 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 [1]. 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 [1]. 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 [4] 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 [5]. Kruppa [6] 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 [7] 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. [8] 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. [9] 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 [1] 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 [10] 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 [1]. 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 ,ac length 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 :Iα3 123 ,, xsxsxsα (2.1) The notation for the differentiation of the curve in relation to the arc length s is α d, d α s α 2 d d2, α s α 3 3 d d α s 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 κn b s btn b (2.5) The vectors are called collectively the Frenet- Serret frame. The Frenet–Serret formulas along α are given by ,,,tnb , , . sκ sκτ sτ tn nt bn (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 ,uvR 12 RR 0, (1,2 r r r u R R) de- note to partial derivatives of the surface . The unit sur- face normal vector field of the surface is given by R R 12 12 RR NRR (2.8) The first fundamental form coefficients of the surface are given by R ,;,1,2 pqp q gpqRR (2.9) The second fundamental form coefficients of the surface are given by 11 1112122222 ,, ,,,LLLRNRNR N (2.10) Let , r us 1, 2r in the 12 uu -plane defines a curve on the surface which can be written as R 21 , usuαRs 2 u (2.11) Then the three derivatives of the curve are given by α 11 2 u RαR (2.12) 11 112122221 122 2uuu u 22 u u RRRRαR 222 uu (2.13) 33 12111 12 11 1 11212122222 22 112121221 12 2 3 33 uuuu uuu uu uu uu u αRR R RR R RR R (2.14) The projection of the curvature vector onto the unit normal vector field of the surface is given by α R 22 12 11112 1 2 uu 22 2 12 ,2Lu LLu RR RR α (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 is the gradient vector of the surface , p f , then the unit surface normal vector field of the surface is given by f Nf (2.16) Let 123 ,, xsxs sax (2.17) be a curve on the surface with constraint 123 ,, 0fxxx then we have Copyright © 2011 SciRes. AM  M. A.-L. SOLIMAN ET AL. 1021 123 123 123 ,, , ,, , ,, . xx xx xx α α α (2.18) 112233 d0 d ffxf xfx s (2.19) 22 22 11 122233 3 2 12 1 213 1 3232 3 112233 d d 2 0 f xfx fx s xxfxxfxx fxf xfx (2.20) The projection of the curvature vector onto the unit normal vector field of the surface α is given by 222 123 ,η ff f αf (2.21) where 2 22 11 1222333 12 1213 1 32323 2 η xfx fx xxfxxfxx 3. Transversal Intersection Curves Consider the intersecting implicit and parametric surfaces and 123 ,, 0fxxx 12 ,;Ru uR 0,fRR 324 such that, 12 . Then the intersection curve of these surfaces can be viewed as a curve on both surfaces as 112 c,uc 0cuc 123 123 s,s,s ;,,0,sx xxfxxxα 121123 24 s,s ;,. uu cuccucαR Then we have 12 s,s ,1,2,3 i isxRuui where Then the surface 123 12 s,s,,.uu RRRR can be expressed as 123 12 ,,,huufRRR0 (3.1) Thus the intersection curve is given by 12 12 1123 24 s,s ;,0, , suu huu cuccu c Rα (3.2) 3.1. Tangential Direction Differentiation (3.1) yields 1212 0hhuu (3.3) where , i i h hu then we have 1 2 2 21 ,uu hh h 0 (3.4) Since α is the unit tangent vector field of the curve (3.2), then we have 22211112 ,uuuu RR RRα1 1 (3.5) which can be written as 11 22 11 122222 g2gguuuu (3.6) Substituting (3.4) into (3.6) yields 1 22 2 221112121221 2 1 22 2 12111 212122 g2 , g2 hhhhg hg hhhhgh u ug . (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 ζ tζRR (3.8) 3.2. Curvature and Curvature Vector The curvature vector is given by differentiation (3.8) with respect to s as follows 2 3 122 211122 12 12 21212211 122112 , 2 hh hh h hhhhhh h ζζ ζζζ ζ ζζ RR R α R R (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 2 2 2 3 2 2 ,, , , , ,. , κ ζζ ζζζ n ζζ ζζζ ζζ ζζζ ζ ζζ ζζζ ζ bζζζζζζ (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. 