Collision detection: Difference between revisions

Revision as of 14:46, 21 November 2024

Revision as of 21:21, 22 November 2024

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[[Image:Billiards balls.jpg|right|200px|thumb|Billiards balls hitting each other in a virtual space are a classic example where collision detection computations are needed]]Collision detection is closely linked to calculating the [[Euclidean distance|distance]] between objects, as two objects (or more) intersect when the distance between them reaches zero or even becomes negative.<ref>{{Cite book |url=https://www.csun.edu/~ctoth/Handbook/HDCG3.html |title=Handbook of discrete and computational geometry |date=2018 |publisher=CRC Press, Taylor & Francis Group, a Chapman & Hall book |isbn=978-1-4987-1139-5 |editor-last=Goodman |editor-first=Jacob E. |edition=3rd |series=Discrete mathematics and its applications |___location=Boca Raton London New York |chapter=39 |editor-last2=O'Rourke |editor-first2=Joseph |editor-last3=Tóth |editor-first3=Csaba D.}}</ref> Negative distance indicates that one object has penetrated another. Performing collision detection requires more context than just the distance between the objects.

Accurately identifying the points of contact on both objects' surfaces is also essential for the computation of a physically accurate [[collision response]]. The complexity of this task increases with the level of detail in the objects' representations: the more intricate the model, the greater the computational cost.<ref name=":col0">{{Cite journalbook |lastlast1=Andrews |firstfirst1=Sheldon |last2=Erleben |first2=Kenny |last3=Ferguson |first3=Zachary |date=2022-08-02 |titlechapter=Contact and friction simulation for computer graphics |date=2022-08-02 |title=ACM SIGGRAPH 2022 Courses |chapter-url=https://dl.acm.org/doi/10.1145/3532720.3535640 |journal=SIGGRAPH ’22 Courses |language=en |publisher=ACM |pages=1–172 |doi=10.1145/3532720.3535640 |isbn=978-1-4503-9362-1}}</ref>

Collision detection frequently involves dynamic objects, adding a temporal dimension to distance calculations. Instead of simply measuring distance between static objects, collision detection algorithms often aim to determine whether the objects’ motion will bring them to a point in time when their distance is zero—an operation that adds significant computational overhead.<ref name=":col1">{{Cite journalbook |lastlast1=Hadap |firstfirst1=Sunil |last2=Eberle |first2=Dave |last3=Volino |first3=Pascal |last4=Lin |first4=Ming C. |last5=Redon |first5=Stephane |last6=Ericson |first6=Christer |chapter=Collision detection and proximity queries |date=2004-08-08 |title=CollisionACM detectionSIGGRAPH and2004 proximityCourse queriesNotes |chapter-url=https://dl.acm.org/doi/10.1145/1103900.1103915 |journal=SIGGRAPH ’04 Courses |language=en |publisher=ACM |pages=15 |doi=10.1145/1103900.1103915 |isbn=978-1-4503-7801-7}}</ref><ref name=":col0" />

In collision detection involving multiple objects, a naive approach would require detecting collisions for all pairwise combinations of objects. As the number of objects increases, the number of required comparisons grows rapidly: for <math>n</math> objects, <math display="">{n(n-1)}/{2}</math> intersection tests are needed with a naive approach. This quadratic growth makes such an approach computationally expensive as <math>n</math> increases.<ref name=":col1" /><ref name=":0">{{Cite journalbook |lastlast1=Cohen |firstfirst1=Jonathan D. |last2=Lin |first2=Ming C. |last3=Manocha |first3=Dinesh |last4=Ponamgi |first4=Madhav |date=1995 |titlechapter=I-COLLIDE: anAn interactive and exact collision detection system for large-scale environments |urldate=http://portal.acm.org/citation.cfm?doid=199404.1994371995 |journaltitle=I3D '95: Proceedings of the 1995 symposium on Interactive 3D graphics - SI3D '95 |chapter-url=http://portal.acm.org/citation.cfm?doid=199404.199437 |language=en |publisher=ACM Press |pages=189–ff. |doi=10.1145/199404.199437 |isbn=978-0-89791-736-0}}</ref>

Due to the complexity mentioned above, collision detection is computationally intensive process. Nevertheless, it is essential for interactive applications like video games, robotics, and real-time physics engines. To manage these computational demands, extensive efforts have gone into optimizing collision detection algorithms.

