur 2: Sctri curvs on sphr in cas n = 5, d = π/8 and d = π/..2 Sctri curvs with strographic projction W obtain similar curvs on th surfac of a sphr with th strographic projction of th sctri. Lt th curv b in plan z = R and projct it from point N(0,0,R) into th sphr with cntr (0,0,0) and radius R (Figur 3). N z P P Figur 3: Strographic projction. Figur 0: Sctri curv on sphr in cas n = 3 and d = π/6. In this cas on can asil gain that th strographic projction of a gnral point P(,, R) from th plan is P (c,c, 2c), whr c = R 2 /( R 2 ). Thus th strographic projction of th curv with quation (3) is n (α) = c(α)cosα U n (cosα), n (α) = c(α)sinα U n (cosα), z n (α) = 2c(α), (9) whr R 2 c(α) = Un 2, (cosα) + α R2 [0,2π]. Figur : Sctri curvs on sphr in cas n =, d = π/6 and d = π/3. Figurs and 5 giv som ampls in cas n = 3,, 5, 6, 7 and 0 whr R = and in th figurs w rotatd th curvs around ais z for bttr visualization. 5

5 KoG La szlo N mth: Sctri Curvs on th Sphr Figur : Strographic projction of sctri in cas n = 3, 5 and 7. Figur 5: Strographic projction of sctri in cas n =, 6 and 0..3 Rolld sctri on th sphr In this subsction w giv curvs which ar rolld to sphr. Lt th radius of th sphr with cntr (0, 0, 0) b R and lt th plan of sctri b th plan z = R (with point S(0, 0, R)) according to Figur 6. z Lt P with paramtr α b on of th points of th sctri dfind b (3), tak th plan Π incidnt to th ais z and paralll to dirction α (thus P is on Π) and lt P0 Π b a point on th sphr so that th lngth of arc SP0 b qual to ρn (α) from polar quation (). In that wa w projct th point of th sctri onto th sphr and th quation sstm of th curvs (α [0, 2π]) is N n (α) = R cos(r(α)) cos α, n (α) = R cos(r(α)) sin α, zn (α) = R sin(r(α)), P S whr r(α) = P Figur 6: Rolling of th sctri onto sphr. 6 (0) Un (cos α) π. R 2 Figurs 7 and 8 show som ampls of th rolld sctri curvs, whr th curvs ar rotatd for bttr visualization.

6 Figur 7: Sctri rolld onto sphr (n = 3,, 7 and R = ). Figur 8: Sctri rolld onto sphr (n = 5, 6, 9 and R =, 2, 2). Rfrncs [] S. GRJANC, Ros surfacs and thir visualizations, Journal for Gomtr and Graphics, 3() (200), 9. [2] E. S. LMIS, Th Ccloid of Cva. 2.7 in Th Pthagoran Proposition: Its Dmonstrations Analzd and Classifid and Bibliograph of Sourcs for Data of th Four Kinds of Proofs, 2nd d. Rston, VA: National Council of Tachrs of Mathmatics, (968), [3] L. NÉMETH, A nw tp of lmniscat, NmE SEK Tudomános Közlménk XX. Trmészttudománok 5. Szombathl, (20), 9 6. [] L. NÉMETH, Ros curvs with Chbshv polnomials, Journal for Gomtr and Graphics, 9(2), (205), [5] L. NÉMETH, A gomtrical proof of sum of cosnϕ, Studis of th Univrsit of Žilina, Mathmatical Sris, 27 (205), [6] T. J. RIVLIN, Chbshv polnomials, Nw York Wil (990). [7] E. V. SHINKIN, Handbook and Atlas of Curvs, CRC Prss (995). [8] E. W. WEISSTEIN, Ccloid of Cva. From MathWorld A Wolfram Wb Rsourc. mathworld.wolfram.com/ccloidofcva.html László Némth -mail: nmth.laszlo@mk.nm.hu Institut of Mathmatics Univrsit of Wst Hungar 900 Sopron, Ad E. u. 5, Hungar 7