Extension of the Villarceau-Section to Surfaces of Revolution with a Generating Conic

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1 Journl for Geometry n Grphics Volume 6 (2002), No. 2, Extension of the Villrceu-Section to Surfces of Revolution with Generting Conic Anton Hirsch Fchereich uingenieurwesen, FG Sthlu, Drstellungstechnik I/II Universität Gesmthochschule Kssel Kurt-Wolters-Str. 3, D Kssel, Germny emil: nhirsch@hrz.uni-kssel.e Astrct. When surfce of revolution with conic s meriin is intersecte with oule tngentil plne, then the curve of intersection splits into two congruent conics. This ecomposition is vli whether the surfce of revolution intersects the xis of rottion or not. It hols even for imginry surfces of revolution. We present these curves of intersection in ifferent cses n we lso visulize imginry curves. The rguments re se on geometricl resoning. ut we lso give in specil cses n nlyticl tretment. Keywors: Villrceu-section, ring torus, surfce of revolution with generting conic, oule tngentil plne SC 2000: 51N05 1. Introuction Due to Y. Villrceu the following sttement it is vli (compre e.g. [1], p. 412, [3], p. 204, or [4]): The curve of intersection etween ring torus Ψ n ny oule tngentil plne τ splits into two congruent circles. We ssume tht r is the rius of the meriin circles k of Ψ n tht their centers re in the istnce, > r, from the xis of rottion. We generlize n replce k y conic which my lso intersect the xis. Uner these conitions it is still true tht the intersection with oule tngentil plne τ is reucile. 2. Extension of the Villrceu-section The following two theorems will e prove y stnr rguments from Algeric Geometry: ISSN /$ 2.50 c 2002 Helermnn Verlg

2 122 A. Hirsch: Extension of the Villrceu-Section on Surfces of Revolution Theorem 1 Let τ e oule tngentil plne τ of surfce Ψ of revolution with conic k s meriin. Then τ intersects Ψ in two congruent conic sections. Theorem 2 The two congruent conics of v := τ Ψ ccoring to Theorem 1 re either rel n of the sme type s the generting conic k, or they re imginry. In the sequel the intersection curve etween Ψ n τ is clle Villrceu-section or riefly v-section. 3. Geometricl Tretment Accoring to clurin s theorem (see e.g. [2], p. 49) n irreucile plne curve of orer n cn possess t most (n 1)(n 2) n := 2 oule points. A surfce of revolution Ψ with generting conic k is n lgeric surfce of orer four. Thus ny plne section is n lgeric curve of orer four which in the irreucile cse cn hve t most 4 = 3 oule points. If the curve hs more thn 3 singulrities, then it splits into t lest two irreucile components. Let k enote the imge of the genertor k uner reflection in the xis of rottion. Then k n k form complete meriin section of the surfce Ψ (see Fig. 1). A oule tngentil plne τ psses through common tngent t of k n k which is not perpeniculr to. (In Fig. 1 t is common inner tngent.) Then τ touches the surfce Ψ t the points k n k. These points re two oule points of the intersection v = τ Ψ. t K k' D' ' K' k Q' D q Q Figure 1: Doule tngentil plne τ of the surfce Ψ of revolution The two coplnr meriin conics k, k intersect ech other in four points. 1 Two of them, 1 The cse k = k with Ψ eing twofol covere quric is exclue here. Furthermore we exclue the trivil cse where the meriin conics touch the rottion xis. In this cse the v-sections coincie with complete meriins.

