A New and Efficient 2D Collision Detection Method Based on Contact Theory Xiaolong CHENG, Jun XIAO a, Ying WANG, Qinghai MIAO, Jian XUE

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1 5th Intenational Confeence on Advanced Mateials and Compute Science (ICAMCS 2016) A New and Efficient 2D Collision Detection Method Based on Contact Theoy Xiaolong CHENG, Jun XIAO a, Ying WANG, Qinghai MIAO, Jian XUE Univesity of the Chinese Academy of Sciences No.19A Yuquan Road, Beijing, , China a xiaojun@ucas.ac.cn Keywods: Contact Collision; Contact Theoy; Entance Block; Contact Cove Abstact. Collision detection is a fundamental poblem in mechanical, obotic, stuctual, and geotechnical engineeing. A new algoithm base on Contact Theoy to solve accuate contact poblem is poposed in this pape. In this method, two polygons (A and B) and a efeence point ae defined, whee the efeence point has paallel movement to polygon A. Then we calculate contact coves, build bounday of Entance Block, and the contact poblem between two geneal polygons is educed to contact between a efeence point and the bounday of Entance Block. i.e., we can get distance between two polygons if they ae sepaated, get contact points if they contact, and get shotest exiting path along with contact points if they ae intesected. Expeiments and analysis show that the poposed method can compute the collision detection poblem accuately and educe the complexity of contact poblem obviously. Intoduction Collision detection is a basic but impotant poblem in compute gaphics, and it is an unavoidable poblem fo analysis of almost all moving geomety objects. Polygons with diffeent positions have diffeent contact points, and thee may be infinite contact points when polygons ae in cetain positions. As a esult [1], this poblem has dawn much attention of scholas fom diffeent fields in the past decades. Seveal stategies have been developed in ode to deal with the poblem [2], such as the common plane [3], the single suface algoithm [4], hieachy teitoy algoithm [5], the maste-slave appoach [6], the mateial depth" appoach [7] and the polygon-paticle algoithm [8]. In addition, thee ae also some stategies fo igid bodies, e.g. the "Optimization Based Animation (OBA)" appoach [9] fo computing gaphics puposes [10, 11] and the Non Smooth Contact Dynamic (NSCD) fomulation [12]. Contact Theoy [1] is the fist theoy that pesents and poves contact poblem in mathematics. It is diffeent fom all existing contact algoithms, since it povides a geneal way to solve contact poblem. In Contact Theoy, a new concept Entance Block is intoduced. By means of Entance Block, the contact poblem between two geneal polygons educes to contact between a efeence point and the Entance Block, which simplifies the complexity of the contact poblem. In this pape, we just give a 2D contact algoithm, but it is still impotant that 2D esult is consevative and can be accepted easily in same cases (e.g., many woldwide majo engineeing pojects ae analyzed by 2D-DDA [13]). We give basic concepts of Contact Theoy in section 2, algoithm in section 3, and two examples in section 4, and conclusion in section 5. Some Base Concepts of Contact Theoy In this section, we define the geomety data stuctue, and give bief explanation of some basic concepts and simple application of Contact Theoy [1]. All geomety data stuctue obey ight-hand coodinate, as shown in figue 1: line segment A (figue 1.a) is epesented by two points (a 0 and a 1 ) with diection a 1 -a 0 ; angle A (figue 1.b) is epesented by two line segments (i.e. e 1 and e 2 ) with the secto otates fom e 1 to e 2 ; polygon A (figue 1.c) is epesented by its points (i.e., a 0 a 4 ), its The authos - Published by Atlantis Pess 95

