Collision Detection with Swept Spheres and Ellipsoids

Transcription

1 Collision etection with Swet Shees and Ellisoids Joit Rouwé Souce code: htt:// Intoduction Toda most games use conex olgons fo collision detection. The ae usuall stoed in a tee like stuctue to make it ossible to quickl eject a lot of olgons when efoming intesection tests. In the end howee, ee intesection test boils down to some imitie esus a olgon test. One commonl used intesection test is that of a swet shee a shee moing along a line o a swet ellisoid an ellisoid moing along a line but not otating with a static olgon. A game chaacte fo examle can be eesented b a collection of shees. When the chaacte is moing though the enionment we need to detect the fist oint fo which the collection of shees intesects with the wold geomet. Afte this oint is detected we also need an indication of whee the chaacte collided so we can comute a sliding diection that eents the chaacte fom getting stuck. As an altenatie to a numbe of shees we can also use a single ellisoid as an aoximation fo the olume of a chaacte. olgons used fo endeing usuall eside somewhee whee the ae not accessible to the collision sstem AG memo fo examle o the ae comessed and inteleaed with othe endeing data like etex colos. This foces us to stoe a seaate set of olgons fo collision. A chea wa of educing the amount of memo needed fo a olgon is to stoe it as a list of etices togethe with a lane equation. Of couse ou get the exta oehead of stoing a lane equation, but when using a S ina Sace atitioning tee fo examle ou need to stoe this lane anwa. In this aticle we will fist look at a function that can conet a olgon consisting of 3 oints into a lane and a list of oints. Afte that we will deie the intesection between a static olgon and a swet shee and that of a static olgon with a swet ellisoid. Finall we show that the same algoithm also woks on olgons consisting of 3 oints in the case that it is not ossible o not desiable to stoe a olgon in.. ojecting 3 oints on a lane A lane is defined b the set of oints fo which C. We call x,, z the lane nomal and C the lane constant. etails on how to ceate a lane equation can fo examle be found in []. 5/8/3

2 5/8/3 The distance of a oint to the lane is: lane C d. To oject a oint on the lane we ceate an othonomal base fo the lane with oigin at C and axes,, V U : U V U x z z x z x x z >,,,, To oject a oint fom wold sace to lane sace use:, V U To oject a oint fom lane sace to wold sace use: C V U x 3. Swet Shee Vesus olgon 3.. Swet Shee Vesus lane The fist ste in detemining if a swet shee collides with a olgon is to detemine the inteal oe which it intesects with the lane of the olgon. We use a swet shee of adius R and cente t t C, whee is the begin osition of the shee, the tanslation of the shee and t is a alue in the ange [, ]. The following image illustates this: R

3 The shee intesects the lane if the distance of the cente of the shee to the lane is equal o less than R: C t R. d lane When is zeo the lane is aallel to the motion of the shee. If d lane > R thee is no intesection, if d lane R the shee intesects oe the whole ange t [, ]. If is not zeo we hae to sole d lane C t R fo t: t t R C R C If both alues ae outside the ange [, ] thee is no intesection. We sot the two alues so that t < t and clam the alues to the ange [, ]. The inteal of intesection is now t [t, t ]. The following function efoms this test: bool laneswetsheeintesectconst lane &inlane, const Vecto3 &inegin, const Vecto3 &inelta, float inradius, float &outt, float &outt // If the cente of the shee moes like: cente inegin t * inelta fo t e [, ] // then the shee intesects the lane if: -R < distance lane to cente < R float n_dot_d inlane.momal.otinelta; float dist_to_b inlane.getsignedistanceinegin; if n_dot_d.f // The shee is moing neal aallel to the lane, check if the distance // is smalle than the adius if Absdist_to_b > inradius else // Intesection on the entie ange outt.f; outt.f; // etemine inteal of intesection outt inradius - dist_to_b / n_dot_d; outt -inradius - dist_to_b / n_dot_d; // Ode the esults if outt > outt SwaoutT, outt; // Eal out if no hit ossible if outt >.f outt <.f 3 5/8/3

4 // Clam it to the ange [, ], the ange of the swet shee if outt <.f outt.f; if outt >.f outt.f; 3.. Swet Cicle Vesus olgon We hae now detemined the inteal oe which the swet shee intesects with the lane of the olgon. To test if the shee intesects with the olgon itself we ae going to look at the oblem in the sace of the lane. The intesection between the shee and the lane is a cicle with a adius that aies along the ath. The cicle has a ositie adius oe ou eious comuted inteal [t, t ]. Outside this inteal the adius is negatie because the shee does not intesect with the lane. The following image illustates this: t t and ae ojected on the lane of the olgon in the same wa as the oints of the olgon see section, the esult in and ectos so that the cente of the swet cicle is descibed b C t t. The adius of the cicle is R t R d lane C t t t 3 with: 3 C R C Thee ae thee cases fo which the shee intesects with the olgon:. The cicle intesects with the olgon at t.. The cicle intesects with a etex on the inteal [t, t ]. 3. The cicle intesects with an edge on the inteal [t, t ]. 4 5/8/3