3 A. Hirsch: Extension of the Villrceu-Section on Surfces of Revolution 123 the points K, K, re locte on the xis of Ψ n they cn e rel or conjugte imginry. The two remining points Q, Q re symmetric with respect to. Uner rottion out they trce oule circle q of Ψ. This circle is symoliclly inicte in Fig. 1 s well s the points K, K. The circle q cn e rel or imginry epening on whether the points Q, Q re rel or conjugte imginry. In the generic cse the oule tngentil plne τ intersects the oule circle q t two oule points D, D. 2 Thus the intersection curve v = τ Ψ contins t lest 4 oule points,, D, D. Hence v splits into two prts. Due to the symmetry of v with respect to t the two components re congruent conics c, c. This proves Theorem 1. A generting ellipse k hs two complex conjugte points t infinity. Therefore Ψ intersects the plne t infinity long two imginry conics. Hence ny component of v-section is either n ellipse or imginry. For prol k the surfce Ψ touches the plne t infinity long rel conic. In the hyperolic cse Ψ hs t infinity two rel perhps coinciing conics. The components c, c of the v-section re in certin reltion to the meriins k, k : For non-intersecting meriins τ, n re rel; for intersecting k, k the plne τ is imginry n, re complex conjugte. Hence lso in these cses the components of v-section re either imginry or of the sme type s k. The well-known Villrceu-section is the intersection of ring torus with oule tngentil plne. It consists of two congruent circles 3 ccoring to the presente theorems (see Fig. 2). We consier few other specil exmples: For torus Ψ the points Q, Q of intersection of the meriin circles k, k re conjugte imginry n t infinity. The oule circle q of Ψ is the solute circle. At ring torus the oule tngentil plne τ touches t the rel points, n it intersects the solute circle q t the points D, D. Thus the v-section splits into two congruent circles tht pss through the points n of tngency n the solute points. If k, k re equilterl hyperols with xes prllel to, their points Q, Q of intersection re rel n t infinity. Thus q is rel conic. When the inclintion of τ is greter thn 45 then the oule points D, D of v re rel points t infinity; the v-section consists of two congruent hyperols (see Fig. 7). It turns out tht otherwise v consists of imginry curves. If k, k re prols with xes perpeniculr to, then they shre prt from the finite points K, K of intersection their infinite point Q. The surfce Ψ touches the plne t infinity long q. This shows tht the v-section consists of two congruent prols with their xis perpeniculr to. 4. Anlyticl Tretment Any v-section of torus with generting imginry circle k is oviously imginry. However, even n pple-shpe so-clle spinle torus Ψ tht intersects the rottion xis hs no rel oule tngentil plne τ. Hence, the v-section consists of two imginry circles. Such cses re noticele s they provie the possiility of eling with imginry structures n their visuliztion. Imginry structures cn e hnle well through nlyticl equtions. 2 Exmples with the specil cse Q = Q re presente s Cses 7 n 8. 3 The remrkle property tht these circles re isogonl trjectories of the prllel circles on Ψ is not consiere here (compre [5], vol. I, pp ).

4 124 A. Hirsch: Extension of the Villrceu-Section on Surfces of Revolution In orer to present the nlyticl tretment s simply s possile we choose crtesin coorinte system (; x 1,, ) with the xis of rottion s -xis. The generting conic k is specifie in the -plne. An nlysis of the v-section of these surfces of revolution shows certin reltionship mong them tht hs ecome evient only fter inclusion of imginry elements. nly Cse 9 shows n exmple with generting ellipse. A r x 4 x 1 Figure 2: v-section of ring torus (Cse 1, : r = 2 : 1) Cse 1: Generting circle k(, r), > r : In the eqution of the meriin k : ( ) r 2 = 0 (1) we replce 2 y x2 1 + x2 2 n get fter seprting the terms which re liner in Ψ : ( r 2 ) ( 1 + 2) = 0. (2) A plne through the x 1 -xis oeys the eqution = m. We sustitute this in eq. (1) n otin

