Abhyankar, Shreeram S.

Bajaj, Chanderjit

English

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Purdue University Purdue e-Pubs Department of Computer Science Technical Reports Department of Computer Science 1986 Automatic Rational Parameterization of Curves and Surfaces II: Cubics and Cubicoids Shreeram S. Abhyankar Purdue University, ram@cs.purdue.edu Chanderjit Bajaj Report Number: 86-592 Abhyankar, Shreeram S. and Bajaj, Chanderjit, "Automatic Rational Parameterization of Curves and Surfaces II: Cubics and Cubicoids" (1986). Department of Computer Science Technical Reports. Paper 511. https://docs.lib.purdue.edu/cstech/511 This document has been made available through Purdue e-Pubs, a service of the Purdue University Libraries. Please contact epubs@purdue.edu for additional information. AUTOMATIC RATIONAL PARAME1ERlZATION OF CURVESAND SURFACES II: CUBICS AND CUBICOlDSShreernm S. AbhyankarChanderjit BajajCSD·1R·592April 1986Revised February 1987Automatic Parameterization ofRational Curves and Surfaces II:Cubics and CubicoidsShreeram S. Abhyanko.rt and Chanderjit BajaftDepartment of Computer Science,Purdue University,West Lafayette. IN 47907.AbstractCubicoids (degree 3 surfaces) always have a parameterization in tenos of rational functions,(a polynomial divided by another). On the other hand cubics (degree 3 plane curves) do notalways have a rational parameterization. However they always have a parameterization of thetype which allows a single square root of rational functions. [n this paper we describe algorithmsto obtain rational and special parametric equations for the cubics and cubicoids. given the impli-cit equations. These algorithms have been implemented on a VAX-8600 using VAXIMA.t Departmr:nt of Mathematics, Purdue University. Research supported in part by grants from ONR and NSF.*Research supported in part by NSF grant ocr 85-21356.-2-1. IntroductionRational curves and surfaces can be represented by implicit or parametric equations. Arational cubic (degree three) curve has an implicit equation given by C(x,y) = ax3 + by3 + cx2y+ dxy2 + ex2 + fy2 + gx:y + hz + iy + j = 0, and rational parametric equations given byx = u(t)lw (I) and y = 11 (t)/w(t), where u, 11 and w are no more than cubic polynomials. Further arational cubicoid (degree three) surface has an implicit equation given by C (X,y,Z) = ax3 + by3 +cz3 + tJx2y + ex2z + fxy2 + gyzz + hxz2 + iyz2 + jx:yz + kx2 + ly2 + mz2 + nxy + oxz +pyz + qx +ry + sz + t = 0, with corresponding rational parametric equatioI1'i x=u(s,t)lq(s,t),y = .... (s.t)/q (s,t), and z = W (s,t)/ q(s,t). The rational parametric form of representing a surfaceallows greater ease for transformation and shape control, Tiller (1983), Mortenson (1985). Theimplicit form is preferred for resting whether a point is above, on, or below the surface, whereabove and below is determined relative to the direction of the surface nonnal. As both formshave their inherent advantages it becomes crucial to be able to go efficiently from one form to theother. especially when surfaces of an object are automatically generated in one of the tworepresentations.Cubicoids (degree 3 surfaces) are rational t , that is, have a parameterization in tenns ofrational functions (ratio of two polynomials). On the other hand cubics (degree 3 plane curves)are not all rational. However they always have a parameterization of the type which allows a sin~gle square root of rational functions. In § 2 and § 3 of this paper we describe algorithms to obtainrational and special parameDic equations for the cubics and cubicoids. given the implicit equa-tions. Polynomial parameterizations are also obtained whenever they exist for the cubics andcubicoids. Higher degree curves and surfaces in general are not rational.The reverse problem of converting from parametric to implicit equations, called implicitiza-tion has been considered computationally by various authors in the past. see Collins (1971) andSederberg et. al., (1985). However as yet no conect closed fonn solution is known for implici-tizing rational surfaces or in general, implicitizing higher dimension paramettic algebraicvarieties.t Exccpl possibly the cubic conc and the cubic cylinders with nonsingulllC cubic generoting curvcs.