 Applied Mathematics, 2011, 2, 1019-1026 doi:10.4236/am.2011.28141 Published Online August 2011 (http://www.SciRP.org/journal/am) Copyright © 2011 SciRes. AM Intersection Curves of Implicit and Parametric Surfaces in 3 Mohamed Abdel-Latif Soliman, Nassar Hassan Abdel-All, Soad Ali Hassan, Sayed Abdel-Naeim Badr Department of Mathematics, Faculty of Science, Assiut University, Assiut, Egypt E-mail: sayed_badr@ymail.com Received June 9, 2011; revised June 30, 2011; accepted July 7, 2011 Abstract We present algorithms for computing the differential geometry properties of Frenet apparatus and higher-order derivatives of intersection curves of implicit and parametric surfaces in 3 for transversal and tangential intersection. This work is considered as a continuation to Ye and Maekawa . We obtain a classification of the singularities on the intersection curve. Some examples are given and plotted. t,n,b,κ,τ Keywords: Geometric Properties, Frenet Frame, Frenet Apparatus, Frenet-Serret Formulas, Surface-Surface Intersection, Transversal Intersection, Tangential Intersection, Dupin Indicatrices 1. Introduction The intersection problem is a fundamental process needed in modeling complex shapes in CAD/CAM system. It is useful in the representation of the design of complex ob-jects, in computer animation and in NC machining for trimming off the region bounded by the self-intersection curves of offset surfaces. It is also essential to Boolean operations necessary in the creation of boundary repre-sentation in solid modeling . The numerical marching method is the most widely used method for computing intersection curves in . The Marching method in-volves generation of sequences of points of an intersec-tion curve in the direction prescribed by the local differ-ential geometry [2,3]. Willmore  described how to ob-tain the unit tangent, the unit principal normal, the unit binormal, the curvature and the torsion of the transversal intersection curve of two implicit surfaces . Kruppa  explained that the tangential direction of the intersection curve at a tangential intersection point corresponds to the direction from the intersection point towards the intersec-tion of the Dupin indicatrices of the two surfaces. Hart-mann  provided formulas for computing the curvature of the transversal intersection curves for all types of in-tersection problems in Euclidean 2-space. Kriezis et al.  determined the marching direction for tangential intersec-tion curves based on the fact that the determinant of the Hessian matrix of the oriented distance function is zero. Luo et al.  presented a method to trace such tangential intersection curves for parametric-parametric surfaces employing the marching method. The marching direction is obtained by solving an undetermined system based on the equilibrium of the differentiation of the two normal vectors and the projection of the Taylor expansion of the two surfaces onto the normal vector at the intersection point. Ye and Maekawa  presented algorithms for computing all the differential geometry properties of both transversal and tangentially intersection curves of two parametric surfaces. They described how to obtain these properties for two implicit surfaces or parametric-implicit surfaces. They also gave algorithms to evaluate the higher-order derivative of the intersection curves. Aléssio  studied the differential geometry properties of inter-section curves of three implicit surfaces in for trans-versal intersection, using the implicit function theorem. 34In this study, we present algorithms for computing the deferential geometry properties of both transversal and tangentially intersection curves of implicit and Paramet-ric surfaces in as an extension to the works of . 