5 The esults of all these tests will gie a closest intesection faction the alue of t fo the fist collision and a collision oint. This collision oint can be coneted back to 3 as descibed in section. It can be used to detemine a collision esonse once the shee hits the olgon. We will now deie the thee tests in the next sections Static Cicle Vesus olgon In the fist test we test if the cicle at t defined b cente C t and adius R t intesects with the olgon. This is the case when:. The cicle cente is inside the olgon. In this case C t is the collision oint and t the intesection faction.. The closest oint fom C t to an of the edges is less than o equal to R t. In this case this closest oint is the collision oint and t the intesection faction. The following function combines both tests. It assumes that the etices ae invetices, inumvetices odeed counte clockwise. The cente of the cicle is incente and the adius of the cicle squaed is inradiussq. The collision oint will be etuned in outoint. bool olgoncicleintesectconst Vecto *invetices, int inumvetices, const Vecto &incente, float inradiussq, Vecto &outoint // Check if the cente is inside the olgon if olgoncontainsinvetices, inumvetices, incente outoint incente; // Loo though edges bool collision false; fo const Vecto * invetices, * invetices inumvetices - ; < invetices inumvetices;, // Get faction whee the closest oint to this edge occus Vecto _ * - *; Vecto _cente incente - *; float faction _cente.ot_; if faction <.f // Closest oint is float dist_sq _cente.getlengthsquaed; if dist_sq < inradiussq collision tue; 5 5/8/3

6 else outoint *; inradiussq dist_sq; float len_sq _.GetLengthSquaed; if faction < len_sq // Closest oint is on line segment Vecto oint * _ * faction / len_sq; float dist_sq oint - incente.getlengthsquaed; if dist_sq < inradiussq collision tue; outoint oint; inradiussq dist_sq; etun collision; The olgoncontains function checks if a oint is inside the olgon: bool olgoncontainsconst Vecto *invetices, int inumvetices, const Vecto &inoint // Loo though edges fo const Vecto * invetices, * invetices inumvetices - ; < invetices inumvetices;, // If the oint is outside this edge, the oint is outside the olgon Vecto _ * - *; Vecto _oint inoint - *; if _.mx * _oint.my - _oint.mx * _.my >.f 3.4. Swet Cicle Vesus Vetex In the second test, we test if the swet cicle intesects with an of the etices of the olgon. Fo ee etex we test if thee is a t fo which the distance between the cente of the cicle and the etex equals the adius of the cicle: C t R t Soling fo t we get a quadatic equation at bt c with: 6 5/8/3

7 7 5/8/3 3 c b a Let t be the smallest solution that lies in the inteal [, cuent closest faction]. If thee is a solution we stoe t as the cuent closest intesection faction and as the cuent collision oint Swet Cicle Vesus Edge In the thid and final test we test if the swet cicle intesects with an edge of the olgon. Fo ee edge to we will test if thee is a t fo which the cicle and the edge touch. Fist we test if the cicle intesects with the infinite line that the edge is at of. The distance between a oint and an infinite line though and is: d edge The cicle and the infinite line touch when: t R t C d edge Soling fo t we get a quadatic equation c bt at with: 3 c b a Let t be the smallest solution that lies in the inteal [, cuent closest faction]. If thee is no solution we continue with the next edge. A oint on the edge is gien b: f

8 The closest oint fom the cicle cente to the edge is when the diection fom the cente of the cicle to the oint is eendicula to the edge: C t Soling fo f : f t If f is inside the ange [, ] thee is an intesection, we stoe t as the cuent closest intesection faction and as the cuent closest intesection oint Imlementation This section will combine the esults of the eious sections into the full swet shee esus static olgon test. Fist of all we will combine the tests fom sections 3.4 and 3.5 since the shae a lot of exessions. The following function tests a swet cicle with the edges and etices of the olgon. The swet cicle taels fom inegin to inegin inelta. ina, in and inc ae a, b, and c of the quadatic equation of the cicle adius. The collision oint will be etuned in outoint and the faction in outfaction. bool SwetCicleEdgeVetexIntesectconst Vecto *invetices, int inumvetices, const Vecto &inegin, const Vecto &inelta, float ina, float in, float inc, Vecto &outoint, float &outfaction // Loo though edges float ue_bound.f; bool collision false; fo const Vecto * invetices, * invetices inumvetices - ; < invetices inumvetices;, float t; // Check if cicle hits the etex Vecto b * - inegin; float a ina - inelta.getlengthsquaed; float b in.f * inelta.otb; float c inc - b.getlengthsquaed; if FindLowestRootInInteala, b, c, ue_bound, t // We hae a collision collision tue; ue_bound t; outoint *; // Check if cicle hits the edge Vecto * - *; 8 5/8/3