5 A. Hirsch: Extension of the Villrceu-Section on Surfces of Revolution 125 x 4 A Figure 3: Position of the oule tngentil plne ( A) (m 2 + 1) r 2 = 0. (3) This plne is tngent to Ψ if the iscriminnt D = 2 (m 2 + 1)( 2 r 2 ) vnishes. This results in τ : = m with m = r 2 r 2. (4) Due to (4) we eliminte in (2). After simplifiction the top view v 1 of v oeys v 1 : ) (x + 2 r 2 x r ( 1 + x2 2 ) = 0. (5) We introuce crtesin coorinte system (; x 1, x 4 ) in τ (see Fig. 3 showing front view). Then ue to x 4 = 1 + m 2 we otin from (5) fter some computtion the eqution of the v-section v : [ (x1 r) ] [ (x 1 + r) ] = 0 (6) consisting of two circles with rius (see Fig. 2). x 1 x 4 i A r rel imginry oule point Figure 4: v-section of orn torus (Cse 2, : r = 1 : 2)

6 126 A. Hirsch: Extension of the Villrceu-Section on Surfces of Revolution Cse 2: Generting circle k(, r), < r : In this cse the slope m of τ is imginry ccoring to (4). Fig. 4 shows the meriin section of Ψ in the -plne. ut t the sme time it shows lso its imge uner the imginry ffine trnsformtion (, ) (, i ) which trnsforms the meriin circles into equilterl hyperols. After this trnsformtion τ hs rel imge. Also the x 4 -xis in τ is imginry. Nevertheless the eqution (6) shows two symmetric hyperols which re isplye in Fig. 4 s if x 4 woul e rel xis. Note tht this uxiliry view oes not correspon to the imginry trnsforme front view. Therefore there is no orer line connecting the two views of the point of contct. Cse 3: Generting imginry circle k(, r = iρ), ρ R : Agin the slope m of τ is imginry ccoring to (4). The two circles of the v-section ccoring to (6) hve complex conjugte centers. The top view in Fig. 5 shows t the sme time the imge uner the imginry scling (, ) (, i ). The uxiliry view in Fig. 5 shows the imge uner (x 1, x 4 ) (ix 1, x 4 ) which gives equilterl hyperols constituting the v-section. Cse 4: Generting equilterl hyperol k(, r), > r : In nlogy to Cse 1 we compute k : ( ) 2 3 r 2 = 0, Ψ : ( 1 + x2 2 x r 2 ) ( 1 + x2 2 ) = 0, τ : = m, m = ir/ 2 r 2. The top view of the v-section Ψ τ oeys ) 2 v 1 : ( r 2 x r ( 1 + 2) = 0. We introuce in the complex plne τ crtesin coorinte system (; x 1, x 5 ) (in the unitry sense). Due to x 5 := 1 m 2 we replce in the eqution of v 1 y x 5 n otin ( v : x r 2) 2 4r 2 1 = 0 which cn e ecompose into (x 1 ± r) = 2 escriing two congruent circles in the imginry plne. These circles re isplye in Fig. 6. Cse 5: Generting hyperol k(, r), < r : Following the computtions of Cse 4 we get rel plne τ (see Fig. 7). v consists of two hyperols. When we set x 5 := i 1 m 2 = then the projection of v into the x 1 x 5 -plne oeying r2 2, (x 1 ± r) 2 5 = 2 (7) consists of two equilterl hyperols which re isplye in Fig. 7.

7 A. Hirsch: Extension of the Villrceu-Section on Surfces of Revolution 127 i A r x 5 x 1 Figure 5: v-section of n imginry torus (Cse 3, : r = 2 : i) i A r x 5 x 1 Figure 6: v-section of surfce of revolution with generting hyperol, rottion xis is intersecte (Cse 4, : r = 2 : 1)

