TANGENCY AMONG THREE CYLINDERS A HYPERBOLOID AND A TORUS

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1 TANGENCY AMONG TREE CYLINDERS A YPERBOLOID AND A TORUS Pul ZSOMBOR-MURRAY McGill University, Cnd A chllenging exercise in solid modeling is exmined nd developed s n opportunity to rise students consciousness in geometric thinking by considering how to extrct dditionl, required design prmeters from n initil specifiction. This involves the formultion nd solution of vriety of problems in nlyticl geometry nd inevitble digression into some simple ppliction of polynomil discriminnts nd the notion tht pitflls s well s dvntges ccompny symmetric configurtion. Keywords: Cylinder, yperbol, Torus, Tngency... hl , version - 5 Mr 009. SOLID MODELING A TABLE Look t ig.. The problem t hnd is to extrct the necessry design informtion from the following specifictions so tht 3D model of this smll ptio cocktil tble my be ssembled from five simple prts, very esily constructed individully with Solid Works once the design informtion is vilble. The tble top is 500mm dimeter by 5mm thick disc nd sits on Three legs tht re cylinders of 50mm outer dimeter nd inclined t the ngle of the sloping edges of regulr tetrhedron with one bse horizontl. Legs mutully touch, ech pir on point t hlf their totl sloping leg height, i.e., legs re beveled t n ngle so their ellipticl ends lie in horizontl plnes on the bottom of the tble top nd the floor. Verticl distnce between floor nd under surfce of tble top is 500mm. A ring in the shpe of torus or perfect doughnut holds the legs together nd touches them. The smllest circulr section of this torus hs dimeter of 50mm. The tsk will be to mp the ellipticl footprints of the legs onto the under surfce of the tble top nd to the floor nd to find the inner rdius of the doughnut hole. igure : A Round Tble with Three Legs eld by Ring. Expecttion The student is expected to produce nd deliver:- A nice CAD drwing of the hyperbol nd the minor circulr torus section, of rdius R, in tngency s ws done in the circle nd ellipse in ig. 5,

2 hl , version - 5 Mr 009 An nlysis to find the minor (meridin) circle centre rdius x of the torus nd the points of tngency of this circulr section with the hyperbol on the xil section of the hyperboloid swept by the tble legs nd Of course, the entire five piece solid model.. NEAT DESCRIPTIVE GEOMETRY The region where the three legs nd torus touch re described constructively in the top nd elevtion view drwing shown in ig. while ig. 3 shows one of the leg xes in end or point view. These three illustrtions, including ig., were tken from n rticle tht ppered recently in IBDG []. These my help if you cn red multiview scled drwings nd hve lerned little descriptive geometry.. Touching Legs nd Torus Now the leg xis geometry nd tht of the hyperbolic externl envelope of the legs will be ddressed so tht the legs nd the torus cn be constructed. irst the mutul tngency of the three legs will be considered using two pproches. In the first, conic coefficient mtrix nd its dul, cofctored form will be used. In the second, the reltion between tngency nd double roots is shown thus providing simple introduction to discriminnt methods. The enveloping hyperboloid will be found by identifying the ruling line, on leg cylinder, tht is furthest from the hyperboloid xis.. Slope of the Legs To find the slope of the legs, from the specifiction tht their cylindricl xes re prllel to three intersecting edges of regulr tetrhedron nd their slope is the ngle θ between n edge nd the plne formed by the remining two intersecting edges, imgine unit cube with the three intersecting edges on its fce digonls thus. OA, OB, OC, O(0, 0, 0) A(,, 0), B(0,, ), C(, 0, ) igure : Leg in True View nd One-Sheet Externl yperbolid of Revolution igure 3: End View of Leg