A commonly used approach towards accelerating the required computations is to divide the process into two phases: the '''broad phase''' and the '''narrow phase'''.<ref name=":col1" /><ref>{{Cite book |lastlast1=Akenine-Möller |firstfirst1=Tomas |url=https://www.realtimerendering.com |title=Real-time rendering |last2=Haines |first2=Eric |last3=Hoffman |first3=Naty |last4=Pesce |first4=Angelo |last5=Iwanicki |first5=Michał |last6=Hillaire |first6=Sébastien |date=2018 |publisher=CRC Press, Taylor & Francis Group |isbn=978-1-138-62700-0 |edition=4th |series=An A K Peters book |___location=Boca Raton London New York}}</ref> The broad phase aims to answer the question of whether objects might collide, using a conservative but efficient approach to rule out pairs that clearly do not intersect, thus avoiding unnecessary calculations.

Objects that cannot be definitively separated in the broad phase are passed to the narrow phase. Here, more precise algorithms determine whether these objects actually intersect. If they do, the narrow phase often calculates the exact time and ___location of the intersection.

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During the broad-phase, when the objects in the world move or deform, the data-structures used to cull collisions have to be updated. In cases where the changes between two frames or time-steps are small and the objects can approximated well with an [[axis-aligned bounding box]]es the [[sweep and prune]] algorithm<ref name=":0" /> can be a suitable approach.

Several key observation make the implementation efficient: Two bounding-boxes intersect [[If and only if|if, and only if]], there is overlap along all three axes; overlap can be determined, for each axis separately, by sorting the intervals for all the boxes; and lastly, between two frames updates are typically small (making sorting algorithms optimized for almost-sorted lists suitable for this application). The algorithm keeps track of currently intersecting boxes, and as objects moves, re-sorting the intervals helps keep track of the status.<ref>{{Cite book |last=Ericson |first=Christer |url=https://realtimecollisiondetection.net/books/rtcd/ |title=Real-time collision detection |date= 22 December 2004|publisher=Elsevier |isbn=978-1-55860-732-3 |edition=Nachdr. |series=Morgan Kaufmann series in interactive 3D technology |___location=Amsterdam Heidelberg |pages=329–338}}</ref>

=== Pairwise pruning ===

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==== Collision detection between convex objects ====

According to the [[Hyperplane separation theorem|separating planes theorem]], for any two disjoint [[convex set|convex]] objects, there exists a plane so that one object lies completely on one side of that plane, and the other object lies on the opposite side of that plane. This property allows the development of efficient collision detection algorithms between convex objects. Several algorithms are available for finding the closest points on the surface of two convex polyhedral objects - and determining collision. Early work by [[Ming C. Lin]]<ref name=":1">{{cite web |author=Lin, Ming C |year=1993 |title=Efficient Collision Detection for Animation and Robotics (thesis) |url=https://wwwx.cs.unc.edu/~geom/papers/documents/dissertations/lin93.pdf |archive-url=https://web.archive.org/web/20140728124049/https://wwwx.cs.unc.edu/~geom/papers/documents/dissertations/lin93.pdf |archive-date=2014-07-28 |publisher=University of California, Berkeley}}

</ref> that used a variation on the [[simplex algorithm]] from [[linear programming]] and the [[Gilbert-Johnson-Keerthi distance algorithm]]<ref>{{Cite journal |lastlast1=Gilbert |firstfirst1=E.G. |last2=Johnson |first2=D.W. |last3=Keerthi |first3=S.S. |date=1988 |title=A fast procedure for computing the distance between complex objects in three-dimensional space |url=https://graphics.stanford.edu/courses/cs448b-00-winter/papers/gilbert.pdf |journal=IEEE Journal on Robotics and Automation |volume=4 |issue=2 |pages=193–203 |doi=10.1109/56.2083}}</ref> are two such examples. These algorithms approach constant time when applied repeatedly to pairs of stationary or slow-moving objects, and every step is initialized from the previous collision check<ref name=":1" />.

The result of all this algorithmic work is that collision detection can be done efficiently for thousands of moving objects in real time on typical personal computers and game consoles.