Intersecting surfaces of special types

Intersecting Surfaces of Special TypesMyung-Soo KimDept.of Computer SciencePOSTECHPohang790-784,Korea.AbstractThe paper reviews new approaches to computing the in-tersection curve of two surfaces of special types.We con-sider algorithms for intersecting a torus with a natural quadric or another torus.After that,we review a topo-logical technique that computes the intersection curve of a sphere and a surface of revolution.Finally,we consider the intersection of two ruled surfaces.1.IntroductionIntersecting two surfaces is a basic geometric opera-tion in geometric and solid modeling,e.g.,for supporting Boolean operations in a solid modeler.There has been a considerable amount of previous work on the intersection of free-form surfaces[9,23,25].Nevertheless,finding al-gorithms that can compute the intersection curve of two ar-bitrary rational surfaces reliably without user intervention is extremely difficult[9].In particular,when two surfaces have singular intersections,general algorithms have serious drawbacks in robustness.Robustness,accuracy and/or efficiency are often im-proved by using a special purpose algorithm customized for the particular class of intersection problem being solved. For intersecting two quadric surfaces,there are many spe-cialized algorithms[7,17,18,20,21,24,26,27,28,29]that provide better solutions(in efficiency and robustness)than general surface intersection algorithms.Algebraic meth-ods[7,17,18,26,28,29](based on symbolic manipulation of surface equations)are general in the sense that they can handle all types of quadric surfaces.However,when the algebraic algorithms are implemented usingfloating-point arithmetic,it is very difficult to ensure their robustness.In the case of intersecting two natural quadrics,the sit-uation is much better.There are reliable geometric algo-rithms that can intersect two natural quadrics efficiently and robustly[20,21,24,27].In particular,Miller and Gold-man[21]classify necessary and sufficient geometric con-ditions that correspond to all possible types of degener-ate/singular intersections.All computations employed in these algorithms have clear geometric meanings.More-over,they can be carried out efficiently and robustly.To-gether with similar geometric algorithms for computing the planar sections of natural quadrics[11,22],these al-gorithms[20,21,24,27]can support efficient and robust Boolean operations for CSG objects constructed by planes and natural quadrics.A natural question is how to extend the geometric coverage to include torus.There are algebraic methods for intersecting two arbi-trary cyclides[4,10,19].Plane,natural quadrics,and torus are special types of cyclide.Therefore,these algorithms can be used in intersecting a torus with other simple sur-faces(plane,natural quadrics,and torus).As in the case of intersecting two quadric surfaces,algebraic methods are general,but they have limitations in robustness.Therefore, we need geometric algorithms that can guarantee efficiency and robustness at the same time.Kim et al.[13]present an algorithm that intersects a torus with a sphere.Based on a configuration space(C-space)transformation,this approach can be extended to the intersection of a torus with a cylinder,a cone,or another torus.The C-space technique is applicable to closed sur-faces that enclose valid solids;thus it is difficult to deal with a spindle(self-intersecting)torus.Kim et al.[16]present a topological technique that can intersect a sphere with an arbitrary surface of revolution(which may self-intersect). Moreover,the intersection curve can be represented exactly in a closed-form equation represented by a square root func-tion.Heo et al.[8]simplify the problem of intersecting two ruled surfaces to the construction of an implicit curve in the plane.A ruled surface is linear in one parameter direc-tion.Thus it is easy to eliminate the linear parameters of two ruled surfaces simultaneously,while reducing the prob-lem to a single equation in two remaining parameters.The construction of an implicit curve in the plane is significantly more efficient and robust than the intersection of two ruled surfaces in the space.The rest of this paper is organized as follows.In Section 2,we consider an example which exemplifies the funda-mental difficulty of general surface intersection.Section3 reviews the C-space transformation approach and Section4 shows how to detect degenerate conic sections in the inter-section of a torus and a natural quadric surface.In Section5, we explain an algorithm that intersects a sphere with a sur-face of revolution.The intersection of two ruled surfaces is discussed in Section6.Finally,in Section7,we conclude this paper.2.Limitation of General Surface IntersectionAmong numerous methods for intersecting two general free-form surfaces,subdivision-based methods produce themost reliable solutions,in general.Nevertheless,when ap-plied to almost tangential surfaces,a subdivision algorithm has serious drawbacks in efficiency and robustness due to acascading sequence of recursive subdivisions.Figure1(a)shows two bi-cubic B-spline surfaces that in-tersect in four closed loops.Figures1(b)and1(c)are the results of scaling the two surfaces of Figure1(a)along the -direction by factors0.1and0.01,respectively.