HELICAL TUBES TOUCHING ONE ANOTHER OR THEMSELVES

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1 15 TH INTERNATIONAL CONFERENCE ON GEOMETRY AND GRAPHICS 0 ISGG 1-5 AUGUST, 0, MONTREAL, CANADA HELICAL TUBES TOUCHING ONE ANOTHER OR THEMSELVES Peter MAYRHOFER and Dominic WALTER The Univerity of Innbruck, Autria ABSTRACT: Helical pipe urface a model for tube and teel cable are not only of interet in technical cience but alo in knot theory a well a in molecular biology and polymer cience. Mainly the double helical tube a the tructure of the DNA molecule which contain the human genetic information i very popular. In thi paper ome pecial helical pipe urface are tudied from the geometric point of view. Keyword: Double helix, multiple coaxial helice, helical tube, toru rotoid, rotoidal tube. 1. INTRODUCTION A helical tube i generated when a circle i moved along a helix o that it plane tay perpendicular to the helix all the time. Thi motion i called a Frenet motion. In CAD ytem thi way of urface generation i uually implemented a one rail weep command. Obviuoly the tube radiu mut not exceed the (contant) radiu of curvature of the helix to prevent elf interection of the tube. If two tube T 1, T touch each other not only in dicrete point but along continuou curve their correponding helice h 1, h mut belong to the ame crew motion, which mean that they have a common crew axi and the ame crew parameter q = p in their parametric repreentation h = [ r co( t), r in( t), pt], h = [ r co( t+ w), r in( t+ w), qt] (w denote the angle by which the helice h 1, h are rotated againt each other). Therefore they all have the ame pitch (i.e. height per one turn) of πp. For a proof of thi tatement we tart with a fixed point P 1 h 1 belonging to a parameter value t = t 0 and a variable point P h belonging to t 0 + w +. Now we compute the tangent vector v1 = [ r1in( t0), r1co( t0), p], v = [ rin( t0 + w+, rco( t0 + w+, q] in P 1, P and the vector n = PP 1. If n i a common normal of h 1, h then the dot product v 1 n and v n hould be zero. Thi lead to the equation rr in( w + p = and rr in( w + q = which imply q = p. The cae of different ign of p and q never lead to tube touching along a curve and can be excluded. Thu q= p i a neceary and ufficient condition for the exitence of common normal of helice h 1, h which are needed for the contruction of touching tube. If thi condition i fulfilled we can find common normal of coaxial helice h 1, h by interecting the normal plane of h 1 in an arbitrary point with h (Figure 1). A the common normal are not unique, there mut be one that give the minimal ditance between h 1 and h. The um of the tube radii mut not exceed thi ditance. The mathematical condition for the exitence of common normal i given by the in( w+ p equation =, (1) rr where denote the running parameter for helix h and w i the angle by which the two helice are rotated againt each other. The formula for Paper 151

2 the normal ditance between h 1 and h i: dit = r rr co( w + + r + p () 1 1 p Fig. 3: Sinc function. Fig. 1: Common normal of two helice in frontal and perpective view.. A SPECIAL SINGLE HELICAL TUBE Let u look at a ingle tube T - baed on a helix h with Radiu r - which touche itelf and ha no central hole (Figure ). It hould be remarked here that Maritan, Micheletti, Trovato, and Banavar [1] computed the lope of a ingle helix h with radiu r = 1 whoe radiu of curvature equal the radiu of a elf-touching tube around h. Thi tube i different from that one decribed here a it doe not contain the crew axi. Fig. : Single elf-touching tube containing the crew axi, frontal and top view. Without retriction of generality we can chooe r = 1. Then the condition for elf touching i in( given by formula (1): = p (3) The left ide of equation (3) i the ocalled ine cardinal function (hort inc function). The abolute minimum of the inc funtion can be computed with Maple a (rounded). Therefore elf touching i only poible for parameter value of The Tube T will have no central hole (i.e. will contain the helical axi if the half normal elfditance of the helix h in formula () equal it radiu r = 1: co( + p = 0 (4) Solving the equation (3) and (4) numerically yield three olution for p and of which only one {p , } will give the deired reult within one turn ( 0 π ). The other olution belong to value of greater than π. They have to be excluded becaue the tube mut touch itelf within the firt turn. Otherwie it would interect itelf. 3. TWO HELICAL TUBES TOUCHING EACH OTHER ALONG A STRAIGHT LINE A helix h with radiu r = 1 and lope p = 1 ha a pitch of π. Let h 0 be a rotated copy of h (rotation axi = helical axi, rotation angle = π). Then the pair (h, h 0 ) i the claic double helix. Tube T and T 0 with radii r = r 0 = 1 touch each other along the helical axi (Figure 4). Fig. 4: Double helical tube.