2 edges (i.e., e 0 e 4 ), and its aea. The basic concepts of Contact Theoy ae elations of two polygons, efeence point, contact cove and Entance Block. The elations of two polygons ae sepaated, contacted and intesected. Moeove, the contact elation is what we concentate on. The efeence point is a key concept in Contact Theoy. In Contact Theoy, two polygons ae named as polygon A and polygon B, and the efeence point has paallel movement to polygon A. Then we can easily locate polygon A by the efeence point. In addition, the efeence point is used to calculate Entance Block and detect cuent contact. The contact cove contains contact infomation of two geomety objects, and can ovelap with each othe. Contact coves ae calculated by the coe fomula expessed as fomula (1), whee A and B epesent polygons, while p epesents efeence point. E( A, B) = B A + p (1) Fo example, giving point a and line segments B (figue 2), thei contact cove is a line segment (fom p to p ') with B fixed. We can detemine the cetain position of point a by evey point on the contact cove. Fo ovelapping (figue 3), we can get two contact coves if efeence point is on the bottom contact cove E( aa, bb ). i i+ 1 j j + 1 The Entance Block is the key concept of Contact Theoy. It consists of contact coves, and contains all contact infomation of A and B with B fixed. We can undestand Entance Block in a simple way that a geomety object composed by the positions of efeence point when two polygons contact o intesect. To detect cuent contact, we only need the bounday of Entance Block. Theefoe, befoe detecting contact, we need to build bounday of 2D Entance Block. The bounday is a contact cove system, which consists of contact edges o angles (figue 4). Taking advantages of Contact Theoy, we detect cuent contact easily that what we do is to select the contact edges with shotest distance to efeence point. Thus, contact computation is simplified with efeence point and Entance Block: fo sepaated case (figue 5.a), we get the shotest distance between the efeence point and the Entance Block, which is also the shotest distance between two polygons; fo contact case (figue 5.b), we get contact points and contact locations of two polygons; fo intesected case (figue 5.c), we get the contact points and shotest exiting path, which is the line between efeence point and its pojective point on the Entance Block. a. Line Segment b. Convex and Concave Angles c. Polygon Figue 1 Geomety Shapes Figue 2 contact cove of point and line segment Figue 3 Ovelapped Contact Coves Figue 4 Entance Polygon of two Tiangles 96

3 a. Sepaated Condition b. Contacted Condition c. Intesected Condition Figue 5 Judging Relations of Polygons with Refeence Point and Entance Block 2D Algoithm to Detect Contact with Contact Theoy Entance Block is complex, but we only need its bounday to detect cuent contact. Thee ae 3 phases to detect the cuent contact coves: 1) define efeence point and two polygons (A and B); 2) calculate contact coves and compose bounday of Entance Block; 3) detect cuent contact coves with efeence point. We will give the phases in detail next. (1)Algoithms fo Calculating Contact Coves Fo 2D contact algoithm, we conside angle-edge and angle-angle contact. Moeove, the esult is both contact edges. Contact edges of Angle-Edge and Edge-Angle ae calculated by fomula 2 (figue 6) and fomula 3 (figue 7), while contact edges of Angle-Angle is calculated by fomula 4 (figue 8). EaB (, ) = { b + t* ( b b) a+ p: t [0,1]} (2) Whee vetex of angle A is point a, points of edge B ae b 1 and b 2, and p is the efeence point. EAb (, ) = { b ( a + t* ( a a)) + p: t [0,1]} (3) Whee points of edge A ae a 1 and a 2, vetex of angle B is point b, and p is the efeence point. Figue 6 Angle-Edge contact Figue 7 Edge-Angle contact u u uu uu EAh (, ) : e h (0,0,1) (, ) : (0,0,1) Ee B e h E( A, B) = E( e, B ): othes E( A, h ): othes 1 2 (4) a. Angle-Angle Contact b. Angle-Angle Contact Figue 8 Contact coves of Angle-Angle Contact (2)Calculate Bounday of Entance Block With the fomulas (2), (3) and (4), we give the pocesses to calculate bounday of Entance Block hee. Accoding to the above desciption, we only need to calculate valid (i.e., fo angle-angle contact, angles must not intesect if thei vetices move to one point; fo angle-edge contact, they must not intesect if angle s vetex moves onto edge) contact edges to solve 2D contact poblems. To judge intesection, thee ae two institute ways: using vecto calculation o using angle calculation. We use judging algoithms by vecto calculation, which is given as fomula (5) and (6), as shown in figue 9 and