(This ex-ample is taken from Figure7.11of J.-J.Choi[3].G.Elber atTechion,Israel,originally suggested testing this example.) The intersection curve of Figure1(a)has the same topolog-ical structure as that of Figure1(c).Nevertheless,it is sig-nificantly more difficult to deal with the almost tangential intersection case of Figure1(c).In Figure1(c),the intersection curve is composed of many tiny curve segments generated in the recursive sub-division.The input surfaces are also subdivided into many small patches.Due to numerical errors occurring in the re-peated subdivision,it is extremely difficult to connect the intersection curve segments in a correct topology.Many tiny loops may be generated erroneously.Thus it is hard to tell how many connected components the intersection curve actuallyhas.(a)(b)(c)Figure1.Tangential Surface Intersection.3.C-space TransformationA torus is the envelope surface of a sphere moving along a circle.Given a torus and a surface,their intersection curve can be characterized as the envelope of the inter-section curves generated by the moving sphere and the given surface.Kim et al.[13]showed that the problem of intersecting a torus and a sphere can be transformed to the intersection of the main circle of the torus and two offsets of the sphere(see Figure2).The left column of Figure2 showsfive different configurations of the main circle of a torus with respect to the volumetric region bounded by two offset spheres.The right column shows the intersection curves corresponding to the configurations of the left column.When the main circle is totally contained in the volumet-ric region bounded by two offset spheres,the intersection curve has two loops.Each tangential intersection between the main circle and an offset sphere corresponds to a sin-gular point of the intersection curve.Each segment(except the whole circle)in the volumetric region corresponds to a connected component of the intersection curve.When the minor radius of the torus is larger than the ra-dius of the sphere,we cannot apply this technique since the inner offset sphere reverses its surface orientation.In this case,we may consider the sphere as a moving object and the torus as an obstacle.As we reduce the sphere to its cen-ter,the torus is offset to two tori.Depending on the region where the sphere center locates,we can tell the topological type of the intersection curve(see Figure3).The C-space approach can be extended to the intersection of a torus witha cylinder,a cone,and another torus[12].4.Degenerate Conic SectionsConic sections embedded in a torus must be circles of special types(see Figure4):(i)profile circles,(ii) cross-sectional circles,and(iii)Yvone-Villarceau circles. When a torus intersects another surface in a conic section, the conic section must be a circle.It is easy to classify all circles embedded in natural quadrics(sphere,cylinder,and cone).By comparing these circles with those embedded in a torus,we can detect and compute all conic sections(i.e., circles)in the intersection curve of a torus and a natural quadric.Kim and Kim[14]present a simple algorithm that de-tects and constructs all degenerate circles in the intersection of a torus and a natural quadric.Consider a torus(with minor radius,major radius,center,and axis direc-tion),and a sphere(with center and radius).TheFigure2.Torus and SphereIntersections.Figure3.Torus and SphereIntersections.(a)(b)(c)Figure4.Circles Imbedded in a Torus.1p2p1Nrp2N1p1rr(a)(b)Figure5.Detecting Yvone-Villarceau Circlestorus and the sphere intersect in two Yvone-Villarceau cir-cles if and only if they satisfy the following two conditions:(2) The circle itself is constructed as a circle of radius,with center atin the intersection curve.The end point (respectively,)is located on the circle if and only if the two segments and are connected at their com-mon end point(and respectively,).For example,the intersection curve of Figure7consists of two closed loops.Similarly, referring back to Figure6,the intersection curve consists of one isolated(tangential)point and two closed loops.When ,,is a closed loop inside(but never in-tersects the circle itself),the intersection curve has two closed loops,each of which is a smooth deformation of (i.e.,homeomorphic to).In general,the surface of revolution may self-intersect and the sphere may intersect the axis of rotation.Figure8 shows four such examples.In each example,the grey re-gion represents the area inside the circle which is at the same time outside the reflected circle(with respect to the axis of rotation).Note that each segment in the grey re-gion corresponds to a closed loop of the intersection curve.Assume that the sphere of radius has its center at and the surface(of revolution)is obtained by ro-tating a plane curve about the-axis.