Fast Ray Tetrahedron Intersection using Plücker Coordinates

Transcription

1 Fast Ray Tetrahedron Intersection sing Plücker Coordinates Nikos Platis and Theoharis Theoharis Department of Informatics & Telecommnications University of Athens Panepistemiopolis, GR Ilissia, Greece {nplatis Abstract We present an algorithm for ray tetrahedron intersection. The algorithm ses Plücker coordinates to represent the ray and the edges of the tetrahedron and employs a robst and efficient test to determine the intersection. The algorithm is highly optimized and provides a significant performance increase over related algorithms. 1 Introdction Tests for intersection of directed lines with graphics primitives are at the heart of many rendering and processing algorithms. The crrent litteratre does not treat explicitly the ray tetrahedron intersection problem, even thogh tetrahedral meshes are sed increasingly for the representation of volmetric models [GCMS01]. In this paper we investigate efficient soltions to this problem. Notation and Conventions In the ray tetrahedron algorithms that follow, the ray will be assmed an infinite directed line, determined by a point P and a direction L. The vertices of the tetrahedron will be marked V 0, V 1, V 2, V, and its faces by the index of the opposite vertex: F (V 0 V 1 V 2 ), F 2 (V 1 V 0 V ), F 1 (V 2 V V 0 ), F 0 (V V 2 V 1 ). We will se the notation V0 i, V1 i, V2 i to refer to the vertices of face F i ; for example, the vertices of face F 2 above are V0 2 = V 1, V1 2 = V 0 and V2 2 = V. Similarly, the edges of face F i will be sbscripted by the index of the opposite vertex: e i 0(V1 i V2 i ), e i 1(V2 i V0 i ), e i 2(V0 i V1 i ). We will assme that the tetrahedron is oriented so that the otward-pointing normal of face F i is directed away from vertex V i. If the ray intersects the tetrahedron, in general there will be two intersection points, and. Special cases with one or infinite intersection points arise if the ray intersects one or more edges; these cases can be handled niformly by the general case as shown in Figre 1, bt in some applications it may be beneficial to mark them explicitly. For the two intersection points, we reqire that the algorithms compte: 1

2 V V 0 V 1 V 2 (a) (b) (c) Figre 1: Ray tetrahedron intersection: (a) two, (b) one, (c) infinite intersection points. Their Cartesian coordinates and. Their barycentric coordinates enter 1, enter 2 and leave 1, leave 2 with respect to the faces that they intersect, F enter and F leave, sch that P k = (1 k 1 k 2)V k 0 + k 1V k 1 + k 2V k 2 for k = enter, leave. Their parametric distance t enter and t leave from the ray origin, sch that P k = P + t k L for k = enter, leave. 2 Related Work 2.1 Approaches for ray tetrahedron intersection A ray tetrahedron intersection test can be solved by a general ray convex polyhedron intersection algorithm. Haines [Hai91] handles this general case with an algorithm that works similarly to the familiar Liang-Barsky [FvDFH96] line clipping algorithm. This algorithm is fairly efficient also for tetrahedra and is sed as a base for or comparisons. It works with the parametric eqation of the ray and comptes the intersection points sing their parametric distance from the ray origin; ths it has the advantage that restrictions concerning this distance (for example, a bond on the maximm valid distance along the ray, sefl in ray tracing) can be applied early. On the other hand, it does not se barycentric coordinates, which conceqently mst be compted as a post-processing for the intersection points; moreover, it can recognize the special cases of intersection mentioned above only throgh these barycentric coordinates. Alternatively, the ray tetrahedron intersection problem can be solved by testing each face of the tetrahedron for intersection with the ray and combining the reslts; several efficient ray triangle intersection algorithms exist, and it is important to se one that can better adapt to this specific problem. For instance, Möller and Trmbore [MT97] provide a very efficient sch algorithm, which also ses the parametric eqation of the ray and at the same time optimizes the comptation of barycentric coordinates; however, the ray tetrahedron intersection algorithm that we constrcted sing this algorithm did not perform well. In the next section we present the ray triangle intersection algorithm that forms the basis of or ray tetrahedron intersection algorithm. 2