3This paper is organized as follows: Section 2 briefly introduces some notations, definitions and reviews of differential geometry properties of curves and surfaces in . Section 3 derives the formulas to compute the prop-erties for the transversal intersection. Section 4 derives the formulas to compute the properties for the tangential intersection. Some examples of transversal and tangen-tially intersection are given and plotted in Section 5. Fi-nally, conclusion is given in Section 6. 3 1020 M. A.-L. SOLIMAN ET AL. 2. Geometric Preliminaries [1, 11-13] Let us first introduce some notation and definitions. The scalar product and cross product of two vectors and are expressed as a c,ac and respectively. The ,aclength of the vector is a,.aaa 2.1. Differential Geometry of the Curves in 3 Let be a regular curve in with arc-length parameterization, 3:Iα3 123,,sxsxsxsα (2.1) The notation for the differentiation of the curve in relation to the arc length s is αd,dαssα 2dd2,αss α 33ddαss α. Then from elementary differential geometry, we have sαt (2.2) sκ αn (2.3) 2,κs αα (2.4) where is the unit tangent vector field and tα is the curvature vector. The factor is the curvature and is the unit principal normal vector. The unit binormal vector is defined as κnbsbtnb (2.5) The vectors are called collectively the Frenet- Serret frame. The Frenet–Serret formulas along α are given by ,,,tnb,,.sκsκτsτ tnntbn (2.6) where is the torsion which is given by τ,τκbα (2.7) provided that the curvature does not vanish. 2.2. Differential Geometry of the Parametric Surfaces in 3 Assume that is a regular parametric surface. In other words where 12,uvR12RR0, (1,2rrruRR) de-note to partial derivatives of the surface . The unit sur-face normal vector field of the surface is given by RR1212RRNRR (2.8) The first fundamental form coefficients of the surface are given by R,;,1,2pqp qgpqRR (2.9) The second fundamental form coefficients of the surface R are given by 11 1112122222,, ,,,LLLRNRNR N (2.10) Let ,rus 1, 2r in the 12uu -plane defines a curve on the surface which can be written as R21,susuαRs2u (2.11) Then the three derivatives of the curve are given by α11 2uRαR (2.12)  11 112122221 1222uuu u22u u  RRRRαR222uu (2.13)   3312111 1211 1 1121212222222112121221122333uuuu uuuuu uuuuu   αRR RRR RRRR (2.14) The projection of the curvature vector onto the unit normal vector field of the surface is given by αR 221211112 1 2uu22 212,2Lu LLu RRRRα (2.15) 2.3. Differential Geometry of the Implicit Surfaces in 3 Assume that 123,, 0fxxx is a regular implicit sur-face. In other words 0f, where 12 3,,ffff is the gradient vector of the surface f, ppffx, then the unit surface normal vector field of the surface f is given by fNf (2.16) Let  123,,sxsxs sax (2.17) be a curve on the surface f with constraint 123,, 0fxxx then we have Copyright © 2011 SciRes. AM M. A.-L. SOLIMAN ET AL. 1021123123123,, ,,, ,,, .xxxxxxxxx  ααα (2.18) 112233d0dffxf xfxs   (2.19)  222211 122233 3212 1 213 1 3232 3112233dd20ffxfx fxsfxxfxxfxxfxf xfx    (2.20) The projection of the curvature vector onto the unit normal vector field of the surface αf is given by 222123,ηfff fαf (2.21) where  22211 122233312 1213 1 323232ηfxfx fxfxxfxxfxx   3. Transversal Intersection Curves Consider the intersecting implicit and parametric surfaces and 123,, 0fxxx 12,;Ru uR0,fRR 324 such that, 12 . Then the intersection curve of these surfaces can be viewed as a curve on both surfaces as 112c,uc0cuc 123 123s,s,s ;,,0,sx xxfxxxα  121123 24s,s ;,.suu cuccucαR Then we have  12s,s ,1,2,3iisxRuui where Then the surface  12312s,s,,.uu RRRRf can be expressed as 12312,,,huufRRR0 (3.1) Thus the intersection curve is given by 12 121123 24s,s ;,0,,suu huucuccu c Rα (3.2) 3.1. Tangential Direction Differentiation (3.1) yields 12120hhuu (3.3) where ,iihhu then we have 12221,uuhhh0 (3.4) Since α is the unit tangent vector field of the curve (3.2), then we have 22211112,uuuu RR RRα11 (3.5) which can be written as  112211 122222g2gguuuu (3.6) Substituting (3.4) into (3.6) yields   12222211121212212122212111 212122g2 ,g2hhhhg hghhhhghuug . (3.7) The