-3-2. CubicsGeometric Viewpoint:The idea of parametrizing a conic was to fix a point on the conic and take lines through thatpoint. which intersects the conic in only one additional point, Abhyankar and Bajaj (1986a). Theconic was thus rationally parametrized by a pencil of lines with parameter t corresponding to theslope of the lines. A cubic curve is a curve which intersect most tines in three points. However ifwe consider a singulllr cubic curve then lines through the singular point, (3 double point), give arational parameterization for the cubic curve as again these lines of slope t intersect the cubic inonly one additional point Such is Dot the case for non-singular cubic curves and thus theycorrespond to the cubics which do not have a rational parameterization.Algebraic MethodA plane cubic curve is given byMake it nonregular in y ( by eliminating the y:J teoo through a coordinate transfonnation). Iflhere exists a real point at infinity then linear transformations suffice. Recall that, points atinfinity are given by the degree fonn of C (x;y), (terms of highest degree). For conics, we havereal points at infinity only in the case of parabolas and hyperbolas. However for cubics, we arelucky as all cubics have a real point at infinity. The reason being: the degree form always has areal root as it is of degree 3 and complex roots occur in conjugate pairs. The degree fonn ondehomogenizing (y = 1). gives f (x) = ax3 + bx2 + ex + d which always has a real root (ifa :t:. 0). When a is zero, the cubic C (x, y) is already nonregular in x. Thus to make C (x ,y) non.regular in y we may use a linear transformation given byx~cU+~Yy ->y;<+5Y ..... (2.1)To make the y3 term to be zero, we set its coefficient to (a ~3 + b p2 + C ~ + d) = 0 by takingS = I, which we may. Now the transformed cubic, in a somewhat rearranged fashion, is given byC(x,y) = (za + v)y2 + (px2 + qx + r)y + (li3 +!X2 + mX + n)which is the usual quadratic equation. Using the old Indian method of Shreedharacharya (5th-4-century), of solving the quadratic equation, ("multiply by 4 times the coefficient of the squareterm and add the square of the coefficient of the unknown, and the rest follows"), we obtain4(uX" + V)zy2 + 4(uX' + V)(pi'2 + qi' + r lY" + 4(ui' + v)(a3 + rz2 + mX +n) = 0which on completing the square becomesIf we lety' = [2(ui" + vlY + (px2 +qx +s)]then equation (2.2) becomes of the type..... (2.2)deg. g(X) S 4 ..... (2.3)We only need to analyze if we.can obtain a parametrization for i' and y. for then using transfor-mations (2.1) and (2.2) we obtain the parameterization for.x and y. To do this we considerseveral cases as follows: g (x') has only one distinct root, g (X) has two distinct roots...etc. In thecase of multiple roots, we may use the following genernl method to get rid of memoSupposedy" = [II (x - ~i )2]Q(X)i "" 1d = 1 or 2so each root Il.: occurs an even number of times and Q(z) has no multiple roots. Then if we let[ . ]•• yY = d.II (x - ~il, • Ithen equation (2.3) reduces toy"2 = O(i)..... (2.4)..... (2.5)If deg. Q(x) S 2, then the above equation (2.5) is a conic and a rational parametrization isalways possible, Abhyankar and Bajaj (1986a). This then, together with trnosfonnations (2.1),(2.2) and (2.4), gives a parameterization for;:t" and y of the original curve. Otherwise, g (.i) haseither 3 or 4 distinct roots, and a rational parametrization is not possible. Funher, it can be proved-5-that these are the only cases for which the cubic curve does not have a rational parametrization,Abhyankar and Bajaj (1986b). However, by solving the above equation (2.5), quadratic in y"••we always have a parameterization for the cubic of the type that allows a single square root ofrational functions. The rational parameterization obtained is global, and of degree at most 3 withthe parameter t ranging from (-00,00) and spanning the entire curve.3. CubicoidsGeometric ViewpointIf we intersect a cubic surface with a plane we get a cubic curve in general. However if weintersect it with a tangent plane then something special happens, namely, we get a singular cubiccurve or a reducible curve (either a straight line and a conic, or three straight lines). In generalwe obtain a singular cubic curve as there are only a finite number of real straight lines on a cubicsurface, see Henderson (1911). In either case the intersection curve is rational and can beparameterized by a single parameter, say s .Now to obtain a rational parametrization of the cubicoid we take