2 12 1 , h uh u ζζ (3.11) Differentiation (3.13) we obtain 12 22 12 11 12 2 1212111121 12 221 212122 1 12 21 2 1 222 ,, ,, . ζ hh u hh hhhh hhh uu uu u uh u ζζ ζζ ζ ζζ ζζ ζRR RR RR RR (3.12) Differentiation (3.12) we obtain 2 1222 112 11 2 22 12 4322 222 21 122 222 3 11 12 21 2 22 11 111 122 1 12 22 2 3 12 , ,, , 2 , 2, , , uu u uu u hhh uu h h hh h u u uu h u h hh u u ζζ ζζζ ζ ζζζζζ ζζ ζζζζ ζζ ζζ ζ ζζ ζζ ζ ζζ ζζ ζ 2 12 4322 1121211 112112 22 12121221 22 2112 112 112 111 111211 121 112 2222 122 122 122 1 1 2 22 212 22 2 1 2 ,, , 2, 2 2 2 2 h uh hhh hhhh hhh hhh hhh h uh u u u 2 ζζζζζ ζζ ζζζζ ζRR RR RR RR RRR RRR RRR RR 1 222 122 1112222 11 112221121 12 12 2 2 2. 2 h hhh hh uu h R RRR RRR (3.13) Substituting 11122 and 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 ,RuuR; 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 12 12 RR f fRR which can be written as 12 12 ,AA f fRR RR (4.1) Then we can write 23 32 11212 31 13 21212 12 21 31212 , , . fRRRR fRRRR fRRRR A A A (4.2) Since 12 ,,1,2,sususi i xR 3, u i i then we have 11 22 ii u xR R (4.3) 4.1. Tangential Direction Projecting the curvature vector onto the two unit nor- mal vectors of both surfaces yields α 12 12 ,, α RR fRR α f (4.4) Using (2.15) (2.21) and (4.4) we obtain 2 22 11 1222333 12 1 213 1 32323 2' 12111 1212 2 22 2 2ff 2u fxf xf x fxx xxxx ALuLuLu RR (4.5) Substituting (4.3) into (4.5) yields 2 1112 22 11 2 22 2aa 0,0 uu uu au (4.6) where 22 12 1112 11111221 2 312231 3311211231 11311 22 12 1 21122 2 312231 3312231 222222 2222223 112 2 1 21122 3312 2 3 2 12121212 1 12 1231221 2, 2, aA Lff ffff aA Lff ffff aA Lff ff f RRR R RRRRRR RRR R RRRRRR RRRR RR RRRRRR 23 321331 23121 213121 2 f. RR RRRR RR 3 3 R R Copyright © 2011 SciRes. AM  M. A.-L. SOLIMAN ET AL. 1023 R ], R This can be written in a matrix form as follows T ijij ij afRRH (4.7) where and 123T 123 [],[ ijij ij ij fff RRR fR 123T [] iiii RRR 11 1213 12 22 23 13 2333 ff ff ff H . is the Hessian matrix of the surface Solving (4.6) for 1 2 u u yields 2 121211 22 12 11 (a ) ,aa uBu Ba a (4.8) Substituting (3.7) and (4.7) into (4.8) we obtain 1 22 1111222 1 22 2111222 2 2. uBBgBgg uBgBgg (4.9) Then the unit tangent vector field of the intersection curve is given by 12 12 B B RR tRR (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 [1] 2 121122 Δ,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. P 0Δ Lemma 2. The surfaces and intersect at the point and at its neighborhood, if and fh ΔP0 22 2 111222 0.aaa(Tangential intersection curve). Lemma 3. The point is an isolated contact point of the surfaces P and , if hΔ0 . Lemma 4. The surfaces and have contact of at least second order at the point , if . (Higher-order contact point). h P1112220aaa 4.2. Curvature and Curvature Vector Differentiation (4.6) and using (4.9) we obtain 121 2 1112 22 1211 12 11 12 , 2;0 uBu a aBaB a au aBa aBa , . (4.11) where TTT 11 111 TT T 22 22 11 1213 123 122223 13 23 33 () , HH H, ijijiji jj ijijijij iii iiii iii au u fff fff fff tHRfRRHR RHR fRRHRRHRR QR QtH (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 21 324 au aua (4.13) where 2111231222 232 42 111121 12 2221222112 ,, ,2B, 2,, ,, aBg gaBgg auB BB RR RR RRRRRRRR Solving the linear system (4.11) and (4.13) yields 34 4 1 32 412 2 32 , B aa aB uaaB aaa uaa (4.14) The curvature vector of the intersection curve is obtained by substituting 11 ,,uuu 2 , and into (2.13). 2 u 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 1 u and by differentiation 12 u 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 22 12 12 22 90, ,3sin,3cos; 02 fxx uu uu R (5.1) as shown in Figure 1. Transversal intersection: Using (3.1) yields 22 12 9cos 0huu (5.2) Copyright © 2011 SciRes. AM  1024 M. A.-L. SOLIMAN ET AL. The intersection curves (0, 1, 0) Figure 1. Transversal and tangential intersection. Differentiation (5.1) and (5.2) we obtain 122 21 222 1112111 222 2 122 222 2 2222 2 2, 9sin2,1,0,0, 18cos 2,0, 36sin2, 1, 0, 0,30, cos,sin, 30,sinsinu,cos , 30,coscosu, sin. huhu huhhh hu uu u u R RR R R 2 (5.3) Using (3.8) and (5.2), we obtain 2112 222 22 2 sin tan ,, 1sinu31sin31sin cos 0. uuuu uu u t 2 , (5.4) Using (3.12) and (5.2), hence 221212 1222 222 2222 2 22 3 12 2 2 18sincos ,6cos ,6sin, 2cos 26sin26cos 2 ,, cos1 sin1 sin1 sin 18cos1 sin, 72 sin ,. 