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performance tuning - Faster way to compute the distance from a point to a surface in 3D


I am trying to compute the shortest distance between a point and a triangle in 3D


distance[point_, {p1_, p2_, p3_}] := Module[{p, s, t, sol},
  p = s*p1 + (1 - s)*(t*p2 + (1 - t)*p3);
  MinValue[{(point - p).(point - p),
   0 <= s <= 1, 0 <= t <= 1}, {s, t}]];

but it seems to be quite slow, is there any way to make it faster?




Answer



Well, you can use the undocumented RegionDistance which does exactly this as follows: (This answer, as written, only works for V9 as noted by Oska, for V10 see update below)


here is a triangle in 3D


region = Polygon[{{0, 0, 0}, {1, 0, 0}, {0, 1, 1}}];

Graphics3D[region]

Mathematica graphics


Now suppose you want to find the shortest distance from the point {1, 1, 1} in 3D to this triangle just do the following:


Load the Region context



Graphics`Region`RegionInit[];

Then


RegionDistance[region, {1, 1, 1}]

Mathematica graphics


As a bonus, you can get the exact point on the triangle that is closest to the given point as follows:


RegionNearest[region, {1, 1, 1}]

Mathematica graphics



Visualize it


Graphics3D[{region, Darker@Green, PointSize[0.03], Point[{1, 1, 1}], 
  Red, PointSize[0.03], Point[{1/3, 2/3, 2/3}]}]

Mathematica graphics


Update for Version 10


The above undocumented functions used in this answer now works out of the box in V10 so no need to load the Region context as I did above. Otherwise everything works as is. Also, now you can use the new Triangle function in place of Polygon above.


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