unit tangent vector field of the intersection curve is given by substituting (3.7) into (2.12) as follows 21 12;ζhhζtζRR (3.8) 3.2. Curvature and Curvature Vector The curvature vector is given by differentiation (3.8) with respect to s as follows  23122211122 121221212211 122112,2hh hhh hhhhhh h  ζζ ζζζζζζ RRRαRR (3.9) The unit principal normal vector field, the curvature and the unit binormal vector are given by using (2.3) (2.4) and (2.5) as follows 222322,,,,,,.,κ ζζ ζζζnζζ ζζζζζ ζζζζζζ ζζζζbζζζζζζ (3.10) 3.3. Torsion and Higher-Order Derivatives Equation (3.7) can be written as Copyright © 2011 SciRes. AM 1022 M. A.-L. SOLIMAN ET AL. 2121,huhuζζ (3.11) Differentiation (3.13) we obtain 12 221211 1221212111121 12221 212122 11221 21222,,,,.ζhhuhhhhhhhhhuuuuuuhu   ζζζζζζζζζζRR RRRR RR (3.12) Differentiation (3.12) we obtain  21222 1121122212432222221122 222311 1221 22211111 122112 222312,,, ,2,2,,,uuuuu uhhhuuhhhhhuuuuhuhhhuu    ζζζζζζζζζζζ ζζζζζζζζζζ ζζζζζζζζζζ ζ21243221121211 11211222 12121221 222112 112 112 111111211 121 1122222 122 122 12211222 212 22212,, ,2,2222huh hhhhhhhhhhhhhhhhhuhuuu2   ζζζζζ ζζζζζζζRR RRRR RRRRRRRRRRRRR1 222122 1112222 11112221121 1212222.2hhhhhhuu h RRRRRRR (3.13) Substituting 11122and 2 into (2.14) we obtain the third-order derivative vector of the intersection curve. Hence the torsion can be obtained by (2.7). ,,,,uuuuu   uWe can compute all higher-order derivatives of the in-tersection curve by a similar way. 4. Tangentially Intersection Curves Assume that the surfaces and 123,, 0fxxx 12,RuuR; 1123 24,cuccu cP are intersecting tangentially at a point on the curve (3.2) then the unit surface normal vector field of both surfaces are parallel to each other. In other words 1212RRffRR which can be written as 1212,AAffRR RR (4.1) Then we can write 23 321121231 132121212 2131212,,.fRRRRfRRRRfRRRRAAA (4.2) Since12,,1,2,sususiixR 3,uiithen we have 11 22iiuxR R (4.3) 4.1. Tangential Direction Projecting the curvature vector onto the two unit nor-mal vectors of both surfaces yields α1212,,αRRfRRαf (4.4) Using (2.15) (2.21) and (4.4) we obtain   22211 122233312 1 213 1 323232'12111 1212222 22ff2ufxf xf xfxx xxxxALuLuLu   RR (4.5) Substituting (4.3) into (4.5) yields 21112 22112222aa 0,0uuuuau    (4.6) where    22121112 1111122123122313311211231 1131122121 21122231223133122312222222222223112 21 211223312 23212121212112 12312212,2,aA LffffffaA LffffffaA Lfffff      RRR RRRRRRRRRR RRRRRRRRRRR RRRRRRRR23 32133123121 213121 2f. RR RRRR RR33RR Copyright © 2011 SciRes. AM M. A.-L. SOLIMAN ET AL. 1023R],RThis can be written in a matrix form as follows Tijij ij afRRH (4.7) where and123T123[],[ijij ij ijfff RRR fR123T[]iiiiRRR11 121312 22 2313 2333fffffffffH. is the Hessian matrix of the surface f Solving (4.6) for 12uu yields 2121211 221211(a ),aauBu Ba  a (4.8) Substituting (3.7) and (4.7) into (4.8) we obtain 1221111222122211122222.uBBgBgguBgBgg (4.9) Then the unit tangent vector field of the intersection curve is given by 1212BBRRtRR (4.10) From the previous formulas, it is easy to see that, there are four distinct cases for the solution of (4.6) depending upon the discriminant these cases are as the following  2121122Δ,aaaLemma 1. The point is a branch point of the inter-section curve (3.2) if and there is another intersec-tion branch crossing the curve (3.2) at that point. P0ΔLemma 2. The surfaces and intersect at the point and at its neighborhood, if and fhΔP022 2111222 0.aaa(Tangential intersection curve). Lemma 3. The point is an isolated contact point of the surfaces Pf and , if hΔ0. Lemma 4. The surfaces f and have contact of at least second order at the point , if . (Higher-order contact point). hP1112220aaa 4.2. Curvature and Curvature Vector Differentiation (4.6) and using (4.9) we obtain 12121112 221211 1211 12,2;0uBu aaBaB aau aBaaBa  ,. (4.11) where TTT11 111TT T22 2211 1213123 12222313 23 33(),HH H,ijijiji jjijijijijiiiiiiiiiiauufffffffff  tHRfRRHR RHRfRRHRRHRR QRQtH (4.11) Since the curvature vector is perpendicular to the tangent vector, then we have ,αα 0. Using (2.12) (2.13) and (4.9) we obtain 21 324au aua (4.13) where 211123122223242 11112112 2221222112,,,2B,2,, ,,aBg gaBggauBBB RR RRRRRRRRRR Solving the linear system (4.11) and (4.13) yields 34 4132412232,Baa aBuaaBaaauaa   (4.14) The curvature vector of the intersection curve is obtained by substituting 11,,uuu2, and into (2.13). 