3 Thi reult i not new. It ha been publihed by Staiak and Maddock []. We can give a very imple explanation for it from the geometric point of view: A normal projection of h, h 0 onto a plane parallel to the helical axi how two ymmetric graph of the ine function which ha a lope of 1 at it point of inflection. The projection of a plane normal to the helix h in uch a point i tangent to the projection of h 0. Therefore the only common normal of h, h 0 interect the helical axi at right angle (Figure 5). Hence all point of contact of the tube T and T 0 mut lie on the helical axi. Obviouly thi i alo valid for helice h, h 0 with lope p > 1. Fig. 5 (left): Double helix with lope p = 1. Fig. 6 (right): Double helix with lope p < 1. For a lope p < 1 the projection of a plane normal to the helix h interect the projection of h 0 in two point within one helical turn (Figure 6). That why there exit two common normal of h, h 0 in each point of h. They have different length and lead to different pair of double helical tube. Only the pair baed on the horter common normal will be touching one another - no more along the helical axi but along another helix. Therefore thi double helical tube how a central hole. 4. TWO HELICAL TUBES WITH DIFFERENT RADII TOUCHING EACH OTHER ALONG A HELIX Let h 1, h denote helice with radii r 1, r. They hould be rotated againt each other by 180 degree around their common axi and have the ame pitch πp. Their parametric 3 repreentation hould be given by h = [ r co( t), r in( t), pt], and h = [ r co(), t r in(), t pt]. in( p Formula (1) now yield =. rr A the right ide of the equation i alway poitive and the value of the inc function ha it abolute maximum of 1 at = 0 we get the condition for contact of tube T 1, T around h 1, h : p rr. There are infinitely many olution for thi inequation. Let u therefore conider the pecial cae of equality. For = 0 the endpoint of any common normal of h 1, h belong to the ame parameter value and have therefore the ame z-coordinate. The common normal are horizontal and interect the helical axi at right angle. p r The equation p = rr rewritten a = r1 p implie that our helice h 1, h have invere lope. Figure 7a how a frontal projection of uch pecial helice h 1, h and Figure 7b a perpective view of touching tube baed on them. If we chooe r 1 = r we get an equal lope of 1 for h 1, h which lead to the pecial cae mentioned in Section 3. Fig. 7 (left): Helice with invere lope. Fig. 8 (right): Tube with different radii touching themelve along a helix.

4 5. N > COAXIAL HELICAL TUBES WITH SAME RADII TOUCHING ONE ANOTHER Let u imagine a cable made from n > drilled cord. Each two of them are helical tube of equal radiu, rotated around their common helical axi by an angle w= π / n. Due to rotational ymmetry we need to conider only two adjacent tube of them. Let T 1, T uch tube around helice h 1, h. The condition (1) for the exitence of common normal of h 1, h in( w+ i = p. (5) The function at the left ide of thi equation i dicontinuou at = 0 where it trend to +/- for any w, 0 < w < π (Figure 9). Therefore there will be alway at leat one value p atifying condition (5). Fig. 9: Graph of in(+w)/ for w = π/3. A for all value of p a common normal of h 1, h of minimal length exit, there are infinitely many poibilitie of drilling n > rope together (Figure 10 how everal olution for w = π/3). It make ene to ak for an optimal olution with repect to a certain property of the baic helice h 1, h. O Hara [3] uggeted to minimize the average rope length of the helice. It i defined by the ratio of the arc length for one turn of a helix and the maximal poible radiu of a tube around it. Fig. 10: Triple tube with different lope. Without retriction of generality the radiu of the helice concerned may be et to r = 1. A an example we compute the optimal triple tube with repect to the average rope length ARL. The arc length for one turn of a helix with radiu 1 and crew parameter p i given by L(p) = π 1+ p. The maximal poible tube radiu ρ i given by the half of the minimal normal ditance between two adjacent helice by adapting formula () a co( w+ + p ρ =. (6) in( + w) Subtituting p = according to formula (5) in L(p) and (6) lead to the function + in( w+ ARL() = 4π ( + co( w+ + in( w+ (Figure 11). Fig. 11: Graph of the function ARL(. 4