9 float _dot_delta.otinelta; float _dot_b.otb; float _len_sq.getlengthsquaed; float a _len_sq * a _dot_delta * _dot_delta; float b _len_sq * b -.f * _dot_b * _dot_delta; float c _len_sq * c _dot_b * _dot_b; if FindLowestRootInInteala, b, c, ue_bound, t // Check if the intesection oint is on the edge float f t * _dot_delta - _dot_b; if f >.f && f < _len_sq // We hae a collision collision tue; ue_bound t; outoint * * f / _len_sq; // Check if we had a collision if!collision outfaction ue_bound; The tests ae not efomed on the ange [t, t ] as descibed in section 3., but on the ange [, cuent closest faction]. umeical ound off can geneate solutions that ae slightl lowe than t. Soling oe this lage ange does not gie us an false collisions since the adius of the cicle becomes negatie outside the ange [t, t ] so no solutions ae ossible. The following iece of code finds the lowest solution of a quadatic equation with coefficients ina, in and inc in the inteal [, inueound]. The solution is etuned in outx when the function etuns tue. bool FindLowestRootInIntealfloat ina, float in, float inc, float inueound, float &outx // Check if a solution exists float deteminant in * in - 4.f * ina * inc; if deteminant <.f // The standad wa of doing this is b comuting: x -b /- Sqtb^ - 4 a c / a // is not numeicall stable when a is close to zeo. // Sole the equation accoding to "umeical Reciies in C" aagah 5.6 float q -.5f * in in <.f? -.f :.f * Sqtdeteminant; // oth of these can etun IF, -IF o A that's wh we test both solutions // to be in the secified ange below float x q / ina; float x inc / q; // Ode the esults if x < x Swax, x; 9 5/8/3

10 // Check if x is a solution if x >.f && x < inueound outx x; // Check if x is a solution if x >.f && x < inueound outx x; 4. Swet Ellisoid Vesus olgon 4.. Theo In this section we will exand the swet shee esus static olgon test into a swet ellisoid esus static olgon test. We will assume that the ellisoid does not otate. To test a swet ellisoid with a olgon we hae to tansfom the olgon to the sace whee the ellisoid is a unit shee a shee with adius. We define an ellisoid b its thee othogonal axis e, e, e : x z e e x The cente of the ellisoid will moe accoding to C t t, whee is the begin osition of the ellisoid, the delta tanslation and t is a alue in the ange [, ]. We define a otation / scaling matix that tansfoms a unit shee in the ellisoid: 5/8/3

11 unit > ellisoid ex e e z unit > ellisoid unit unit > ellisoid When tansfoming the ellisoid b the matix moing along C t. unit unit unit > ellisoid unit t unit we get a unit shee and with We need to tansfom the lane of the olgon with lane equation C with: unit > ellisoid, this leads to the C unit > ellisoid unit > ellisoid C T unit > ellisoid T T Whee T unit > ellisoid >. indicates the tansose of unit ellisoid With this tansfomed lane we can detemine the inteal of intesection between the unit shee and the lane as befoe. If thee is an intesection we need to tansfom ou olgon to the sace of the unit shee. ojecting a oint fom lane sace to wold sace can be witten in matix fom: lane wold lane wold U x V C Let lane wold and tansfomed lane wold esectiel be the matix that takes oints fom the untansfomed lane to wold sace and fom the tansfomed lane to wold sace. The tansfomation needed fo the olgon is: 5/8/3