3 (a) s (b) s (c) r r r s Figre 2: Relative orientation of two lines: Looking towards the direction of r, (a) s goes conterclockwise arond r, (b) s goes clockwise arond r; (c) s intersects r. 2.2 Ray Triangle Intersection sing Plücker Coordinates Plücker coordinates [Eri97, Sho98, Sto91, TH99] are a way of specifying directed lines in three-dimensional space sing six-dimensional vectors. Given a ray r determined by point P and direction L, its Plücker coordinates are given by the six-vector π r = {L : L P } = {U r : V r } (1) In or context, their important property is that, given two rays r and s, the permted inner prodct indicates their relative orientation (Figre 2): π r π s = U r V s + U s V r (2) π r π s > 0 s goes conterclockwise arond r π r π s < 0 s goes clockwise arond r π r π s = 0 s intersects or is parallel to r This property is the basis of a ray triangle intersection test. Sppose we are given a ray r and a triangle (V 0, V 1, V 2 ) with edges e 0 (V 1 V 2 ), e 1 (V 2 V 0 ), e 2 (V 0 V 1 ). Then r intersects iff it has the same orientation (cw or ccw) relative to all its edges or it intersects at most two of its edges (Figre ). Ths the following hold: r intersects (enters) π r π ei 0 i and j : π r π ej 0 r intersects (leaves) π r π ei 0 i and j : π r π ej 0 r is coplanar with π r π ei = 0 i () This test is sed sccessflly in [AC97] (and in [SF01] as the eqivalent 4 4 determinant method) to speed p ray tracing of trianglar meshes. It is robst and efficient since it reqires few floating point operations, no division, and relies only on sign comparisons; moreover, calclations for an edge can be shared among neighboring triangles. Additionally, it is important that if the ray and the triangle are not coplanar, the permted inner prodcts π r π ei provide directly the (nscaled) barycentric coordinates of the intersection point P k with respect to the vertices V i [Jon00]. Ths setting w k i = π r π ei and k i = w k i / wi k, (4) i=0

4 (a) (b) (c) Figre : (a) The ray enters the triangle; (b) The ray leaves the triangle; (c) The ray and the triangle do not intersect. the Cartesian coordinates of the intersection points can be compted as P k = k 0V 0 + k 1V 1 + k 2V 2. Sbseqently, their parametric distance from the ray origin can be trivially compted by solving for t k a coordinate of the eqation P k = P + t k L for which L is non-zero. Ray Tetrahedron Intersection sing Plücker Coordinates.1 Basic Algorithm The above ray triangle intersection test leads to a simple ray tetrahedron intersection algorithm. In the following, referring to face F i of the tetrahedron, we will denote by πj i the Plücker coordinates of edge ei j and by σi j the sign of the permted inner prodct π r πj i for the given ray r: 1, if π r π σj i = sign(π r πj) i j i > 0 = 0, if π r πj i = 0 1, if π r πj i < 0 The algorithm tests each face of the tetrahedron in trn and determines if it is F enter or F leave according to eqation (); if either of them has been fond, the relevant sign tests need not be performed for the remaining faces. F enter = nil F leave = nil for i =, 2, 1, 0 do Compte σ i 0, σ i 1 and σ i 2 if ((σ i 0 0) or (σ i 1 0) or (σ i 2 0)) if ((F enter == nil) and (σ i 0 0) and (σ i 1 0) and (σ i 2 0)) F enter = F i else if ((F leave == nil) and (σ i 0 0) and (σ i 1 0) and (σ i 2 0)) F leave = F i end for This algorithm has the advantage that it can compte the barycentric coordinates of the intersection points with little additional work, sing eqation (4). 4