4 With fomulas (2)-(6), we now give the pocesses (figue 11) to build the bounday of Entance Block. Thee ae thee steps: 1) select disjoint convex angles fom polygon A and B, and calculate the fist angle-edge contact cove; 2) select next angle along the edge by ight-hand system, and calculate next contact edge; 3) epeat step 2 until we get the stating two angles. Fo algoithm in this pape, we do not conside staight angles fo that omitting staight angles will educe computing and does not change shape of Entance Block. u uu AngleEdgeFlag( A, h) = { n * e 0 1 } { n * e 0 2 } (5) Whee n = (0,0,1) h AngleAngleFlag = AngleEdgeFlag( A, h ) AngleEdgeFlag( A, h ) (6) 1 2 Figue 9. Angle-Edge Intesection Judgment select disjoint convex angles aa ii and bb jj fom block AA and BB Figue 10. Angle-Angle Intesection SSSSSS ii cc, jj dd, SS [] ee ii+1 h jj +1 nn bo ee ii+1 h jj +1 nn bo Yes SS SS EE(aa ii, h jj +1 ), jj (jj + 1)%mm Yes SS SS EE(ee ii+1, bb jj ), ii (ii + 1)%n SS SS EE(aa ii, h jj +1 ), SS SS EE(ee ii+1, bb jj ), SS SS EE(aa (ii+1)%nn, h jj +1 ), jj (jj + 1)%mm bo ii = cc, jj = dd Yes End Figue 11. Flow chat of Contact Algoithm fo Convex Polygons (3)Detect cuent contact Afte Entance Block built, we use the efeence point and Entance Block to detect cuent contact. cuent contact means that the contact is happening, as shown in figue 12 and 13 that two contact edges woks at the same time. To detect cuent contact, we calculate distance between the efeence point and contact edges one by one accoding to fomula (8), and collect contact edges with shotest distance by fomula (9). Thee ae two things left to be done: (1) we must check the collected contact edges to get if any two contact edges can compose an angle-angle contact cove; (2) ecalculate and eselect contact edges fo the ones that the edge of Angle-Edge contact cove contains staight angles. dist( e, p) = v *( p e ) (8) 1 Whee v = nomalize( e (0,0,1)), and e 1 is the stat point of contact edge e. min R = e : ( fabs( dist( e, p )),fo all dist( e, p) 0 (9) e S 98

5 Whee S contains all calculated contact coves. Analysis and Expeiments In this section, we analyze the algoithm complexity and give two examples to explain the pocess of getting cuent contacts with the poposed algoithm. Usually, accuate collision detection algoithms need to check each Angle-Edge and Angle-Angle cases, which has a complexity of 2 * Om ( * n) + Om ( * n) = Om ( * n) if polygon A and B have n and m points sepaately. But in the same condition, the complexity of the poposed algoithm is just Om ( + n) + Om ( + n) = Om ( + n). Now we give two examples. The fist one shows how to detect contact fo convex polygons without paallel edges (figue 12), while the second one uses convex polygons with 10 points (figue 13). Fo the two examples, we define polygon A with blue colo, polygon B with yellow colo, and efeence point being the cente of polygon A. The contact coves of Angle-Edge (angle of A to edge of B) ae in ed colo, while contact coves of Edge-Angle (edge of A to angle of B) ae in blue colo. (1)Expeiment fo Convex Polygons without Paallel Edges The case of convex polygons without paallel edges is the simplest example. The Entance Block is shown in figue 12(a); contact coves of Angle-Edge ae shown in figue 12(b); contact coves of Edge-Angle ae shown in figue 12(c). If we set polygon A at position of the hollow tiangle in figue 12(d), we can get an Edge-Angle contact cove with efeence point and Entance Block. b. Contact Coves of c. Contact Coves of d. Selecting woking a. Entance Block Angle-Edge Edge-Angle Contact