2u 4.3. Torsion If we have a branch point, then we can compute the torsion by taking the limit of the torsion of transversal intersection curve at this point. If we have tangential intersection curve, then we can compute 1u and by differentiation 12u u and 2.uSubstituting 11122,,,,uuuuu , and 2into (2.14) we obtain the third-order derivative vector of the intersec-tion curve. Then we can obtain the torsion by using (2.7). u 5. Examples Example 1. Consider the intersection of the implicit and the parametric surfaces 221212 2290,,3sin,3cos; 02fxxuu uuR (5.1) as shown in Figure 1. Transversal intersection: Using (3.1) yields 22129cos 0huu (5.2) Copyright © 2011 SciRes. AM 1024 M. A.-L. SOLIMAN ET AL. The intersection curvesP (0, 1, 0) Figure 1. Transversal and tangential intersection. Differentiation (5.1) and (5.2) we obtain 122 21222 1112111222 2122222 22222 22, 9sin2,1,0,0,18cos 2,0,36sin2,1, 0, 0,30, cos,sin,30,sinsinu,cos ,30,coscosu, sin.huhuhuhhhhuuuuu  RRRRR2 (5.3) Using (3.8) and (5.2), we obtain 2112222222sin tan,,1sinu31sin31sincos 0.uuuuuuut2, (5.4) Using (3.12) and (5.2), hence 221212122222222222223122218sincos ,6cos ,6sin,2cos 26sin26cos 2,,cos1 sin1 sin1 sin18cos1 sin,72 sin,.1sinuuuuuuuuu uuuuuuuuuuζζζζζ, (5.5) Using (2.4), (2.5), (3.12), (3.13) and (5.4) then we have 1222222221222 222322212 22222sin cos,,91 sin31sin31sin2sin cos,,321sin1sin21sin21sin,3cos2sin tan1,0,.2321sinuuuuuuuuuu uκuuu uuu22222,,u    αnb (5.6) Using (3.15) and (3.16) hence 21122222121222 2222sin ,,1sinu9cos 1sinsin cos,.9(1 sin)9(1 sin)uuuuuuuuuuu 22uu (5.7) Using (3.17) and (5.7) hence 22217222122 2227222sin2 3cos,91 sin(2sintancoscos 2).81 1sinuuuuuuu uuuu   (5.8) Using (2.7) and (2.14) yields  22221127722222222222 2272223(23cos)sin2u6usin,,27(1 sin)27(1 sin)4sincossin1 sinsin27 1sinuu uuuuuu uuu   α (5.9) 12 222223422222253221 221222412212224tan4sin10 cossin42cosu4cos sin7cossincos sincossin2 cossin3 cossin2 costan6 costanuu uuuuuuu uuuuuuuu uuuuuuuu 2 (5.10) Tangentially intersection: The surfaces are intersect-ing tangentially at the points . Consider the 0,1, 0Ppoint 10,1, 0,P using (4.7) (4.8) (4.9) and (5.3), then we have 12 3122, 0,18,13,, .23aa aBu u12  (5.11) Then this means that the point is a branch point (Figure 1). From (4.10) and (5.11), we obtain Δ0,1P11,0,22t (5.12) Using (2.13) and (4.14) hence Copyright © 2011 SciRes. AM M. A.-L. SOLIMAN ET AL. 10252110,1, 0,610,1, 0,,611,0,0, .22uuκ nbα (5.13) Using (5.10) at , we obtain 10,1, 0P24222π2222122 2233122 22522221241221221lim42 sin2cossin42cos22costan42 cossin32 cossin72cossin10 2cossin2cossin4 2tan2 2cossin6 2costan0uuuuuuu uuuuu uuuuuuuuuuuu uu2u (5.14) Example 2. Consider the intersection of the implicit and the parametric surfaces 22212312 2290,R,3sin,3cos,0 2fxxxuu uu (5.15) as shown in Figure 2. At 1, 0x12// .fRR0, Using (4.7) and (5.15), we have Δ this means that the surfaces are intersect-ing tangentially in a curve as (Figure 2). Then from (4.8) and (4.9), we have 1210, 0,3uuB (5.16) Using (4.10) we have 20, cos,sinuut22 (5.17) Using (5.16) hence 1120, 0uuuu   (5.18) Using (2.4) and (2.13) hence the curvature vector and the curvature are given by 10x Figure 2. Tangential intersection. P (0, 3, 0) Figure 3. Tangential intersection. 