5 It can also readily discern, if reqired, the special cases of bondary intersections (Figre 1(b,c)), which correspond to some of the σj i being zero. Compared to Haines algorithm, it has the drawback that it can accomodate restrictions concerning the parametric distance along the ray only as a post-processing step. As we will show, this basic algorithm can be greatly optimized and adapted to the problem being solved. Some simple enhancements are possible by nrolling the for loop and changing the sccessive tests as follows: As soon as F enter and F leave are determined, the algorithm may terminate. The inner test needs to be performed for at most three of the for faces of the tetrahedron. If none of the three faces is intersected, the forth will not be intersected either; otherwise it will be F enter or F leave depending on which face is not already fond. Comptations of the permted inner prodcts can be resed for the faces that share each edge. Care shold be taken to adjst the sign of the prodct according to the edge orientation within each face; for example, edge V 0 V 1 is sed for face F (V 0 V 1 V 2 ) whereas V 1 V 0 is sed for F 2 (V 1 V 0 V ), and π r π V0 V 1 = (π r π V1 V 0 ), see eqations (1),(2). This is important when the permted inner prodcts are not calclated in advance, since only one mst be compted for each pair of opposite directed edges; it is even more important when ray-tracing tetrahedral meshes, where each edge is shared by mltiple tetrahedra..2 Optimizations While the enhancements mentioned in the previos paragraph reqire few modifications of the basic algorithm, frther optimizations are possible at the expense of increased code complexity. Or aim is to se as little information as possible in order to decide whether a face is intersected by the ray. We notice that the inner test examines all σj i at once; however, if two of them do not agree, the third one need not be examined at all and the comptation of the relevant Plücker coordinates and permted inner prodct can be avoided. Care mst be taken since any two (bt not all three) σj i may be zero. Ths the inner test for face F i becomes: Compte σ i 0 and σ i 1 {Check if σ i 0 and σ i 1 agree, or they are zero} if ((σ i 0 == σ i 1) or (σ i 0 == 0) or (σ i 1 == 0)) Compte σ i 2 {Find the sign (orientation) σ i of the face} σ i = σ i 0 if (σ i == 0) σ i = σ i 1 if (σ i == 0) σ i = σ i 2 {At this point, σ i will be eqal to σ i 0 or σ i 1, whichever is non-zero;} {in that case it mst agree with σ i 2, or σ i 2 can be zero.} 5

6 {If both σ i 0 and σ i 1 are zero, σ i will be eqal to σ i 2} {and it mst be non-zero.} if ((σ i 0) and ((σ i 2 == σ i ) or (σ i 2 == 0))) if (σ i > 0) F enter = F i else F leave = F i When one face intersected by the ray has been fond, the above test can be simplified for the remaining faces: edges shared with the intersected face will have conforming sign and need not be tested again; frthermore, the σ i j for the remaining edges shold satisfy σi j 0 if F leave is not fond or σ i j 0 if F enter is not fond. When one face intersected by the ray has been fond and only two faces remain ntested, it is possible to determine the other one by performing only a sbset of the sign tests normally reqired. The choice depends on the sign of their common edge: sppose that F leave mst be determined between F 1 (V 2 V V 0 ) and F 0 (V V 2 V 1 ); if π r π V2V < 0, then F leave = F 1, otherwise F leave = F 0 (nless all three σj 0 are zero, in which case also F leave = F 1 ). The last two optimizations will be especially sefl when ray-tracing tetrahedral meshes. In this setting, having fond a tetrahedron intersected by the ray, connectivity information of the mesh provides the tetrahedron intersected next and F enter is already known for it (this holds nless the ray leaves the last tetrahedron throgh an edge, in which case the next tetrahedron cannot be determined directly).. Revisiting Geometry The ray tetrahedron intersection problem presents a geometric property which appears as a potentially sefl mechanism to gide the intersection algorithm. Unfortnately, in practice it does not accelerate the flly optimized algorithm of the previos section. We describe this property here for completeness and for its geometric interest. Sppose that the first face examined is F (V 0 V 1 V 2 ). If it is intersected by the ray, the algorithm contines as described in the previos section. If not, the ray hits its spporting plane inside one of the six regions formed by the lines along the edges of F (Figre 4); these regions correspond to different ranges of barycentric coordinates on this plane with respect to V 0, V 1 and V 2. Depending on the region and the ray direction, the ray may enter or leave the tetrahedron only throgh a specific face; if it does not, there is no intersection at all. For example, if it enters the plane in region (E), it may enter the tetrahedron only throgh F 1 (V 2 V V 0 ); if it enters the plane in region (A), it may leave the tetrahedron only throgh F 0 (V V 2 V 1 ). We note that the ray enters the plane if 2 j=0 (π r πj ) > 0 (it is easy to show that this is eqivalent to the test L N < 0, where N is the face normal). 6