Coves Figue 12. convex polygons without paallel edges (2)Expeiment fo Convex Polygons with 10 edges In eal computing, it is often to meet the case of calculating contact with cicles. The algoithms calculate the case fo cicles as well if we discetize cicle into line segments. In ode to facilitate the demonstation, we just discetize cicle into 10 line segments. The Entance Block is shown in figue 13(a); contact coves of Angle-Edge ae shown in figue 13(b); contact coves of Edge-Angle ae shown in figue 13(c). If we set polygon A at position of the hollow cicle in figue 13(d), we can get two Angle-Edge contact coves and one Edge-Angle contact cove in two diections. Moeove, we need to detemine which exiting path along with the coesponding contact coves is to be used next. It should be mentioned that this is an Angle-Angle contact accoding to the contact edges. a. Entance Block b. Contact Coves c. Contact Coves of Angle-Edge of Edge-Angle Figue 13. convex polygons with paallel edges d. Selecting woking Contact Coves Conclusion Contact Theoy has opened a new filed fo solving contact poblem. The algoithms poposed in this pape ealize it fo 2D convex polygons, and povide a fast and obust algoithm to detect contact. In futhe wok, we will wok to calculate the cases of convex polygons and stat to design algoithms fo 3D case. Moeove, as we see in this pape, bounday of Entance Block contains all contact infomation, but we only need a small pat of bounday of Entance Block to find the needed contact coves. Theefoe, we will ty to find a way to calculate smallest patial bounday of 99

6 Entance Block that contains needed contact infomation in futhe wok. Acknowledgement Thanks a lot fo D. Shi detailed instuctions and supevision. This wok is suppoted by the National Natual Science Foundation of China (No ), Pesident Fund of UCAS, Youth Innovation Pomotion Association CAS ( ). Refeence [1] G.H. Shi. Contact theoy. Aticle Special Issue On Rock Factues And Discontinuities: Modeling And Analysis Science China Technological Sciences. 2015, 58(9): [2] W. P. Computational Contact Mechanics. J. Wiley & Sons, New Yok, [3] Cundall P A. Fomulation of a thee-dimensional distinct element model-pat I: a scheme to detect and epesent contacts in a system composed of many polyhedal blocks. Int J Rock Mech Min, 1988, 25: [4] Benson D J, Hallquist J O. A single suface contact algoithm fo the post-buckling analysis of shell stuctues. Comput Method Appl M, 1990, 78: [5] Zhong Z H, Nilsson L. A contact seaching algoithm fo geneal contact poblems. Comput Stuct, 1989, 33: [6] Jeleni c G, Cisfield M A. Non-linea maste-slave elationships fo joints in 3D beams with lage otations. Comp Meth Appl Mech Engng, 1996, 135: [7] G. Hiota, F. S., and S. A. An impoved finite-element contact model fo anatomical simulations. The Visual Compute, 19(5): , [8] Li S H, Zhao M H, Wang Y N, et al. A new numeical method fo DEM-block and paticle model. Int J Rock Mech Min, 2004, 41 [9] V. J. Milenkovic and H. Schmidl. Optimization-based animation. In SIGGRAPH '01: Poceedings of the 28th annual confeence on Compute gaphics and inte active te chniques, pages 37-46, New Yok, NY, USA, ACM Pess. [10] D. Baaff. Analytical methods fo dynamic simulation of non-penetating igid bodies. In SIGGRAPH '89: Poceedings of the 16th annual confeence on Compute gaphics and inteactive techniques, pp , New Yok, NY, USA,1989. ACM Pess. [11] D. Baaff and A. Witkin. Lage steps in cloth simulation. In SIGGRAPH '98: Poceedings of the 25th annual confeence on Compute gaphics and inteactive techniques, pages 43-54, New Yok, NY, USA, ACM Pess. [12] M. Jean, V. Acay, and Y. Moneie. Non-smooth contact dynamics appoach of cohesive mateials. R. Soc. Lond. Philos. T ans. Se. A Math. Phys. Eng. Sci., 359(1789): , Non-smooth mechanics. [13] Shi, G.H. Discontinuous defomation analysis - A new numeical model fo the statics and dynamics of block systems. Ph.D. thesis Univ. of Califonia, Bekeley, Calif. 100