8 128 A. Hirsch: Extension of the Villrceu-Section on Surfces of Revolution x 3 A r x5 x1 Figure 7: v-section of surfce of revolution with generting hyperol, rottion xis is not intersecte (Cse 5, : r = 1 : 2) Cse 6: Generting hyperol k(, r = iρ), ρ R : τ is rel plne. Eq. (7) represents two imginry circles with complex conjugte centers. Fig. 8 shows their imge uner the imginry scling (x 1, x 5 ) (ix 1, x 5 ). Cse 7: Generting prol k with p, > 0 : In the sme wy s in previous cses we get The top view of the intersection τ Ψ oeys k : 3 = 2p( ), Ψ : ( 3 + 2p) 2 4p 2 ( 1 + 2) = 0, τ : = m, m = p/2. v 1 : ( x ) = 0

9 A. Hirsch: Extension of the Villrceu-Section on Surfces of Revolution 129 i A r x 5 x 1 Figure 8: v-section of surfce of revolution with generting hyperol, rottion xis is intersecte (Cse 6, : r = 2 : i) which cn e ecompose into the equtions 2 = ±4(x 1 ± ) (8) representing two prols (see Fig. 9) which turn out to e inepenent from the initil prmeter p. Cse 8: Generting prol k with p > 0, < 0 : This time the oule tngentil plne τ is imginry n eq. (8) represents two imginry prols. Fig. 10 shows their imge uner n imginry scling. We omit here the trivil cses where the conic k egenertes. Remrks The eight presente cses cn e rrnge in pirs which re corresponing uner the imginry ffine trnsformtion (x 1,, ) (x 1,, i ). The pirs re 1 4, 2 5, 3 6, n 7 8. Wht is rel in one cse tht is imginry in the corresponing cse. This cn lso e seen y compring the corresponing Figures 2 6, 4 7, 8 5, n 9 10.

10 130 A. Hirsch: Extension of the Villrceu-Section on Surfces of Revolution x3 p A x2 ' x2 x1 Figure 9: v-section of surfce of revolution with generting prol, rottion xis is not intersecte (Cse 7, : p = 2 : 1) x3 p i A x2 x2 x1 Figure 10: v-section of surfce of revolution with generting prol, rottion xis is intersecte (Cse 8, : p = 2 : 1)

11 A. Hirsch: Extension of the Villrceu-Section on Surfces of Revolution 131 Cse 9: Generting meriin ellipse k in generl position: k' k c' x 4 ' c x 1 Figure 11: v-section of Ψ with meriin ellipse k in generl position (Cse 9) As n exmple we specify k s n ellipse with semixes = 2 n = 1. The principl xis is rotte uner ϕ = 45 n the center of the ellipse hs the istnce = 2 to the xis (see Fig. 11). We procee in the sme wy s in Cse 1 n otin the equtions: Ψ : k : = 0, ( 5x ) 2 (6x3 + 20) 2 ( 1 + 2) = 0, 31 τ : = 4 2 3/4 = The top view v 1 of the v-sections oeys We sustitute n otin v 1 : x x2 1 x x4 2 = 0. = x 4 v : x x x x 4 4 = 0 This escries two congruent ellipses (see Fig. 11) with equtions x 1 x ± x x = 0.

12 132 A. Hirsch: Extension of the Villrceu-Section on Surfces of Revolution 5. Conclusions In the present pper we hve performe in simple wy n extension of the term Villrceusection y replcing the ring torus y surfces of revolution with conics s meriins. There re mny other wys for generliztion. It is possile, for exmple, to specify generting conic section not in meriin plne or to trnsform the surfce of revolution into cyclies. References [1] R. ereis: Drstellene Geometrie I. Akemie-Verlg, erlin [2] G. Fischer: Plne Algeric Curves. Americn themticl Society, Provience [3] K. Struecker: Vorlesungen üer Drstellenen Geometrie. Vnenhoeck & Ruprecht, Göttingen [4] G. Weiss: Villrceu-Kreise es Ringtorus, ein elementrer Zugng. IDG 2 (1984). [5] W. Wunerlich: Drstellene Geometrie I. I-Hochschultschenuch. 96, iliogrphisches Institut, nnheim Receive y 23, 2001; finl form ctoer 15, 2002

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