a simple point on it Theintersection of the tangent plane at this simple point with the cubicoid gives a singular cubiccurve, which can be rationally parameterized by the method of § 2. Next consider a variablepoint s on this singular cubic curve. Then consider the tangent plane to the cubic surface at thatpoint We again get a singular cubic curve (or a reducible curve) as the intersection, which canbe parametrized by another parameter t. Thus we get a parametrization of the cubic surface bytwo parameters s and t. Given values of s and t they uniquely define a point on the surface.Hence a cubic surface can be parametrized starting with a simple point. However, with s and t asparameters on singular cubic curves on the cubicoid, any point ex. y • z) on the surface can beshown to correspond to six pairs of (s. t), giving a 6-fold parameterization or a 610ld covering ofthe plane. If reducible curves (conics and straight lines) are obtained as the intersection with thetangent planes, and parameters s and t are chosen on them, one could obtain a lower fold param-eterization.To ensure obtaining say a I-fold parameterization of the cubicoid we need to generate twodifferent rational curves on its surface. Let t and 't correspond to independent parameterizations-6-of the two chosen rational curves. The set ofunique lines defined by endpoints t and t (a variablepoint t on one rational cunre and a variable point 't on the other), intersect the cubic surface inone additional point giving a rational parameterization of the cubicoid. Here a JXlint (x;y ,z) onthe cubic surface can be seen to correspond to a single pair (t ,t) yielding a I-Jold parameteriza-tion or a I-j'o/d covering of the plane.One method of obtaining two different rational curves on the cubic surface is to repeat theabove method for two different simple points on the cubicoid thereby obtaining two differentsingular rational cubic curves. Alternatively. one can generate two non-intersecting straight linesfrom the twenty seven lines on a cubic surface, see Henderson (1911). This using essentially theabove method of intersections with tangent planes to the cubic surface. However, differing withthe second step of the earlier method, the variable point s is now chosen appropriately such thatthe tangent plane to the surface at this points yields a reducible intersection with the cubic sur-face. Each value of s can yield one or three straight lines lying on the same tangent plane. How-ever two specific values of s are chosen to yield two non-intersecting strnight lines of thecubicoid lying on different tangent planes.Algebraic MethodA general cubicoid (degree three) surface has an implicit equation given by+ kx2.+ ly2+ mz2 + nxy + o:rz + pyz + q:r + ry + sz + t = 0Take a simple point (xo,yo,zo) on it. Most points on the cubicoid are simple, so this is not a prob-lem. Bring the simple point to the origin by a simple trnnslation x = Xl + XO, Y = y' + Yo andz=z'+zo.C(xl, y', z') = a'x l + bly' + c'z' + ... terms of higher degree.Next rotate the tangent plane to the surface at the origin, given by the order form (terms oflowest degree). to the z = 0 plane. TItis by using a simple rotation, Xl = x, yl = Y andI 1 a' b' hi h .'=-'--,x--,yw c gIVesc' c cc (x. y. ,) = , + [f ,(x. y) + f ,(x. y), + f OZ']-7-wherejj(x;y) and 8j(x;y) are appropriate terms of degree i. Its intersection with the tangentplane z = 0 is simply.hex, y) + 8'(x,y) = 0 ..... (3.1)which is either a reducible curve or a cubic curve with a double point at the origin. In all casesthe curve (3.1) can be rationally parameterized with a single independent parameter sandrational functions K and Lx = K(x)y = L(x)z=ONow bring a general point specified by parameter s on this parameterized curve to the origin.TItis by a simple translationx=x+K(s)y =y +L(x)z =z..... (3.2)Since this point also lies on the cubic surface C(x,y"z). the surface equation with zero constantterm, is given byC(i", y. Z) = a(s)X + b(a)y + c(s)z + ... terms of higher degree.Next a simple rotationY=i_ 1 _ a(x) _ b(x)_z =--z ---x ---yc(x) c(x) c(x)..... (3.3)makes the tangent plane to the surface at the origin to be the i = 0 plane, resulting again inC(i,j,f) = z' + rf2(X, i) + i,(i, i)f + i oz-2]Its intersection with f = 0 plane will givei2(X, i) + {,(x, i) = 0 ..... (3.4)- 8 -which is a plane curve wilh coefficients involving s. Once