find the collision of a particle with a swept triangle. - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T23:55:37Z http://mathoverflow.net/feeds/question/12763 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/12763/find-the-collision-of-a-particle-with-a-swept-triangle find the collision of a particle with a swept triangle. Julian Mann 2010-01-23T19:24:15Z 2010-01-24T01:23:30Z <p>Given there is triangle: V in 3D space that transforms over time t -> t1 to V1, and a static point P is somewhere in 3d space, how can I determine if P ever collides with V, and if so at what value of t?</p> <p>The transformation of the triangle from V -> V1 over time means the vertices Va,Vb,Vc each move linearly and independently in three dimensions to new vertex positions V1a,V1b,V1c. (Sorry about my notation). There are no constraints on the transformation of the vertices with respect to each other. The triangles at V and V1 representing the transformation may be any 2 triangles in 3d space.</p> http://mathoverflow.net/questions/12763/find-the-collision-of-a-particle-with-a-swept-triangle/12793#12793 Answer by Gerhard Paseman for find the collision of a particle with a swept triangle. Gerhard Paseman 2010-01-23T23:29:59Z 2010-01-23T23:29:59Z <p>Consider the (mostly nontrivial) tetrahedron formed by the three moving points and P. There is a determinantal formula for the volume of the tetrahedron, as well as one for the area of the triangle formed by the three moving points. Now you have two formulas depending on t, one measuring signed tetrahedral volume, the other the signed area of the moving triangle. If the area is nonzero at time t while the volume is 0 at time t, then you have a time when point P is in the plane of the triangle. Then you can do some calculations to see if P is inside the triangle or not. Otherwise, either the volume is nonzero (and thus no intersection between P and the triangle), or the triangle has zero area, and you have to check to see if P is collinear at that time with the other points, and also lies within the segment determined by the three points.</p> <p>If you have more information about the trajectories, I may have more suggestions on how to find time t, if it exists.</p> <p>Gerhard "Ask Me About System Design" Paseman, 2010.01.23</p> http://mathoverflow.net/questions/12763/find-the-collision-of-a-particle-with-a-swept-triangle/12805#12805 Answer by Dan Piponi for find the collision of a particle with a swept triangle. Dan Piponi 2010-01-24T01:23:30Z 2010-01-24T01:23:30Z <p>This type of problem is usually called "continuous collision detection". There is a substantial literature on this subject as you'll discover if you try a <a href="http://www.google.com/search?q=%22continuous+collision+detection%22" rel="nofollow">google search</a> on those key words. It's rare that someone wants to do this for one triangle at a time. The literature has techniques for checking against large sets of triangles as well as generalisations to alternative shapes and more complex motion such as quadratic motion wrt time.</p>