1sin uuuuuu uuu u uuuu uu uu u ζ ζ ζ ζζ , (5.5) Using (2.4), (2.5), (3.12), (3.13) and (5.4) then we have 12 22 222 22 12 22 2 22 3 22 2 12 22 2 2 2sin cos ,, 91 sin31sin31sin 2sin cos ,, 321sin1sin21sin 21sin, 3 cos2sin tan1 ,0,. 2 321sin uu uuu uuu uu u κu uu uu u 2 2 2 2 2 , , u α n b (5.6) Using (3.15) and (3.16) hence 21 12 22 22 12 12 22 22 22 sin ,, 1sinu9cos 1sin sin cos ,. 9(1 sin)9(1 sin) uu uu u uu uu uu 2 2 u u (5.7) Using (3.17) and (5.7) hence 2 22 17 22 2 122 22 27 22 2 sin2 3cos, 91 sin (2sintancoscos 2). 81 1sin uu u u uuu uu u u (5.8) Using (2.7) and (2.14) yields 22 22112 77 22 22 22 22 222 22 7 22 2 3(23cos)sin2u6usin ,, 27(1 sin)27(1 sin) 4sincossin1 sinsin 27 1sin uu u uu uuu uu u α (5.9) 12 22 2 2 234 222222 53 221 22122 24 122122 24tan4sin10 cossin 42cosu 4cos sin7cossincos sin cossin2 cossin3 cossin 2 costan6 costan uu uuu uuuu uu uuuuuu uu uuuuuu 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 3 12 2, 0,18, 1 3,, . 23 aa a Bu u 1 2 (5.11) Then this means that the point is a branch point (Figure 1). From (4.10) and (5.11), we obtain Δ0,1 P 11 ,0, 22 t (5.12) Using (2.13) and (4.14) hence Copyright © 2011 SciRes. AM  M. A.-L. SOLIMAN ET AL. 1025 21 10,1, 0, 6 1 0,1, 0,, 6 11 , 0, 0, . 22 uu κ n b α (5.13) Using (5.10) at , we obtain 10,1, 0P 2 4 22 2 π2 2 22 122 22 33 122 22 5 222212 4 122122 1 lim42 sin2cossin 42cos 22costan42 cossin 32 cossin72cossin 10 2cossin2cossin4 2tan 2 2cossin6 2costan 0 u uu u uuu uu uuu uu uuuuuu uuuu uu 2 u (5.14) Example 2. Consider the intersection of the implicit and the parametric surfaces 222 123 12 22 90, R,3sin,3cos,0 2 fxxx uu uu (5.15) as shown in Figure 2. At 1, 0x 12 // .fRR 0, 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 12 1 0, 0,3 uuB (5.16) Using (4.10) we have 2 0, cos,sinuut 2 2 (5.17) Using (5.16) hence 112 0, 0uuuu (5.18) Using (2.4) and (2.13) hence the curvature vector and the curvature are given by 10 Figure 2. Tangential intersection. P (0, 3, 0) Figure 3. Tangential intersection. 22 22 10, sin, cos, 3 1 0, sin, cos,3 uu uuκ α n (5.19) Using (2.5) (2.7)and (2.14) hence 22 10, cos,sin, 9 1, 0, 0,0. uu τ α b (5.20) Example 3. Consider the intersection of the implicit and the parametric surfaces 22 12 12 (6)90, ,3 3sin,3cos. fx x uu R2 u (5.21) as shown in Figure 3. At the point 0,3, 0P, 12 // .fRR 0, Using (4.7) and (5.21), we have Δ this means that the point is an isolated tangential contact point (Figu re 3 ). P Example 4. Consider the intersection of the implicit and the parametric surfaces 222 31 2 222 122112 0, 1,1, fxxx uuuuuu 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, R PSS on the intersection curve (Figure 4), we have 111 111 22212 112122 222 11122 222 2 12122112 1, 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. h hh hh hh hh RR RRR RRR (5.23) Using (3.8) and (5.23), we obtain 0,1,0t (5.24) Using (3.12) (3.13) and (5.23) we obtain 2, 0,3, 23 (,0,),13 1313 κ α n. (5.25) Using (2.5) (2.7) (2.14) (3.17) and (5.25) we obtain 33 ,19, , 44 32 ,0,,. 52 1313 τ α b3 (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 [1]. 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 [1] 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 [2] 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 [3] 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 [4] T. J. Willmore, “An Introduction to Differential Geome- try,” Clarendon Press, Oxford, 1959. [5] 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 [6] E. Kruppa, “Analytische und Konstruktive Differentialgeometrie,” Springer, Wien, 1957. [7] E. Hartmann, “G2 Interpolation and Blending on Sur- faces,” The Visual Computer, Vol. 12, No. 4, 1996, pp. 181-192. doi: 10.1007/s003710050057 [8] 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 [9] 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 [10] 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 [11] M. P. do Carmo, “Differential Geometry of Curves and Surface,” Prentice Hall, Englewood Cliffs, NJ, 1976. [12] J. J. Stoker, “Differential Geometry,” Wiley, New York, 1969. [13] D. J. Struik, “Lectures on Classical Differential Geome- try,” Addison-Wesley, Reading, 1950.
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