again the curve (3.4) can be rationallyparametrized with a single independent parameter t;< =M(t)j=N(t)1=0..... (35)with coefficients of rational functions M and N also involving s. Finally using (3.5) and thetransfonnations (3.2) and (3.3) above, gives a rational parameterization of the cubic surface forthe original variables x. y and z in terms of rational functions involving both parameters sand t.The rational parameterization we obtain is global, with parameters oS and t both ranging from(-eo,oo) and covering the entire surface.Alternatively consider the plane curve given by (3.4), with coefficients involving s. For cer-tain values of oS. the plane curve is reducible which gives the lines on the cubic surface.Specifically (3.4) is reducible for those values of s for which the two polynomials 12(£. j) and83(.i. i) have a linear or quadratic common factor. One way of obtaining these s values is as fol-lows. Consider 12(:£. y) = 0 the homogeneous equation of degree 2 with coefficients involving8. It has two linear factors i =m 1(8)£ and i =mz(s)£. Substituting either of these into thehomogeneous equation g3(:£, y) = 0 yields a cubic equation of the fonn p (8)£3 = 0 where p (8)is a function of 8. Specific solutions 8 of the equation p (8) = 0 can easily be obtained by usingknown methods for obtaining roots of polynomial equations, see Buchberger et. aI. (1982). With(3.2) and (3.3) and for two appropriate choices of 8 one obtains the linear parametric equations oftwo distinct lines L 1 and L 2 on the cubic surface, viz., xl = t'YI =a1t +b l • z1 =C1t +dl andX2 = 't, yz = az't + b2" %z = cz't + dz·Next consider the straight line passing through a point on each of th.e two distinct lines L Iand Lz. 'This in space is given by two equations..... (3.6)=Xz - XI..... (3.7)and defines a family of straight lines for varying t and't. Substituting for Y and z in tenus of x inthe equation of the cubic surface C (x.y ,%) = 0 yields a cubic equation in x with coefficients in t-9-and 'to viz., g(x,t,'t)=O. However both x =t and x ='t satisfy this equation and thusg(x,t,<) O' I' . . I·' . lfu . f d T th . 6)( )( ) = IS mear In x yle wng;r as a rabona nellaD 0 t an 'to age er WIth (3.x-tz-'[and (3.7) this yields a rational parameterization of the cubic surface in terms of the independentparameters t and't.4. ConclusionFor surfaces of degree higher than three no rational p3I3metric forms exist in general,although parameterizable subclasses can be identified. For low degree curves and surfaces. inthis paper and in Abhyankar and Bajaj (1987a) procedures have been developed and implementedfor parameterizing implicit forms. Various computational issues in extending this approach toparameterize planar curves of higher degree are discussed in Abhyankar and Bajaj (1987b).Currendy efforts are being made to obtain explicit parameterizations of special families of quarticsurfaces and surfaces of higher degree which would prove useful for representin~ blending sur-faces.5. ReferencesAbhyankar, S. S., and Bajaj, C., (1987a)Automatic Parameterization of Rational Curves and Surfaces I: Conics and Conicoids,Computer Aided Design, 19,1,11-14.Abhyankar, S. S., and Bajaj, C., (1987b)Automatic Parameterization of RacionaI Curves and Surfaces 111: General Plane Curves,(in preparation).Buchberger, B., el aI. (1982)Computer Algebra, Symbolic and Algebraic Computacion, Computing Supplemenrum 4,Springer Verlag, Wien New York, 1982. (edited by B. Buchberger, G. Collins, and R.Laos,).Collins, G., (1971)The Calculation of Multivariate Polynomial Resultants, Journal of the ACM, 18,4,515-532.- 10-Henderson, A., (1911)The Twenty Seven Lines upon the Cubic Surface, Cambridge Tracts in Math. And Math.Physics, no 13.Mortenson, M., (1985)Geometric Modeling, John Wiley & Sons, Inc., 1985.Sederberg, T., Anderson, D., and Goldman, R. t (1985)Implicit Representation of Parametric Curves and Surfaces, Computer Vision, Graphics& Image processing, vol 28. 72-84.Tiller, W., (1983)Rational B-splines for Curve and SuIface Representation IEEE Computer Graphics andApplicaJions. 23. 6, 61-69.

Automatic Rational Parameterization of Curves and Surfaces II: Cubics and Cubicoids

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Automatic Rational Parameterization of Curves and Surfaces II: Cubics and Cubicoids

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