3 hl , version - 5 Mr 009 O, A, B, C re cube vertices nd the unit norml to plne ABC is n while u is the unit vector long AO. n = AB AC AB AC = 3 [ ] T u = [ 0] T The inner product n u = sin θ = /3..3 Leg Cotngency Conditions Imgine tht the three legs re lredy in their proper positions. Exmining the lyout in top view projection like the upper illustrtion in ig.. The equilterl tringle in the centre is reproduced in ig. 4 nd represents the three leg xis. Sectioning the legs with successive horizontl plnes proceeding from the floor up one will eventully rrive t level, exctly mid-wy up, where the ellipticl sections of the legs re cotngentil. Only one such ellipse is shown in ig. 4 but two other, congruent ones my be visulized with their mjor xes on the other two sides of the equilterl tringle. It is obvious tht one need not consider ellipse tngency since the right bisectors of the tringle will more simply serve the sme purpose. Now consider the ellipse in stndrd form s shown in ig. 4 in the Crtesin frme on origin O nd xes x, x. One knows the leg cylinder rdius is b = 5mm nd sin θ = b where is the ellipse semi-mjor xis length nd θ is the slope ngle of legs prllel to the concurrent edges of regulr tetrhedron, i.e., sin θ = /3 nd = 5 3/mm. Wht is not known is w the distnce between hyperboloid s xis of symmetry nd ny leg xis..4 Coefficient Mtrix nd Cofctors We hve sitution where the five conditions necessry to define conic, the ellipse in this cse, include four tngent lines, p on points S x p S P O up Q q s r x b w R igure 4: Tringle nd Ellipse nd P, q on points P nd Q, r on points Q nd R nd s on points R nd S. urthermore one knows tht the point (not shown) T (0, b) is on the ellipse, too. In order to review principles used below, refer to []. Consider the three conic coefficient mtrices in the following expression. A 00 A 0 A 0 A 0 A A A 0 A A A A A A 0 A A 0 A A 0 A A 0 A A 00 A A 0 A 0 A A 0 A A 0 A 0 A 00 A A 0 A A 0 A A 0 A 0 A 00 A A 00 A A The first coefficient mtrix is tht of conic tngent line eqution, i.e., pre-multiplying it with row vector of plnr line coordintes nd postmultiplying by the column vector produces the sclr line eqution of the conic. Cofctoring up

4 hl , version - 5 Mr 009 produces the middle mtrix, the coefficient mtrix of the corresponding point conic with coefficients represented s ij. With the four tngent line corner point homogeneous coordintes from ig. 4 we get the four tngent line eqution coefficients. P { : 0 : w}, Q{ : 3w : 0} R{ : 0 : w}, S{ : 3 : 0} p{ 3w : : 3}, q{ 3w : : 3} r{ 3w : : 3}, s{ 3w : : 3} () Now the five constrint equtions, Eq., define the ellipse in terms of coefficients A ij. p : 3w A wA 0 6wA 0 + A 3A + 3A = 0 q : 3w A 00 3wA 0 6wA 0 + A + 3A + 3A = 0 r : 3w A 00 3wA 0 + 6wA 0 + A 3A + 3A = 0 s : 3w A wA 0 + 6wA 0 + A + 3A + 3A = 0 T : A A A + ba 0 A ba 0 A +b A 00 A b A 0 = 0 () Solving for A ij nd cofctoring produces the ellipse point coefficient mtrix, in ij, on the right in expression 3, not surprisingly in stndrd form (w b ) b 3(w b )b b 0 (3) 0 0 3(w b ) With row nd column vectors of vrible homogeneous coordintes of point X{x 0 : x : x } nd with x 0 = in Eucliden spce the ellipse is given by Eq x 3(w b ) + x b = 0 (4) Equting denomintors of x by definition s = 3(w b ) gives w = ± ( + 3b )/3..5 Stndrd orm nd Discriminnt An lternte pproch to symmetriclly plcing the three coplnr, cotngentil ellipticl sections on equilterl tringle sides, formed by top view projection of the leg cylinder xes, is now discussed. Since the solution is esily seen to simplify to stndrd form ellipse with given minor xis length b nd tngent line, sy, QR r with given principl xis intercepts Q( 3w, 0) nd R(0, w), the ellipse nd line, using the negtive reciprocl intercept formultion described in [3], re presented in Eq. 5 in terms of three unknowns, x, x,, where is the semi-mjor xis length. + x + x b = 0, x 3w + x w = 0 (5) Eliminting x between the ellipse nd line expressed in Eq. 5 produces qudrtic in x only. (9b + 3 )x 6 3 wx 9 (b + w ) = 0 This is differentited with respect to x ( + 3b )x 3 w = 0 nd x is eliminted between the qudrtic nd its derivtive yielding function f(, b, w) = b ( + 3b )( + 3b 3w ) = 0 The fctor contining w is solved to gin rrive t w = ± ( + 3b )/3 nd with the given specifictions w = = 5 3/mm..6 rom Ellipse to yperboloid Axis ig. 5 shows the tngent circle of mximum rdius, centred on the minor xis of the ellipse t distnce w from its centre. A leg cylinder genertor on such tngent point is lso genertor of the one sheet hyperboloid of revolution to which the torus is tngent. In fct, the circle is the throt circle of tht hyperboloid.