12 olgon tansfomed lane wold unit > ellisoid untansfomed lane wold Once the olgon has been tansfomed we can oject unit and unit on the tansfomed lane to fom and. ow we follow the same ath as befoe to detemine collision between a moing cicle and a olgon. If a collision oint is found we hae to tansfom it into wold sace b using the following matix: collision wold unit > ellisoid tansfomed lane wold 4.. Imlementation The swet ellisoid moes fom inegin to inegin inelta. The incial axis of the ellisoid ae inaxis, inaxis and inaxis3 which should be othogonal. When thee is a collision the function etuns tue and the collision oint will be in outoint and the cente of the shee is inegin outfaction * inelta when the shee collides. bool olgonswetellisoidintesectconst lane &inlane, const Vecto *invetices, int inumvetices, const Vecto3 &inegin, const Vecto3 &inelta, const Vecto3 &inaxis, const Vecto3 &inaxis, const Vecto3 &inaxis3, Vecto3 &outoint, float &outfaction // Comute matix that takes a oint fom unit shee sace to wold sace // OTE: When colliding with lots of olgons this can be cached atix unit_shee_to_wold; unit_shee_to_wold.column inaxis; unit_shee_to_wold.column inaxis; unit_shee_to_wold.column inaxis3; // Comute matix that takes a oint fom wold sace to unit shee sace // OTE: When colliding with lots of olgons this can be cached atix wold_to_unit_shee unit_shee_to_wold.getinesed; // Comute begin and delta in unit shee sace // OTE: When colliding with lots of olgons this can be cached Vecto3 begin_uss wold_to_unit_shee * inegin; Vecto3 delta_uss wold_to_unit_shee * inelta; // Tansfom the lane into unit shee local sace lane tansfomed_lane; tansfomed_lane inlane.gettansfomedineseunit_shee_to_wold; // etemine the ange oe which the unit shee intesects the tansfomed lane float t, t; if!laneswetsheeintesecttansfomed_lane, begin_uss, delta_uss,.f, t, t // Get matix that tansfoms a oint fom lane sace to wold sace atix lane_to_wold inlane.getlanetowoldatix; // Get matix that tansfoms a oint fom the tansfomed lane to unit shee sace atix tansfomed_lane_to_unit_shee tansfomed_lane.getlanetowoldatix; 5/8/3

13 // Get matix that takes a d olgon etex fom the oiginal sace to the sace of the // tansfomed lane so that the unit shee is still a unit shee atix lane_to_tansfomed_lane tansfomed_lane_to_unit_shee.getinesed * wold_to_unit_shee * lane_to_wold; // The adius of the cicle is defined as: adius^ - distance lane to cente^ // this can be witten as: adius^ a * t^ b * t c float n_dot_d tansfomed_lane.momal.otdelta_uss; float dist_to_b tansfomed_lane.getsignedistancebegin_uss; float a -n_dot_d * n_dot_d; float b -.f * n_dot_d * dist_to_b; float c.f - dist_to_b * dist_to_b; // Get the basis ectos fo the tansfomed lane const Vecto3 &u tansfomed_lane_to_unit_shee.column; const Vecto3 & tansfomed_lane_to_unit_shee.column; // To aoid tanslating the olgon we subtact the tanslation fom the begin oint // and then late add it to the collision esult again Vecto tanslane_to_tansfomed_lane.e, 3, lane_to_tansfomed_lane.e, 3; // Get the equation fo the intesection cicle between the lane and the // unit shee: cente begin t * delta Vecto begin lane::sconetwoldtolaneu,, begin_uss - tans; Vecto delta lane::sconetwoldtolaneu,, delta_uss; // Tansfom the olgon Vecto *tansfomed_etices Vecto *allocainumvetices * sizeofvecto; fo int i ; i < inumvetices; i tansfomed_etices[i] Tansfomxlane_to_tansfomed_lane, invetices[i]; // Test if shee intesects at t Vecto ; if olgoncicleintesecttansfomed_etices, inumvetices, begin delta * t, a * t * t b * t c, outfaction t; outoint unit_shee_to_wold * tansfomed_lane_to_unit_shee * Vecto3 tans; // Test if shee intesects with one of the edges o etices if SwetCicleEdgeVetexIntesecttansfomed_etices, inumvetices, begin, delta, a, b, c,, outfaction outoint unit_shee_to_wold * tansfomed_lane_to_unit_shee * Vecto3 tans; 5. Using olgons Stoed as 3 oints We stoed ou olgons as a list of oints and a lane equation, but the algoithm is the same if olgons ae stoed as a list of 3 oints. To make the algoithm wok we need to make the following changes to the equations in the eious sections: 3 5/8/3

14 Comute the lane equation at un time o stoe it. Set and so the become 3 ectos. Set and. 3 R Tansfom the olgon b the 4x4 matix:. olgon unit > ellisoid collision wold unit > Tansfom the collision esult b the 4x4 matix: ellisoid. 6. Refeences. eell, Gahics Gems III, W.H. ess, umeical Recies in C, Second Edition htt:// 3. T. Schoede, "Collision etection Using Ra Casting", Game eeloe agazine,. 5-57, August ft://ft.gdmag.com/ub/sc/aug.zi 4. T. Akenine-ölle, E. Haines, Real-Time Rendeing htt:// 4 5/8/3