222210, sin, cos,310, sin, cos,3uuuuκ  αn (5.19) Using (2.5) (2.7)and (2.14) hence 2210, cos,sin,91, 0, 0,0.uuτ  αb (5.20) Example 3. Consider the intersection of the implicit and the parametric surfaces 221212(6)90,,3 3sin,3cos.fx xuu  R2u (5.21) as shown in Figure 3. At the point 0,3, 0P,  12// .fRR0, Using (4.7) and (5.21), we have Δ this means that the point is an isolated tangential contact point (Figu re 3 ). PExample 4. Consider the intersection of the implicit and the parametric surfaces  22231 22221221120,1,1,fxxxuuuuuuR3. (5.22) as shown in Figure 4. P (1, 0, 1) Figure 4. Transversal intersection. Copyright © 2011 SciRes. AM M. A.-L. SOLIMAN ET AL. Copyright © 2011 SciRes. AM 1026 At the point 1, 0,1,fRPSS on the intersection curve (Figure 4), we have  111 11122212112122 22211122 2222 121221121, 0, 2,0, 0, 2,24,0,2,0,2, 0, 0,0,2,0,0,2,0,2, 0, 0,0, 0,6,2, 10,12,0,24, 0.hhh hhhh hh    RRRRRRRR (5.23) Using (3.8) and (5.23), we obtain 0,1,0t (5.24) Using (3.12) (3.13) and (5.23) we obtain 2, 0,3,23(,0,),131313 κ αn. (5.25) Using (2.5) (2.7) (2.14) (3.17) and (5.25) we obtain 33,19, ,4432,0,,.521313 τ αb3 (5.26) 6. Conclusions Algorithms for computing the differential geometry prop-erties of intersection curves of implicit and parametric sur-faces in are given for transversal and tangential inter-section. This paper is an extension to the works of Ye and Maekawa . They gave algorithms to compute the dif-ferential geometry properties of intersection curves be-tween two parametric surfaces then they applied it on a simple example for implicit and parametric surfaces inter-section. This paper presented direct and simple formulas to compute all differential geometry properties, which may reduce the time it takes to calculate those properties. The types of singularities on the intersection curve are charac-terized. The questions of how to exploit and extend these algorithms to compute the differential geometry properties of intersection curves between three surfaces in , can be topics of future research. 34 7. Acknowledgements The authors would like to thank the reviewers for their valuable comments and suggestions. 8. References  X. Ye andT. Maekawa, “Differential Geometry of Inter-section Curves of Two Surfaces,” Computer-Aided Geo-metric Design, Vol. 16, No. 8, September 1999, pp. 767- 788. doi:10.1016/S0167-8396(99)00018-7  C. L. Bajaj, C. M. Hoffmann, J. E. Hopcroft and R. E. Lynch, “Tracing Surface Intersections,” Computer-Aided Geometric Design, Vol. 5, No. 4, November 1988, pp. 285-307. doi:10.1016/0167-8396(88)90010-6  N. M. Patrikalakis, “Surface-to-Surface Intersection,” IEEE Computer Graphics & Applications, Vol. 13, No. 1, January-February 1993, pp. 89-95. doi:10.1109/38.180122  T. J. Willmore, “An Introduction to Differential Geome-try,” Clarendon Press, Oxford, 1959.  M. Düldül, “On the Intersection Curve of Three Paramet-ric Hypersurfaces,” Computer-Aided Geometric Design, Vol. 27, No. 1, January 2010, pp. 118-127. doi:10.1016/j.cagd.2009.10.002  E. Kruppa, “Analytische und Konstruktive Differentialgeometrie,” Springer, Wien, 1957.  E. Hartmann, “G2 Interpolation and Blending on Sur-faces,” The Visual Computer, Vol. 12, No. 4, 1996, pp. 181-192. doi: 10.1007/s003710050057  G. A. Kriezis, N. M. Patrikalakis and F.-E. Wolter, “Topological and Differential Equation Methods for Sur-face Intersections,” Computer-Aided Geometric Design, Vol. 24, No. 1, January 1992, pp. 41-55. doi:10.1016/0010-4485(92)90090-W  R. C. Luo, Y. Ma and D. F. McAllister, “Tracing Tan-gential Surface-Surface Intersections,” Proceedings Third ACM Solid Modeling Symposium, Salt Lake City, 1995, pp. 255-262. doi:10.1145/218013.218070  O. Aléssio, “Differential Geometry of Intersection Curves in 4 of three Implicit Surfaces,” Computer-Aided Geometric Design, Vol. 26, No. 4, May 2009, pp. 455- 471. doi:10.1016/j.cagd.2008.12.001  M. P. do Carmo, “Differential Geometry of Curves and Surface,” Prentice Hall, Englewood Cliffs, NJ, 1976.  J. J. Stoker, “Differential Geometry,” Wiley, New York, 1969.  D. J. Struik, “Lectures on Classical Differential Geome-try,” Addison-Wesley, Reading, 1950.