5 b x O x = w r = + b w - b w - b x where Q = S = T = 0 hs been represented by only the qudrtic fctor P x + R = 0 tht leds to pir of imginry solutions. The rel, significnt, double root (x 0) = 0 is omitted. In such cses where the circle centre point is on the ellipse minor xis x = b. A pir of rel roots emerges from the qudrtic fctor if θ is sufficiently smll s in the rbitrry cse illustrted in ig. 5. The ctul sitution with the given design prmeters tht produce x = 0 is summed up in ig. 6 where r = b + w = b + = 5( + 3/). igure 5: yperboloid Throt Circle b hl , version - 5 Mr 009 The stndrd form ellipse nd tngent circle equtions re written s Eq x + x b = 0, r +x +(x +w) = 0 (6) Eliminting x produces qudrtic in x. Tking its derivtive with respect to x yields second, liner eqution in x. Solving these to eliminte x gives the circle rdius r nd then simultneous solution of Eq. 6 finds the two points tht ech support cylinder-nd-hyperboloid genertor. The rdius nd tngent points re given by Eq. 7 where results re expressed s dimensionless rtios. r = + w b, x = ± ( wb ) b x w = b b (7) Als, lthough the solution bove hs been simplified so s to del only with qudrtic eqution, the intersection between two conics dmits four solutions. Referring to [4] one sees comprehensive development of this topic. In the solution Eq. 7 bove the qurtic igure 6: Three Leg Sections on Torus Equtor x x igure 7: Tori nd yperboloid Sections w r P x 4 + Qx 3 + Rx + Sx + T = 0

6 hl , version - 5 Mr Torus nd yperboloid Tngency irst the xil hyperbolic section of the stndrd form one sheet hyperboloid of revolution with throt rdius r given by Eq. 7 is configured s plnr projection on, sy, x = 0. This is done using the point X{x 0 : x : x 3 } = { : r : 0} nd n bsolute point A{ 0 : : 3 } = {0 : cos θ : sin θ} on the chosen plne of projection, x = 0. The chosen hyperbol is written in Eq. 8 long with the coefficients derived with points X nd A. Notice tht α nd β re used to represent the unknown conic coefficients since the more usul nd b were lredy used for the ellipse. β x α x 3 α β x 0 = 0 x α x 3 β = 0 x 0 =, α = r, β = r tn θ (8) As in the cse of finding the rdius r of the circle tht circumscribes the three mutully tngentil ellipticl sections of the leg cylinders, s shown in ig. 6, one must find the loction of the circulr xis of the torus, distnce v from the hyperbol xis on x = 0. The meridil circulr section of the torus is of given rdius R = 5mm nd this circle (x v) + x 3 R = 0 must be tngentil to the hyperbol given by Eq. 8. If one chooses to eliminte x 3 between Eqs. 8 nd the circle, two right nswers re obtined, the first of the two pirs ±(α +R), ±(α R), α = r. The second pir represents tngency of circles inside the hyperbol. If however R is sufficiently lrge or θ is sufficiently smll then the solution produced by eliminting x, tht dmits two seprte points of tngency, pplies, i.e., v = r ( + tn θ) + R ( + cot θ). Sections of n R = 5mm nd n R = 00mm torus re shown on the right nd left of the hyperbol pertining to b = 5mm legs t slopes of θ = sin /3 re shown in ig Conic nd Circle Tngency Applying the nlysis outlined in [4] to the configurtions depicted in igs. 5,6,7, where the circle is centred t P (0, ±p, 0) nd P (±p, 0, 0), respectively, revels the condition where the two contct points t ±x nd ±x 3 collpse to x = 0 or x 3 = 0, respectively. This is expressed in terms of pproprite dimensionless rtios. ( x ) ( ) bw = (9) b Eq. 9, essentilly reprise of Eq. 7, is devoid of rdius r or R. As w is incresed so tht w/ (/b) (b/) > then the single point contct pplies nd the inner rdius of the torus is r = b+w nd the circulr xis of the torus is t x = r+r for the given dimensionl specifiction of the tble leg configurtion, i.e., b = R = 5mm nd w = = 5b. 3. GENERAL CONIGURATION Consider plcing in mutul contct three cylinders of revolution of vrious given rdii nd xil directions. There is no loss in generlity if one chooses the first cylinder xis to lie on the Crtesin frme xis x. This xis is shown in ig. 8 s segment AB of line P. The second cylinder xis on segment CD of line Q is seprted from AB by common norml AC long the Crtesin frme xis x 3. The third cylinder xis cnnot be plced immeditely but its direction is given by line segment AX. When AX is projected on plne norml to this direction the two pirs of prllel ruling lines on the first two cylinders re shown to define the limits of these two cylinders. There re four possible loctions for the circulr section of the third cylinder. These re locted by intersecting line pirs tht re respectively prllel to limit ruling of ech of the first two cylinders nd offset by distnce equl to the rdius of the third cylinder. The third cylinder xis chosen is shown s segment E of line R. Now the other two common normls G nd JK, connecting E to AB nd to CD, cn be found nd points P, Q, R plced on these nd on AC. These three points re where cylinder pirs mke contct.