5 After ubtituting w = π /3 the minimum of ARL( can be computed numerically with Maple. Thi numerical calculation reult in min , p min and ρ for the optimized triple helical tube hown in Figure. Fig. 14: Double rotoidal tube. Fig. : Triple helical tube with optimal average rope length. 6. DOUBLE TORUS ROTOIDAL TUBES A toru rotoid i helical curve wound around a circular arc k of radiu a. It can be generated by two proportional imultaneou rotation with kew perpendicular axe of contant ditance a (ee [4] and Figure 13). It may be interpreted a a helix on a toru urface. For a circle k of fixed radiu a we can compute numerically a maximal tube radiu r for which two helical tube wound around k (in other word a double rotoidal tube) touch each other along the circle k having no other interection (Figure 14). The goal of thi computation i to find a value of r o that r i globally the minimum ditance between the two rotoid. The computation i baed on the ditance function between the two rotoid h 1 and h which erve a the pine curve of the rotoidal tube concerned. The parametric equation of h 1 and h are h = [( a rco( pt))co( t),( a rco( pt))in( t), rin( pt)] h = [( a + rco( pt ))co( t ),( a+ rco( pt ))in( t ), r in( pt )] The ditance between two arbitrary point on h 1 and h depend on a, r, p, t 1, and t where a and p are fixed and r may be replaced by r = ξa (ξ erving only a a caling factor). In order to avoid a quare root expreion we can look at the quare of the ditance intead of the ditance itelf. Finally thi function d could be plotted a a urface D above the [t 1, t ]-plane (Figure 15). After a tranlation of D downward the z-axi by (r) the global minima hould be point of D for which the coordinate plane z = 0 i a tangent plane. Fig. 13: Toru rotoid. 5 Fig. 15: Plot of the urface D. The computation with Maple how, that

6 thee global minima lie on the traight line t 1 = t which mean that the hortet ditance between the rotoid h 1 and h are alway found on common normal interecting the circle k. Unfortunately we could not find a imple formula which decribe the interdependence of the parameter p and the radiu r, but for given a and p the maximum tube radiu r could be computed. Figure 16 how variou double rotoidal tube for fixed a = 10 and parameter value: p 1 = 0 (which i the trivial cae of two tori touching each other along a circle), p = 1, p 3 =, p 4 = 3, and p 5 = 4. The correponding tube radii are r 1 = 5, r , r 3.907, r , and r peudohelical curve h 1, h wound around b (Fig. 17) which can erve a pine curve for pipe urface with radiu r. We call them peudohelical becaue the rotation around b i carried out not uniformly, but with changing peed along the curve. The reaon for thi i that b i not parameterized by arc length (which would not alway be poible or at leat rather difficult to compute). Fig. 17: Peudohelical curve around Bezier curve, r = 0.. Fig. 16: Double rotoidal tube belonging to different parameter value. 7. GENERALIZED DOUBLE HELICAL TUBES AROUND A SPATIAL CURVE Double helical tube could not only be wound around planar curve. It i alo poible to compute uch urface touching each other along a ufficiently mooth patial curve. For example we conider a Bezier curve b of third order given by 4 corner (0 / 0 / 0), (1 / 0 / 0), (1 / 0 / 1) and (1 / 1 / 1) of the unit cube. The parametric repreentation of b i given by Following the ame concept a in Section 7 we could compute numerically a maximum value of the tube radiu r max under the condition that r max i the abolute minimum ditance between the two peudohelice h 1 and h. Figure 18 how thee pipe urface which touch each other along the given Bezier curve b and further what wa truly not expected in at leat one iolated point P with rounded coordinate (0.54 / 0.03 / 0.6). b 3 3 t(1 t) + 3 t (1 t) + t, 0 t 1 3 t (1 t) + t 3 = t 3 After applying the tranformation of a Frenet motion along the curve b to the point (0, 0, r) and (0, 0, -r) we get two 6 Fig. 18: Peudohelical tube of maximal radiu around Bezier curve.

7 REFERENCES [1] A. Maritan, C. Micheletti, A. Trovato, J.R. Banavar. Nature 406 (000), 87 [] A. Staiak and J.H. Maddock. Nature 406 (000), [3] J. O Hara. Ideal bet packing, and energy minimizing double helice. In Proceeding of Statitical phyic and topology of polymer with ramification to tructure and function of DNA and protein, Kyoto 010 [4] G. Glaeer, H. Stachel. Open GL+advanced geometry, Springer New York, 1999 ABOUT THE AUTHORS 1. Peter Mayrhofer, PhD, i aitant profeor at the Unit Geometry and CAD, Univerity of Innbruck, Autria. Hi reearch interet include graphic programming, medical x-ray photogrammetry in orthopaedic and didactic of 3d-modelling. He can be reached by <peter.mayrhofer@uibk.ac.at> or through potal addre: Univerity of Innbruck, Unit Geometry und CAD, Technikertrae 13, A-600 Innbruck.. Dominic Walter, PhD, i a freelance worker at the Unit Geometry and CAD, Univerity of Innbruck, Autria, and deal with reearch on the analyi of mechanim uing algebraic method. Contact by <cab679@gmx.at> or through potal adre: Univerity of Innbruck, Unit Geometry and CAD, Technikertrae 13, A-600 Innbruck. 7