(We may assume this specific configuration by applying trans-lation and rotation to both surfaces.)Kim et al.[16]show that the intersection curve can be represented exactly as follows:Thus the problem of intersecting two ruled surfaces can be essentially reduced to that of constructing an implicit curve in the-plane.Let be a surface parametrized by.The curve is constructed by inter-secting and the-plane.By applying the B-spline subdivision technique to the surface,we can com-pute the intersection curve.This approach provides an effi-cient and robust algorithm,in particular,when the two ruled surfaces intersect almost tangentially.(a)(b)(c)(d)(e)(f)Figure9.Intersection of two cylinders.In(a)-(d),two almost coaxial cylinders are inter-sected,with the angle between the two cylin-ders being,,,.In(e)and(f),the functions for cases(a)and(d)areshown.Figures9(a)–9(d)show a sequence of examples that in-tersect two almost coaxial cylinders with angles of,, ,between the two cylinders,in that order.Fig-ures9(e)and9(f)are the-functions of the examples shown in Figures9(a)and9(d),respectively.It is very difficult to distinguish two intersecting cylinders that appear almost overlapping in Figures9(c)–9(d).Moreover,the-function of Figure9(f)is almostflat.Nevertheless,the computation results are numerically stable and they produce reasonable solutions,which demonstrates the robustness of our inter-section algorithm for two ruled surfaces.Elber et al.[5]show that traditional ray-tracing can be re-duced to a problem of intersecting a free-form surface with ruled surfaces(see Figure10).Thus the development of efficient algorithms for surface intersection can have a sig-nificant impact on graphics rendering.Given two free-form curves in the plane,their bisector curve is defined as the set of points which are at equal distance from the two curves. Choi[3]reduced the problem of computing a bisector curve to that of intersecting two developable surfaces(which are also ruled surfaces).Thus the surface intersection of free-form surfaces(in particular,that for ruled surfaces)has var-ious interesting applications in computer graphics and geo-metricmodeling.Figure10.Ruled Tracing.7.ConclusionThis paper reviewed the author’s recent work in devel-oping algorithms that intersect surfaces of special types.As an envelope surface of a sphere moving along a circle,the torus has a simpler geometric structure than the cone.Con-sequently,geometric algorithms for tori can be designed in a much easier way than those dealing with natural quadrics. Ruled surfaces are also simpler than general free-form sur-faces.The computational procedure for intersecting two ruled surfaces is similar to that of Elber and Kim[6]which computes the bisector curve of two rational curves in the plane.It may also be possible to transform other problems for ruled surfaces to simpler problems of solving a few al-gebraic constraints in two variables.AcknowledgementsThe author owes much to technical collaboration with his colleagues and students,in particular,to Jung-Ju Choi,Ger-shon Elber,Hee-Seok Heo,Ku-Jin Kim,and Ralph Martin.This research was supported in part by the Korean Min-istry of Science and Technology under Grant97-NS-01-05-A-02-A of STEP2000,by KOSEF(Korea Science and Engineering Foundation)KOSEF(Korea Science and En-gineering Foundation)under Grant96-0100-01-01-2and Com Mac.References[1] C.Bajaj,C.Hoffmann,J.Hopcroft,and R.Lynch.Tracingsurface puter Aided Geometric Design, 5(4):285–307,November1988.[2] C.Bajaj and G.Xu.NURBS approximation of sur-face/surface intersection curves.Advances in Computational Mathematics,2(1):1–21,January1994.[3]J.-J.Choi.Local Canonical Cubic Curve Tracing along Sur-face/Surface Intersections.Ph.D.Thesis,Dept.of Computer Science,POSTECH,February1997.[4]J.de Pont.Essays on the Cyclide Patches.Ph.D.The-sis,Cambridge University Engineering Department,Cam-bridge,February1984.[5]G.Elber,J.-J.Choi,and M.-S.Kim.Ruled tracing.TheVisual Computer,13(2):78–94,February1997.[6]G.Elber and M.-S.Kim.Bisector curves of planar rationalcurves.To appear in Computer-Aided Design,1999.[7]R.Farouki,C.Neff,and M.O’Connor.Automatic parsingof degenerate quadric-surface intersections.ACM Trans.on Graphics,3(3):174–203,July1989.[8]H.-S.Heo,M.-S.Kim,and G.Elber.The intersection of tworuled surfaces.To appear in Computer-Aided Design,1999.[9]J.Hoschek and sser.Fundamentals of Computer AidedGeometric Design.A.K.Peters,Wellesley,MA,1993. 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