7 V (E) (A) (F) V0 V1 (C) V 2 1 < 0 0 > 0 (D) 1 > 0 0 < 0 (B) 2 < 0 2 > 0 Figre 4: If a ray does not intersect F (V 0 V 1 V 2 ), it intersects its spporting plane in one of the six regions shown (A F). Using this test is slower than testing the remaining faces as per section.2, for two reasons: first, all three prodcts π r πj, j = 0, 1, 2, mst be compted in order to apply this property, whereas one of them cold potentially be avoided as mentioned earlier; second, a rather involved examination of the barycentric coordinates is reqired in order to determine the region of intersection on the plane. 4 Reslts We tested or algorithms on random pairs of rays and tetrahedra. Mltiple sets of 10,000 random ray tetrahedron pairs were generated, with increasing percentage of intersecting pairs from 0% to 100%; in this way the efficiency of each algorithm when different proportions of the tests scceed was assessed. Five algorithms were compared: the three variations of or algorithm corresponding to Sections.1,.2 and., Haines algorithm adapted for tetrahedra, and an algorithm similar to the one of Section.2 bt sing the Möller/Trmbore ray triangle test. All algorithms prodce, when the test is sccessfl, the two intersection points and, the intersected faces F enter and F leave, the corresponding barycentric coordinates enter 1, enter 2 and leave 1, leave 2, and the parametric distances t enter and t leave. Barycentric coordinates for Haines algorithm were compted as described in [BA88]. Tests were performed on an Intel Celeron PC at 900MHz rnning Linx. Figre 5 attests that or algorithms sing Plücker coordinates otperform the other algorithms in all cases. The optimizations of Section.2 provide an important performance gain, whereas the modifications of Section. have negative impact, as already noted. It is also interesting that Haines algorithm performs comparatively better when few pairs intersect; this is de to the fact that, in some cases, it is able to decide that there is no intersection withot examining all for faces of the tetrahedron, bt it mst process them all to find the intersection points. 7

8 Time per intersection test (µs) Plücker 1 Plücker Plücker Haines Möller/Trmbore Percentage of intersected tetrahedra Figre 5: Reslts for ray tetrahedron intersection tests Time per intersection test (µs) Plücker Plücker 2 Plücker Haines Percentage of intersected tetrahedra Figre 6: Reslts for intersection tests with precompted qantities. In a practical sitation, a spatial data strctre can be sed to identify the tetrahedra along the path of the ray; for these tetrahedra several comptations can be performed once for each ray. We modified the algorithms so as not to inclde sch comptations in the timings (π r πj i and σi j for all i, j in the case of or algorithms, similar relevant qantities for Haines algorithm; the algorithm sing the Möller/Trmbore test was not inclded in this test since it wold reqire storing a large nmber of precompted qantities). The reslts, shown in Figre 6, are remarkably consistent with the previos ones and affirm the potential of or algorithms. In any case, or flly optimized algorithm is the fastest. 8

9 References [AC97] John Amanatides and Kin Choi. Ray Tracing Trianglar Meshes. In Proceedings of the Eighth Western Compter Graphics Symposim, pages 4 52, April [BA88] Rod Bogart and Jeff Arenberg. Ray/Triangle Intersection with Barycentric Coordinates. Ray Tracing News, 1(11), November rtnews5b.html#art. [Eri97] Jeff Erickson. Plücker Coordinates. Ray Tracing News, 10(), December html/rtnv10n.html#art11. [FvDFH96] James D. Foley, Andries van Dam, Steven K. Feiner, and John F. Hghes. Compter Graphics, Principles and Practice. Addison- Wesley, second edition, [GCMS01] [Hai91] Fabio Ganovelli, Paolo Cignoni, Cladio Montani, and Roberto Scopigno. Enabling cts on mltiresoltion representation. The Visal Compter, 17: , Eric Haines. Fast Ray Convex Polyhedron Intersection. In James Arvo, editor, Graphics Gems II, pages Academic Press, [Jon00] Ray Jones. Plücker Coordinate Ttorial. Ray Tracing News, 1(1), Jly rtnv1n1.html#art8. [MT97] Tomas Möller and Ben Trmbore. Fast, Minimm Storage Ray/Triangle Intersection. jornal of graphics tools, 2(1):21 28, [SF01] Rafael J. Segra and Francisco R. Feito. Algorithms to Test Ray- Triangle Intersection. Comparative Stdy. In Vaclav Skala, editor, WSCG 2001 Conference Proceedings, Febrary [Sho98] Ken Shoemake. Plücker Coordinate Ttorial. Ray Tracing News, 11(1), Jly html/rtnv11n1.html#art. [Sto91] Jorge Stolfi. Oriented Projective Geometry. Academic Press, [TH99] Seth Teller and Michael Hohmeyer. Determining the Lines Throgh For Lines. jornal of graphics tools, 4():11 22, Web Information Sample C++ code implementing the ray tetrahedron intersection algorithm of section.2 is availabe online at 9