Gilbert–Johnson–Keerthi distance algorithm: Difference between revisions

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{{Short description|Method of determining minimum distance between two convex sets}}

The '''Gilbert–Johnson–Keerthi distance [[algorithm]]''' is a method of determining the minimum distance between two [[convex set]]s, first published by [[Elmer G. Gilbert]], Daniel W. Johnson, and S. Sathiya Keerthi in 1988. Unlike many other distance algorithms, it does not require that the geometry data be stored in any specific format, but instead relies solely on a [[support function]] to iteratively generate closer [[simplex|simplices]] to the correct answer using the ''configuration space obstacle'' (CSO) of two convex shapes, more commonly known as the [[Minkowski difference]].

"Enhanced GJK" algorithms use edge information to speed up the algorithm by following edges when looking for the next simplex. This improves performance substantially for polytopes with large numbers of vertices.

GJK makes use of Johnson's distance subalgorithmsub algorithm, which computes in the general case the point of a tetrahedron closest to the origin, but is known to suffer from numerical robustness problems. In 2017 Montanari, Petrinic, and Barbieri proposed a new subalgorithmsub algorithm based on signed volumes which avoidsavoid the multiplication of potentially small quantities and achieved a speedup of 15% to 30%.

GJK algorithms are often used incrementally in simulation systems and video games. In this mode, the final simplex from a previous solution is used as the initial guess in the next iteration, or "frame". If the positions in the new frame are close to those in the old frame, the algorithm will converge in one or two iterations. This yields collision detection systems which operate in near-constant time.

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GJK relies on two functions:

* <math>\mathrm{Support}(\mathrm{shape}, \vec{d})</math>, which returns the point on {{math|shape}} which has the highest [[dot product]] with <math>\vec{d}</math>.

* <math>\mathrm{NearestSimplex}(s)</math>, which takes a simplex {{mvar|s}} and returns the simplex on {{mvar|s}} closest to the origin, and a direction toward the origin normal to the new simplex. If {{mvar|s}} itself contains the origin, {{math|NearestSimplex}} accepts {{mvar|s}} and the two shapes are determined to intersect.

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'''if''' dot(A, D) < 0:

reject

s = s ∪ {A}

s, D, contains_origin := NearestSimplex(s)

'''if''' contains_origin:

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==See also==

*[[Minkowski Portal Refinement]]

*[[Hyperplane separation theorem]]

==External links==

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*[https://ora.ox.ac.uk/objects/uuid:69c743d9-73de-4aff-8e6f-b4dd7c010907/download_file?safe_filename=GJK.PDF&file_format=application%2Fpdf&type_of_work=Journal+article "Improving the GJK algorithm for faster and more reliable distance queries between convex objects"], Montanari, Petrinic and Barbieri.

*[https://arxiv.org/pdf/2205.09663.pdf "Collision Detection Accelerated: An Optimization Perspective"], Montaut, Le Lidec, Petrik, Sivic and Carpentier. This research article notably shows how the original GJK algorithm can be accelerated by exploiting Nesterov-type acceleration strategies, contributing to lowering the overall computational complexity of GJK.

*[https://computerwebsite.net/writing/gjk "the Gilbert–Johnson–Keerthi algorithm explained as simply as possible"]

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