7 A B D C D B C D A B X X X XA B D C TL E E C A E E igure 8: Three Contcting Cylinders of Vrious Rdius nd Axil Direction hl , version - 5 Mr 009

8 hl , version - 5 Mr Anlytic Geometry In wht follows the xis of the third cylinder will be defined on the intersection of plne pir. As in the cse of offset lines on the plne of projection, bove, there re four such plnes tht my be pired to locte four possible xes of the third cylinder. ig. 9 shows three touching cylinders of different rdius nd, for clrity, with their xes mutully orthogonl. A sphere of rdius equl to the sum of the rdii of the second nd third cylinders is plced nywhere on the xis of the second. Then plne is shown tngent to the sphere nd prllel to the second nd third cylinder xes. Agin, for clrity, this plne is not on the xis of the third cylinder but it is esy to imgine the two plnes which re. igure 9: Three Contcting Cylinders, Sphere nd Plne 4. CONCLUSION Aprt from providing n interesting nd chllenging exercise in solid modeling this problem goes well beyond the issues of contcting cylinders nd tori. Specificlly, the robust nd efficient computtion of shortest distnce between points in the plne nd in spce nd conics nd qudrics, respectively, is n ctive subject of modern reserch in imge processing for precise cmer ided inspection of industril products, e.g., [5]. ACKNOWLEDGEMENT This reserch on exercises in Geometric Thinking is supported by Nturl Sciences nd Engineering Reserch Council (Cnd) Discovery Grnt. REERENCES [] Schröcker,.-P. & Suzuki, K. (007) Ein Vergleich von Lösungsstrtegien für eine nspruchsvolle Modellierufgbe (Compring Approches to Solving Chllenging Solid Modeling Problem), Informtionsblätter der Geometrie, Univ. Innsbruck, v.6, n., pp.3-5. [] Zsombor-Murry, P.J. (006) Introduction to Conics nd Qudrics, <http//: mcgill.c / pul /ICQ69q.pdf>, p.. [3] Zsombor-Murry, P.J. (006) Line Geometry Primer, <http//: mcgill. c/ pul/lg.pdf>, p3. [4] Zsombor-Murry, P.J. (007) Shortest Distnce from Point to Coplnr Conic, <http//: pul/ SDLPtCQO75d.pdf>, pp.3-5. [5] rker, M., O Lery, P., (006) irst Order Geometric Distnce (The Myth of Smpsonus), Proc. British Mchine Vision Conf., 06-09, Edinburgh, pp ABOUT TE AUTOR Pul Zsombor-Murry, Ph.D., is Associte Professor of Mechnicl Engineering t McGill University, Montrél, Cnd. is reserch interests re Kinemtic Geometry, Robotics, Cmer Aided Inspection. e cn be reched by e-mil: <pul@cim.mcgill.c> or through postl ddress: McGill University, Deprtment of Mechnicl Engineering, 87 Sherbrooke St. W., Montrél (Québec) Cnd 3A K6.